The RO(G)-Graded Equivariant Ordinary Homology of G-Cell Complexes with Even-Dimensional Cells for G = Z=p Kevin K. Ferland L. Gaunce Lewis, Jr. Author address: Department of Mathematics, Bloomsburg University, Bloomsburg, PA 17815 E-mail address: kferland@bloomu.edu Department of Mathematics, Syracuse University, Syracuse NY 13244-1150 E-mail address: lglewis@syr.edu Contents Introduction 1 Part 1. The Homology of Z=p-Cell Complexes with Even- Dimensional Cells 7 Chapter 1. Preliminaries 8 1.1. Mackey functors for Z=p 8 1.2. RO(G)-graded Mackey functor-valued homology 12 1.3. The homology H* of a point 14 1.4. Modules over H* 18 1.5. Generalized G-cell complexes 23 Chapter 2. The main freeness theorem (Theorem 2.5) 25 Chapter 3. An outline of the proof of the main freeness result (Theorem 2.5) * *30 3.1. The freeness results for adding a single cell 30 3.2. Colimits of diagrams of free H*-modules 32 3.3. Completing the proof of the main freeness theorem 36 Chapter 4. Proving the single-cell freeness results 42 4.1. A proof overview for the dimension-shifting theorem (Theorem 3.3) 43 4.2. Simplifying the cell-attaching long exact sequence 44 4.3. Characterizing dimension-shifting long exact sequences 48 4.4. Constructing the comparison dimension-shifting sequence 50 Chapter 5. Computing HG*(B [ DV ; A)in the key dimensions 54 5.1. Using the Universal Coefficient Theorem 54 5.2. Constructing the maps of the comparison sequence 57 Chapter 6. Dimension-shifting long exact sequences 65 6.1. Preliminary observations about dimension-shifting sequences 65 6.2. The reduction to complexity one dimension-shifting sequences 69 6.3. Sequences with minimal complexity and spread 72 6.4. The reduction to sequences of minimal spread 75 6.5. The congruence condition on d(V +P !i-P !0j) 81 Chapter 7. Complex Grassmannian Manifolds 83 Part 2. Observations about RO(G)-graded equivariant ordinary homology 87 v vi CONTENTS Chapter 8. The computation of HS*for arbitrary S 88 Chapter 9. Examples of HS* 101 Chapter 10. RO(G)-graded box products 108 Chapter 11. A weak Universal Coefficient Theorem 113 Chapter 12. Observations about Mackey functors 118 Bibliography 121 Abstract It is well known that the homology of a CW-complex with cells only in even dimensions is free. The equivariant analog of this result for generalized G-cel* *l com- plexes is, however, not obvious, since RO(G)-graded homology cannot be computed using cellular chains. We consider G = Z=p and study G-cell complexes construct* *ed using the unit disks of finite dimensional G-representations as cells. Our main* * result is that, if X is a G-complex containing only even-dimensional representation ce* *lls and satisfying certain finiteness assumptions, then its RO(G)-graded equivariant ordinary homology HG*(X; A)is free as a graded module over the homology H* of a point. This extends a result due to the second author about equivariant compl* *ex projective spaces with linear Z=p-actions. Our new result applies more generall* *y to equivariant complex Grassmannians with linear Z=p-actions. Two aspects of our result are particularly striking. The first is that, even though the generators of HG*(X; A)are in one-to-one correspondence with the cel* *ls of X, the dimension of each generator is not necessarily the same as the dimens* *ion of the corresponding cell. This shifting of dimensions seems to be a previously unobserved phenomenon. However, it arises so naturally and ubiquitously in our context that it seems likely that it will reappear elsewhere in equivariant hom* *otopy theory. The second unexpected aspect of our result is that it is not a purely f* *ormal consequence of a trivial algebraic lemma. Instead, we must look at the homology* * of X with several different choices of coefficients and apply the Universal Coeffi* *cient Theorem for RO(G)-graded equivariant ordinary homology. In order to employ the Universal Coefficient Theorem, we must introduce the box product of RO(G)-graded Mackey functors. We must also compute the RO(G)- graded equivariant ordinary homology of a point with an arbitrary Mackey functor as coefficients. This, and some other, basic background material on RO(G)-graded equivariant ordinary homology is presented in a separate part at the end of the paper. ____________ October 3, 2002. 2000 Mathematics Subject Classification. Primary 55M35, 55N91, 57S17; Secon* *dary 14M15 55P91. Key words and phrases. Bredon-Illman homology, equivariant ordinary homolog* *y, Grass- mann Manifolds. vii Introduction If X is a CW complex with cells only in even dimensions, then its integral o* *rdi- nary homology Hn(X; Z) is a free abelian group in every dimension n. Essentiall* *y, the goal of this paper is to prove a precise version (Theorem 2.1) of the follo* *wing equivariant generalization of this result: Theorem. Let X be a finite G-cell complex having only even-dimensional cells. Then the equivariant ordinary homology HG*X of X is free. Several questions must be addressed to convert this vague assertion into a precise result. The first is what sort of cells are to be used in forming X. Us* *ually, cells of the form G=H x Dn are used to form G-cell complexes. This choice yield* *s a theorem which is trivial to prove, but turns out to be inapplicable to any inte* *resting G-spaces. An alternative type of G-cell (one which occurs naturally in, for exa* *mple, equivariant complex flag manifolds) must therefore be introduced before we can state our main result precisely. The second question is what sort of equivariant homology is intended here. Tied to that is a third question regarding the sense in which HG*X is free. The obvious candidates for the homology theory are Borel homology and Bredon-Illman homology. It isn't hard to obtain a version of the theorem above for Borel homo* *logy, but our interest is in more sensitive theories than Borel theory. A simple example illustrates the difficulties which arise in trying to obtai* *n a theorem of the desired sort for Bredon-Illman homology. Let V be a nontrivial complex representation of G. Its one-point compactification SV is surely the so* *rt of space to which such a freeness theorem ought to apply. Nevertheless, if p is* * a prime dividing the order of G, then the Bredon-Illman homology HGn(SV ; M)of SV with respect to a coefficient system M contains p-torsion unless M is a very unusual coefficient system _ such as one consisting entirely of Z[1=p]-modules. This torsion eliminates the possibility of an interesting freeness result for Z* *-graded Bredon-Illman homology. This example also illustrates the problem with using cells of the form G=HxD* *n. The space SV can be constructed using cells of this form, and its Z-graded Bred* *on- Illman homology can be computed via the chain complex derived from this cell structure. Moreover, it is easy to argue that, if all the cells appearing in th* *is cell structure were even-dimensional, then the Bredon-Illman homology of SV would be torsion-free. Since this homology is not torsion-free, we know that it is not p* *ossible to build even as nice a space as SV out of only even-dimensional cells of the f* *orm G=H x Dn. Indeed, it seems likely that very few spaces can be constructed using only such even-dimensional cells. 1 2 INTRODUCTION There is a simple explanation for these difficulties with Z-graded Bredon- Illman homology. For any reasonably well-behaved coefficient system M, Bredon- Illman homology with coefficients in M is represented by an equivariant Eilenbe* *rg- Mac Lane spectrum in the complete equivariant stable category [13, 16, 17]. The* *re is an RO(G)-graded equivariant homology theory associated to this spectrum. The equivariant analog of the dimension axiom implies that, for this theory, the ho* *mol- ogy HM* of a point vanishes in dimension n for any nonzero integer n. However, this axiom does not force the vanishing of HM* in the dimensions associated to * *non- trivial virtual G-representations. In fact, in those dimensions, HM* is full of* * torsion at the primes dividing the order of G. The reduced Bredon-Illman homology group eHGn(SV ; M)is a part of HM* and so reflects this p-torsion. This explanation for the lack of a good freeness theorem for Z-graded Bredon- Illman homology leads us to both the right homology theory and the right notion of freeness. If the coefficient system M is ring-valued, then its Eilenberg-Mac* * Lane spectrum is a ring G-spectrum, and the associated homology HM* of a point is an RO(G)-graded ring. One may then ask if the equivariant homology HG*(X; M) of a G-space X is free as a module over HM*. Even in the nonequivariant case, this is the sort of freeness one expects when working with a generalized, rather than ordinary, homology theory. Note that, when M is ring-valued, the suspension axiom implies that the re- duced RO(G)-graded homology eHG*(SV ; M)of SV is a free module over HM*. This suggests that we use the unit disks DV of G-representations V as our cells rath* *er than cells of the form G=H x Dn. For such a cell DV , the appropriate meaning of "even-dimensional" is that, for each subgroup H of G, the fixed point space * *V H is an even-dimensional real vector space. Beyond leading to a freeness result * *of the desired form, this choice has the advantage that, at least when G is a fini* *te abelian group, many interesting spaces have well-understood cell structures of * *this sort. For nonabelian groups, cells of the form DV may not suffice for building * *all the spaces we wish to consider. Instead, cells of the form G=H x DV , for a sub- group H of G and a G-representation V , or even cells of the form G xH DW , for a subgroup H of G and an H-representation W , may be needed. Cells of this sort arise naturally in equivariant Morse theory [21] and fit nicely into our approa* *ch to proving equivariant freeness results. The use of RO(G)-graded homology and an alternative type of cell complex leads to one other adjustment in our approach. RO(G)-graded homology theories are implicitly Mackey functor valued rather than just abelian group valued. This additional structure plays a critical role in the proofs of our results. Thus, * *hereafter, we think of equivariant homology as Mackey functor-valued. The Burnside ring Mackey functor A plays much the same role in the category of Mackey functors that Z plays in the category of abelian groups. Thus, A is the generic choice f* *or the coefficients in our ordinary theories. Henceforth, the RO(G)-graded homology of a point with Burnside ring coefficients is denoted by H*. One difficulty arises immediately in trying to prove a freeness theorem for RO(G)-graded equivariant ordinary homology. Unlike Z-graded Bredon-Illman ho- mology, this theory cannot be computed in a straightforward fashion from chain complexes. Thus, the naive algebraic argument used to prove the nonequivariant result must be replaced by an alternative argument. Assume that B is a G-space INTRODUCTION 3 whose RO(G)-graded equivariant ordinary homology is free over H* with even- dimensional generators. Let Y be a G-space obtained from B by adjoining a single even-dimensional cell DV . If we could show that the homology of Y must also be free, then an inductive argument indicates that any finite generalized G-cell complex with only even-dimensional cells has free homology. An obvious tactic f* *or trying to prove the freeness of the homology of Y would be to look at the long * *exact sequence . . .__//HG!(B; A)_//HG!(Y ; A)_//eHG!(SV ;@A)!//_HG!-1(B;_A)//. . . associated to the cell attachment. This is a long exact sequence of modules ove* *r H*, and the reduced homology eHG*(SV ; A)of SV is a free H*-module on one generator. Thus, if the boundary map @! vanished for every !, then the sequence would spli* *t, and HG*(Y ; A)would be a free H*-module having one generator for each generator of the homology of B and one additional generator coming from the cell DV . Moreover, the dimensions of these generators would be the obvious ones. This is the approach to an equivariant freeness theorem taken by the second author in [11]. There it is shown that, if G is a cyclic group of prime order and X is a generalized G-cell complex having even-dimensional cells which are attached in a suitable order, then the homology HG*(X; A)of X is free over H*. Moreover, it is shown that complex projective spaces with linear G-actions have a cell structur* *e of the required sort and so have free homology. There are two obvious defects in this freeness theorem from [11]. The first * *is that, even for G = Z=p, spaces as simple as the Grassmann manifold of complex 3-planes in a typical 6-dimensional complex G-representation V appear not to ha* *ve a cell structure satisfying the appropriate dimensional restrictions. There is,* * there- fore, no reason to expect that the boundary maps in the cell-attaching long exa* *ct sequences for such spaces are zero. Hence, the simple approach of [11] gives us* * no freeness result for the homology of such a G-space. Even worse, for groups as s* *mall as Z=p x Z=p and Z=p2, there are linear actions on CP 2for which all the obvi- ous generalized G-cell structures yield a nonzero boundary map in the long exact sequence associated to attaching the 4-cell to the 2-skeleton. Thus, the approa* *ch taken in [11] cannot be generalized in a useful way to groups larger than Z=p. If the modules in our cell-attaching long exact sequences were Z-graded, rat* *her than RO(G)-graded, then these nonvanishing results would doom our quest for a more general equivariant freeness result. However, as the first author showed i* *n his thesis [5], the additional complexity implicit in the RO(G)-grading allows a ra* *ther strange thing to happen. At least for the group Z=p, rather than producing tors* *ion in the homology of Y , a nonzero boundary map in the cell-attaching long exact sequence simply forces the generators of the homology of Y to appear in unexpec* *ted dimensions. This bit of near magic implies that, if X is a finite generalized Z* *=p- cell complex having only even-dimensional representation cells, then the homolo* *gy HG*(X; A)of X with Burnside ring coefficients is free over H*. There is a one-t* *o-one correspondence between the cells of X and the generators of HG*(X; A). However, the dimension-shifting forced by a nonzero boundary map depends so subtly on that map that very little can be said about the dimensions of the generators of HG*(X; A). Our impression is that, for most spaces, some completely different l* *ine of argument, such as shifting to cohomology and looking at cup products, will be needed to determine the dimensions of the generators. 4 INTRODUCTION Given this freeness result for finite complexes, it is natural to seek an an* *alogous result for infinite complexes. For the classical nonequivariant freeness result* * and the main result in [11], the transition to infinite complexes is elementary because* * such a complex X can be described as a colimit of finite complexes whose homologies are direct summands of the homology of X. However, the dimension-shifting in our new freeness result makes extending it to an infinite complex X much tricki* *er. One can, of course, still describe X as a colimit of finite complexes. Unfortun* *ately, the homology of a typical finite subcomplex is no longer a direct summand of the homology of X. Moreover, it is quite easy to construct a diagram of finitely generated free H*-modules whose colimit is obviously not a free H*-module. There is no obvious reason to believe that these purely algebraic diagrams cannot be realized as the homology of the diagram of finite subcomplexes from which a G- space X is constructed. The only way to get around this algebraic difficulty seems to be to impose a condition on X which, in essence, implies that the generator associated to any * *one cell of X participates in only finitely many dimension shifts. In [5], the firs* *t author worked with cohomology, rather than homology, and this extra condition took the form of an obvious equivariant analog of a finite-type assumption. The condition imposed here is weaker than that in [5] and is best understood by looking at the hypotheses of the freeness theorem in [11]. Those hypotheses require that, if a* * cell of the form DW is attached after a cell of the form DV and the dimension of W is greater than the dimension of V , then the dimension of W Gmust be at least * *as large as that of V G. The extra condition imposed here is that, for each cell D* *V of X, this dimensional restriction from [11] can be broken only finitely many times by cells DW added after DV . Our freeness result differs from the classical nonequivariant result and the* * result in [11] in that it is not an immediate consequence of a purely algebraic result* *. In Theorem 3.35 of [5], the first author shows that one can take the boundary map @ : eHG*(SV ; A)//_HG*-1(B;oA)f a legitimate cell-attaching long exact sequence and construct a long exact sequence . . .__//HG!(B; A)_//D!___//eHG!(SV ;@A)!//_HG!-1(B;_A)//_. . . of H*-modules in which D* is not a free H*-module. In order to show that HG*(Y * *; A) is not such a non-free H*-module, it is necessary to consider the homologies of* * B, Y , and SV with coefficients other than Burnside ring coefficients and to exam- ine the long exact homology sequences associated to certain short exact coeffic* *ient sequences. The obvious way to obtain the homology of B with respect to some other coefficients would be a Universal Coefficient Theorem. Since no such resu* *lt existed for RO(G)-graded equivariant ordinary homology and cohomology at the time [5] was written, ad hoc arguments were used to circumvent the need for this result. These arguments are cumbersome and also very unlikely to be extendible * *to groups other than Z=p. One of our primary goals in writing this paper was to el* *im- inate the need for such ad hoc arguments. Unfortunately, the Universal Coeffici* *ent Theorem for equivariant ordinary cohomology seems inherently less powerful that the corresponding result for nonequivariant ordinary cohomology in that it appl* *ies only to finite, rather than finite-type, complexes. This weakness was the prima* *ry motivation for our shift from cohomology, which is used in [5], to homology. The equivariant Universal Coefficient Theorem for going from homology to cohomology INTRODUCTION 5 is just as powerful as its nonequivariant analog. Thus, the results in [5] can* * be recovered from our results via that theorem. One of the particularly attractive aspects of the main freeness theorem in [* *5] is that, since it applies when the cell-attaching boundary maps are nonzero, it* * is reasonable to hope that this result could be extended to groups other than Z=p. However, the proof given in [5] is highly computational and requires a thorough understanding of the multiplicative structure of H*. It is therefore most unlik* *ely that this argument could be extended to groups more complex than Z=p. A second primary goal in preparing this paper was to replace the arguments in [5] with o* *ther, more easily extended arguments. With the exception of the argument presented in Section 6.3, our arguments are significantly less computational and require a m* *uch less complete understanding of the multiplicative structure of H*. Unfortunatel* *y, in that critical section, we must use the freeness of the homology of complex proj* *ective spaces with linear Z=p-actions (proven in [11]) to construct a few model long e* *xact sequences. It seems clear that establishing the freeness of the homology of com* *plex projective spaces with linear actions is an unavoidable prerequisite to obtaini* *ng a general freeness result like ours for larger groups. Since the cell-attaching m* *aps for these spaces tend to be nonzero for larger groups, this is, for the moment, a s* *erious obstruction. This paper is divided into two parts. The first part, containing Chapters 1 through 7, presents our freeness result and its proof. The second part, contain* *ing Chapters 8 through 12, supplies background information on RO(G)-graded equi- variant ordinary homology. That background is needed in Part 1, but, since it i* *s of independent interest, it has been separated out to make it more accessible. Chapter 1 supplies basic information about Mackey functors, equivariant ordi- nary homology, and G-cell complexes needed to understand the statement of our main freeness theorem. Our freeness theorems for both finite and infinite compl* *exes are stated in Chapter 2. That chapter also contains some examples motivating the somewhat mysterious finiteness hypothesis contained in our freeness result for * *infi- nite complexes. The proofs of our freeness results are quite long. Chapter 3 pr* *ovides an overview of the entire argument, and Chapter 4 deals with the process of add* *ing a single cell to a G-space with free equivariant homology. Chapters 5 and 6 fi* *ll in some key technical details postponed in Chapter 4. The last chapter of Part 1 is devoted to complex Grassmann manifolds with linear actions by an abelian group. It contains our proof of the freeness of the equivariant ordinary homolo* *gy of a complex Grassmann manifold with a linear Z=p-action. We must invoke a weak form of the Universal Coefficient Theorem for RO(G)- graded equivariant ordinary homology in the proof of our main freeness result. * *The primary purpose of Part 2 is to provide the information needed for the use of t* *his theorem. In particular, the first two chapters in this part describe the RO(G)- graded equivariant ordinary homology HS* of a point with an arbitrary Mackey functor S as coefficients. This information about HS*leads to several observati* *ons about some curious connections among the equivariant Eilenberg-Mac Lane spectra for various Mackey functors (see Corollaries 9.3 and 9.6). Chapter 10 discusses the properties of the category of RO(G)-graded Mackey functors for any finite group G. In particular, the box product of RO(G)-graded Mackey functors is introduced there. Unfortunately, there is still no published proof of a Univer* *sal Coefficient Theorem for RO(G)-graded equivariant ordinary homology. The best 6 INTRODUCTION approach for obtaining this result seems to be via an equivariant generalization of the Universal Coefficient Theorem for E1 -ring spectra and their E1 -modules contained in [3]. This will be provided in [15]. However, this generalization c* *annot be applied to RO(G)-graded equivariant ordinary homology until it is shown that equivariant Eilenberg-Mac Lane spectra have the required E1 structures. It is widely acknowledged that the required E1 structures exist, at least when the gr* *oup G is finite. However, since there is no published proof of the existence of th* *ese structures, Chapter 11 contains a short ad hoc proof of the weak form of the Universal Coefficient Theorem for equivariant ordinary homology needed in this paper. The last chapter contains some elementary observations about short exact sequence of Z=p-Mackey functors. Part 1 The Homology of Z=p-Cell Complexes with Even-Dimensional Cells CHAPTER 1 Preliminaries 1.1.Mackey functors for Z=p Mackey functors for a finite group were first introduced by Green [6]; a more abstract approach was also given shortly thereafter by Dress [2]. Subsequently, several other approaches have been given, a survey of which can be found in [20* *]. Here we mainly use a slight variant of the approach of Green. This approach is usually refered to as the elementary approach and is particularly convenient fo* *r the group G = Z=p. In Section 1 of [11], a tutorial is given on Mackey functors for Z=p. Except as indicated below, we adopt the notation used there. A Mackey functor M for G = Z=p consists of an abelian group M(G=G), a Z[G]-module M(G=e) and two maps æ : M(G=G) __//_M(G=e)and ø : M(G=e) ___//M(G=G). The maps æ and ø are required to be G-equivariant with respect to the trivial action on M(G=G). Moreover, the composite æ O øPis required to be the trace of the G-action on M(G=e); that is, (æ O ø)(x) = g2Ggx for all x 2 M(G=e). The maps æ and ø are called the restriction and transfer, respectively. As in [11],* * M is displayed in a diagram M M(G=G)____ _________________UU_____________________ æ_________________________________ø__________* *_______________________ ~~__________________________________ M(G=e)WW_____ ___________________________________________ ____________________________________________* *___________________ ` where ` denotes the G-action. Whenever M(G=e) is a p-fold direct sum Cp of copies of an abelian group C and G acts on M(G=e) by permutations, ` is replaced by the notation perm. For G = Z=2, ` is replaced by -1 to indicate an action via multiplication by -1. When the G-action is trivial, ` is omitted from the diagr* *am. A map f between two Mackey functors M and N consists of two homomor- phisms, fG : M(G=G) __//_N(G=G) and fe: M(G=e) ___//N(G=e). The homomorphism fe is required to be a G-map, and the two maps fG and fe are required to commute with the restriction and transfer maps in the obvious w* *ay. The category M of Mackey functors is a complete and cocomplete abelian category. Kernels, cokernels, and so forth are defined levelwise. For easy reference, we recall from [11] the particular Mackey functors that * *are of interest to us. In the following diagrams, C denotes an abelian group, and d 8 1.1. MACKEY FUNCTORS FOR Z=p 9 denotes an integer prime to p. A[d] AG=e L R Z___Z________ ZVV _Z______VV_______Z_______VV_______C_VV_ _________________VV___________________________________________________* *______________________________________________________________@ (d,p)________________________________(0,1)________________________________* *M________________________________O____________________________@ ~~__________________________________~~____________________________~~__* *____________~~______________~~__________________________ Z Zp_YY_______________________ZZ 0 ___________________________________________________ __________________________________________________ perm Here, M and O denote the diagonal map and the folding map, respectively. If G = Z=2, then the two additional Mackey functors L- R- Z=2_____ 0______VV______ _____________________UU_______________________________* *______________________________________________________________@ 0 __________________________________ß___________________* *_______________0____________________________________0_________@ ~~____________________________~~__ Z_YY__________________ZYY_____________________________* *_______ ______________________________________________________* *________________________________________________ ______________________________________________________* *_______________________________________ -1 -1 are also of interest. In the display of L- , ß denotes the usual projection. * *For our computations, it is useful to have standard names for the generators of the* *se Mackey functors. In each of A[d], L, L- , R, and R- , ' denotes the generator at G=e. In A[d], L, and L- , ø(') is denoted by ø (or by ~øif there is a danger th* *at it will be confused with the transfer map ø). Note that ø generates L and L- at G=G. A[d] is generated at G=G by ø and one other element, which is denoted ~. The significance of the d in the notation A[d] is that æ(~) = d'. The generator* * of R at G=G is denoted , and is chosen so that æ(,) = '. Mackey functors of the form A[d] are projective and play an especially impor- tant role in our work. In particular, A[1] is just the Burnside ring Mackey fun* *ctor A, which plays a role in M similar to the role played by Z in the category Ab of abelian groups. It is sometimes useful to employ an alternative set of generato* *rs of A[d] at G=G. These alternative generators are denoted oe and ~ and are given by oe = a~ + bø and ~ = p~ - dø, where a and b are integers such that ad + bp = 1. Note that æ(oe) = ' and that * *~ is a generator of the kernel of æ. The equations ~ = doe + b~ and ø = poe - a~ can be used to convert back to the standard basis. In Examples 1.1(b) of [11], the second author shows that A[d1] ~=A[d2] if d1* * is congruent to d2 mod p. In fact, the converse is also true. Lemma 1.1. Let d1 and d2 be integers prime to p. Then, A[d1] ~=A[d2] if and only if d1 d2 mod p. Proof. Our comments prior to the lemma indicate that it suffices to prove that d1 d2 mod p if A[d1] ~= A[d2]. Suppose that f : A[d1]___//A[d2]is an isomorphism. Let {~1, ø1, '1} and {~2, ø2, '2} be standard generators for A[d1]* * and A[d2], respectively. We may assume that fe('1) = '2, and hence fG (ø1) = ø2. Wr* *ite 10 1. PRELIMINARIES fG (~1) = x~2+ yø2. The map fG is described by the 2 x 2 matrix x0y1. Since fG is an isomorphism, it follows that x = 1. The equation d1'2 = fe(æ(~1)) = æ(fG (~1)) = (d2x + py)'2 therefore gives the desired congruence. The Mackey functor AG=e is a particular instance of a construction due to Dr* *ess [2]. In general, any Mackey functor M determines a Mackey functor MG=e, and the restriction æ and transfer ø for M determine maps bæM: MG=e __//_M and bøM: M __//_MG=e. The following diagram displays the values of MG=e, bæM, and bøM. MG=e ___bøM___//M___bæM___//_MG=e M(G=e)____ø__//_M(G=G)__æ__//_M(G=e) ___________________________________________________________* *_______UU_______________________UU_______________________UU___@ M ____________________________________æ______________________* *______________M____________________________________O__________@ ~~________________~~________________~~_____________________* *___________eO________________eM________________ M(G=e)pWW_____//_M(G=e)___//M(G=e)p. ___________________________________________WW_______________* *_____________________WW____________________________________ ___________________________________________________________* *______________________________________________________________@ perm ` perm To define the maps eOand eM, we select a generator g 2 G and assume that g acts on M(G=e)p by moving each summand to its successor (mod p). Then, for any (x1, x2, . .,.xp) 2 M(G=e)p and any y 2 M(G=e), X eO(x1, x2, . .,.xp) = gk-1xk and eM(y) = (y, g-1y, . .,.g1-py). 1 k p The maps eOand eMare referred to as the twisted folding and diagonal maps, resp* *ec- tively. Observe that, for each h 2 G, there is a map ~h: MG=e __//_MG=eof Mackey functors which is given by the action of h on M(G=e) at G=G and by a combination of the action of h on each summand and a permutation of the summands at G=e. The Mackey functors A, AG=e, L, and R are each characterized by a universal mapping property. In describing these properties, we denote the abelian group of maps from a Mackey functor M to a Mackey functor N by M(M, N). Note that evaluation at G=e gives a homomorphism vG=e: M(M, N) __//_HomG(M(G=e), N(G=e)). Lemma 1.2. Let M be a Mackey functor. (a) The map M(A, M) __//_M(G=G) sending a map f : A __//_M to fG (~) is an isomorphism of abelian groups. (b) Denote by (1, 0, 0, . .,.0) the element of Zp = AG=e(G=e) which is one in* * the first coordinate and zero in the others. Then, the map M(AG=e, M) ___//M(G=e) sending f : AG=e___//M to fe((1, 0, 0, . .,.0)) is an isomorphism of abelian gr* *oups. (c) The map vG=e: M(L, M) ___//HomG(L(G=e), M(G=e)) = M(G=e)G is an isomorphism. 1.1. MACKEY FUNCTORS FOR Z=p 11 (d) The map vG=e: M(M, R) ___//HomG(M(G=e), R(G=e)) = Hom (M(G=e)=G, Z) is an isomorphism. Recall from [10] that the category M carries a symmetric monoidal product which is denoted and called the box product. This product plays much the same role in M as the tensor product plays in Ab. Given two Mackey functors M and N, their box product M N is described by the diagram M N [(M(G=G) N(G=G)) _(M(G=e)__ N(G=e))]= ____________________UU____________________* *___ (æM æN,tr)____________________________________i2__* *__________________________________ _~~_______________________________ M(G=e) __N(G=e)WW___ ___________________________________________ __________________________________________* *_____________________ `M `N Here, tr denotes the trace map of the action `M `N of G on M(G=e) N(G=e). The equivalence relation is determined by the Frobenius relations mG øne æmG ne and øme nG me ænG , where mH 2 M(G=H) and nH 2 N(G=H) for H = e, G. The Burnside ring Mackey functor A is the unit for the box product. In general, box products are difficult to compute, but only a few simple cas* *es are needed in our work. For easy reference, the particular box products of inte* *rest to us are recorded in Table 1.1. All of the results displayed there can easily* * be extracted from Examples 1.2 of [11]. Of course, the L- and R- entries in this table apply only if G = Z=2. ______________________________________________ | | |A[d2] |L | R | |L- |R- | |______|_|_______|___|______|________|___|___|__ | A[d1] |A|[d1d2]L| | R | |L- |R- | |_______|_|_______|_|_______|________|___|___|_ | L | |L |L | L | 0 |L- | L- | |_______|_|_____|___|_______|________|___|____ | | R | |R |L | R | |L- |R- | |_______|_|_____|___|_______|________|___|___|_ | | | | 0 ||| 0 |0 | |______|_|_____|____|_______|________|____|___| | L- | |L- |L- | L- | 0 | L |L | |______|_|_____|___|________|________|____|___| | R- | |R- |L- | R- | 0 | L |L | |______|_|_____|___|_______|_________|____|___| Table 1.1. Box Products Proposition 1.3 of [11] characterizes a map f : M N ___//P out of a box product. The map f determines and is determined by a pair of homomorphisms FG : M(G=G) N(G=G) ___//P (G=G) 12 1. PRELIMINARIES and Fe: M(G=e) N(G=e) ___//P (G=e) which commute with restriction in the obvious way, preserve the G actions at G=* *e, and respect the Frobenius relations. This characterization is useful for unders* *tand- ing multiplicative structures given by maps out of box products. The box product can be used to define the notion of a ring in M. Specificall* *y, a Mackey functor ring is a Mackey functor S, together with structure maps j : A __//_S and OE : S S___//S which specify the unit and multiplication respectively, and which satisfy the u* *sual coherence diagrams. Equivalently, a Mackey functor ring is a Mackey functor S such that æ is a ring map between the rings S(G=G) and S(G=e), and ø is an S(G=G)-module map via æ. From this, A and R are easily seen to be Mackey functor rings. For any two Mackey functors M and N, there is a Mackey functor of äm ps" from M to N which provides a right adjoint to the box product construc- tion. This Mackey functor is given by M(M,_N)___ ____________________UU____________________* *____ (bæN)*__________________________________(bøN)*_* *_________________________________ ~~__________________________________ M(M, NG=e).VV_ ________________________________________ __________________________________________* *________________________ ` Here, (bæN)* and (bøN)* are derived from the maps bæNand bøNrelating N and NG=e, and the action ` comes from the self maps of NG=e associated to the various ele* *ments of G. As indicated in Section 1 of [10], the adjunction relating and is* * an isomorphism M(M N, P ) ~=M(M, natural in each of the three Mackey functors M, N, and P . 1.2. RO(G)-graded Mackey functor-valued homology Throughout this paper, we work with RO(G)-graded, Mackey functor-valued equivariant ordinary homology theories (see [16-18]). For the overview of such theories presented in this section, G can be any finite group. This sort of hom* *ology theory is determined by its coefficient system, which is a Mackey functor. For any G-space X, virtual G-representation !, and Mackey functor M, the Mackey functor-valued homology of X in dimension ! with Mackey functor coefficients M is denoted HG!(X; M). The connection between this notion of equivariant ordinary homology and the older notion introduced by Bredon [1] and Illman [9] is that the Bredon-Illman homology of X in dimension n with respect to the covariant coefficient system derived from M is just HGn(X; M)(G=G); that is, the value of HG*(X; M) associated to the trivial G-representation of dimension n and the orb* *it G=G. The equivariant ordinary homology theory associated to M is most easily de- fined in terms of the equivariant Eilenberg-Mac Lane spectrum HM (see [13, 14, 1.2. RO(G)-GRADED MACKEY FUNCTOR-VALUED HOMOLOGY 13 16]). If ! is represented by the formal difference V - W of G-representations V and W and K is a subgroup of G, then the value of the Mackey functor HG!(X; M) at G=K is given by HG!(X; M) (G=K) = [ V 1 G=K+ , W X+ ^ HM]G , where [?, ?]G denotes maps in the G-stable category. The restriction and transf* *er maps for HG!(X; M) come from the stabilization of space-level maps between or- bits and the transfers associated to those space-level maps regarded as equivar* *iant covering spaces (see Corollary V.9.4 and Proposition V.9.9 of [17]). At times, * *we work with the reduced homology eHG!(X; M)of a based G-space X. This can be viewed either as the homology of the pair (X, *) or as the collection of equiva* *riant stable homotopy groups [ V 1 G=K+ , W X ^ HM]G . The properties of equivariant ordinary homology used in this paper all fol- low easily from this spectrum-level definition. In particular, equivariant ordi* *nary homology satisfies the following axioms: (i)Additivity: Disjoint unions of G-spaces are carried to direct sums. (ii)Exactness: Cofibre sequences of G-spaces are converted to long exact sequences (iii)Exactness with respect to coefficients: Any short exact sequence of co* *ef- ficient Mackey functors yields a long exact sequence in homology. (iv)Suspension: HeG!(X; M)~=eHG!+V( V X ; M)for any G-representation V , element ! of RO(G), and based G-space X. (v)Dimension: For n 2 Z, regarded as the trivial G-representation of dimen- sion n, ( HGn(*; M)~= M ifn = 0, 0 otherwise. Remark 1.3. (a) The exactness of homology with respect to coefficients fol- lows directly from the observation that the passage to Eilenberg-Mac Lane spect* *ra converts a short exact sequence of Mackey functors into a fibre sequence of G- spectra. (b) Note that the dimension axiom says nothing explicit about the homology of a point at the nontrivial elements of RO(G). In fact, as illustrated in Chapter* * 8, the dimension axiom determines HG!(*; M)for every ! 2 RO(G), but computing HG!(*; M)can be highly nontrivial. One of the ways in which the Burnside ring Mackey functor A plays much the same role in the category of Mackey functors as that played by the integers in * *the category of abelian groups is that equivariant ordinary homology with Burnside * *ring coefficients is universal among equivariant ordinary homology theories in the s* *ame way that integral homology is universal among nonequivariant ordinary homology theories. In particular, the equivariant ordinary homology H* of a point with Burnside ring coeffficients is a ring, and the homology HG*(X; M) of any G-space with any coefficients is a (graded) module over H*. This follows from the fact * *that HA is a ring spectrum and, for any Mackey functor M, HM is a module spectrum over HA (see Proposition 5.4 of [14]). The complexity of the representation ring RO(G), compared to that of Z, makes it much harder to visualize the homology HG*(X; M) of a G-space X than to visualize the homology of a nonequivariant space. For G = Z=p, this difficul* *ty 14 1. PRELIMINARIES can be reduced somewhat by employing a simple observation about equivariant homology theories. Let |! |and |!G |denote the real dimensions of an element ! of RO(G) and its fixed set !G , respectively. Assume that ! and !0 are ele- ments of RO(G) such that |! |= |!0|and |!G |= |(!0)G|. Then the action map H!0-! HG!(X; M)___//HG!0(X; M)is an isomorphism by Proposition 8.12. More- over, H!0-! ~=A[d] for some integer d prime to p (see Definition 1.4 and Propos* *ition 1.7 below). Thus, HG!0(X; M)is derivable from HG!(X; M) by a straightforward al- gebraic process. In fact, frequently HG!0(X; M)and HG!(X; M) are isomorphic. It follows that one can, essentially, plot HG*(X; M) in the plane by assigning * *the Mackey functor HG!(X; M)to the point with integer coordinates (|!G |, |!)|. Str* *ictly speaking, in order to form this plot, one must select a representative ! of each collection of elements of RO(G) having a common pair (|!G |, |!)|of dimensions. However, the uncertainty implicit in this selection process is frequently immat* *erial. 1.3.The homology H* of a point Here, we provide some information about the additive and the multiplicative structure of H* . As indicated at the end of the previous section, it is almost possible to display H* by plotting it in the plane. Our first step in describin* *g H* is introducing the machinery needed to describe the uncertainty in this plot. * *At the heart of this machinery is a function d out of the subgroup RO0(G) of RO(G) consisting of those ! 2 RO(G) such that |! |= |!G |= 0. This function may be regarded as a homomorphism from RO0(G) to the quotient group (Z=p)x = 1 of the multiplicative group (Z=p)x of nonzero elements of Z=p. When so regarded, d is well defined and uniquely determined. Unfortunately, we often need to think * *of d as a function from RO0(G) to Z whose values are integers prime to p. When so regarded, d is not a homomorphism, is not uniquely determined, and is not even obviously well-defined. These problems with d are tied to picking a representiv* *e of each element ! of RO0(G) as a formal difference V - W of two G-representations V and W . For p = 2, no difficulties arise in selecting V and W . For an odd pr* *ime p, every element of RO0(G) can be written as a formal difference of complex G- representations. This suffices to ensure that d is at least well-defined as a f* *unction into Z. Definition 1.4. If p = 2, then d : RO0(G) __//_Z is the constant map to 1 2 Z. If p is odd and ! 2 RO0(G) is nonzero, select nontrivial irreducible com* *plex G-representations j1, j2, . . . , jn and i1, i2, . . . , in such that ! = j1 + j2 + . .+.jn - (i1 + i2 + . .+.in). For each i, take dito be the least positive integer such that the complex power* * map z 7! zdi is a G-map from the unit circle Sji C of ji to Sii. Note that di must be prime to p since both ji and ii are nontrivial G-representations. Let Y d(!) = di. i Also, let d(0) = 1. Observe that this gives a well-defined map from RO0(G) to Z whose values are integers prime to p. Typically, we denote d(!) by d!. It is easy to verify the following key properties of the function d. 1.3. THE HOMOLOGY H* OF A POINT 15 Lemma 1.5. (a) When regarded as a map into (Z=p)x = 1, d is a homomor- phism and is independent of the choices made in its definition. (b) Let j be a nontrivial irreducible complex G-representation, and let jk be its k-fold complex tensor power for some integer k relatively prime to p. Then j - jk 2 RO0(G) and dj-jk k mod p. Thus, when regarded as a map into (Z=p)x = 1, d is surjective. Remark 1.6. (a) The integer d! depends on the choices made in its definition in several ways. First, it depends on the ordering of the ji and ii. Second, * *if a nontrivial irreducible representation j is inserted in both of the ji and ii li* *sts, but at different places in those lists, then d! is changed. These two dependencies * *vanish if d is regarded as a function into (Z=p)x . The most serious dependency comes, however, from the identification of complex representations with their conjugat* *es in RO(G). Replacing one of the ji or ii with its conjugate changes the sign of d! in (Z=p)x . Passing to the quotient group (Z=p)x = 1 eliminates this chang* *e. The appearance of k, instead of k, in Lemma 1.5(b) is a reflection of this si* *gn problem. (b) The point of the map d is that, if ! 2 RO0(G), then H! ~=A[d!]. It usually suffices to think of d as a map into (Z=p)x = 1, since this value determines * *the isomorphism class of A[d!]. However, in picking generators of either of the sta* *ndard forms for A[d!](G=G), the integral value of d! is implicitly used. (c) The function d defined here is not the same as that defined in [11], beca* *use here we are working with homology rather than cohomology. The values of the two functions are multiplicative inverses in (Z=p)x = 1. (d) If p = 3, then (Z=p)x = 1 is trivial, and the map d : RO0(G) ___//Z can be taken to be the constant map to 1 2 Z. However, this masks, rather than eliminates, the sign problems implicit in replacing complex representations by * *their conjugates. The additive structure of H* depends on whether p is even or odd. Thus, we give a two part proposition (which is a special case of Theorem 8.1) and two tables describing that structure. Since H! almost always depends only on the pair (|!G |, |!)|, the tables displayed on the next two pages are likely to be * *more enlightening than the proposition. Proposition 1.7. (a)Let p be odd and ! 2 RO(G). Then 8 >>>A[d!]if ! 2 RO0(G), >>> G >>> R if |! |= 0 and |! |> 0, >>> L if |! |= 0 and |!G |< 0, >< G H! = > if8|! |6= 0 andG|! |= 0, G >>> ><|! |< 0, |! |> 0, and |! |is even >>>if or >>> >: >>> |! |> 0, |!G | -3, and |!G |is odd, : 0 otherwise. 16 1. PRELIMINARIES (b) Let p = 2 and ! 2 RO(G). Then 8 >>>A[d!]if ! 2 RO0(G), >>> G G >>> R if |! |= 0, |! |> 0, and |! |is even, >>> R- if |! |= 0, |!G | -1, and |!G |is odd, >>> >>> L if |! |= 0, |!G |< 0, and |!G |is even, < L if |! |= 0, |!G | -3, and |!G |is odd, H! = > - G >>> if |!8|6= 0 and |! |= 0, >>> >|! |< 0, |!G |> 0, and |!G |is even >>> < >>>if or >>> >: G G >>: |! |> 0, |! | -3, and |! |is odd, 0 otherwise. Remark 1.8. Even for p = 2, the display in part (a) of the proposition above correctly describes H! if |!G |and |! |are either both even or both odd. This i* *s the part of H* that matters for almost every aspect of our arguments. Thus, the best way to follow the remainder of the paper is to focus on the odd prime case. |! | V .. . .. . . .. . .<.Z=p> . .<.Z=p> . . . L L A[d!] R R . .>.|!G | . . . . . . .. . . . .. .. Figure 1.1. H* for p odd 1.3. THE HOMOLOGY H* OF A POINT 17 |! | V .. . . . .. .. . .<.Z=2> . .<.Z=2> . . .L- L L- L R- A R- R R- R . .>.|!G | . . . . . . .. . . . .. .. Figure 1.2. H* for p = 2 In order to characterize the projective objects in the category of H*-module* *s, we need to describe HG*(G=e ; A). Corollary 1.9. Let ! be an element of RO(G). Then ( HG!(G=e ; A)= AG=e if|! |= 0, 0 otherwise. Proof. This follows immediately from the observation that, for any G space X and any ! 2 RO(G), HG!(G=e x X ; A)~=HG!(X; A)G=e. For convenience, we recall here from Theorems 4.3 and 4.9 of [11] the portion of the multiplicative structure of H* that is needed for our arguments. We need to understand this structure only in those dimensions ! for which |!G |and |! |* *are either both even or both odd. In these dimensions, it does not matter whether or not p is 2. Proposition 1.10. There exist elements 8 >>>'! 2 H! (G=e) for|! |= 0, >>>~ , ~ø, ~ 2 H (G=G)for! 2 RO (G), >>> ! ! ! ! 0 ><,! 2 H! (G=G) for|! |= 0, |!G |> 0, and|!G |even, G |< 0, and|!G |even, >>>~ø!2 H! (G=G) for|! |= 0, |! >>>ffl! 2 H! (G=G) for|! |< 0 and |!G |= 0, >>> -1 G >:fflÆ ~! 2 H!-Æ (G=G)for! 2 RO0(G), |Æ|< 0, and|Æ |= 0, ! 2 H! (G=G) for|! |> 0, |!G | -3, and|!G |odd of H* which additively generate H! (G=G) (or H! (G=e) as appropriate) in their dimensions. Moreover, these elements satisfy the relations: (a) '!'!0= '!+!0 (b) æ(~!) = d!'! 18 1. PRELIMINARIES (c) ~ø!= ø('!) if |!G | 0 (d) ~! = p~! - d!ø~! (e) æ(,!) = '! (f) ~!,Æ = d!,!+Æ (g) ,!,Æ = ,!+Æ( G| 0 (h) ~ø!,Æ = ~ø!+Æ if|(! + Æ) p,!+Æ if|(! + Æ)G|> 0 (i)~!fflÆ = ffl!+Æ (j)ffl!fflÆ = ffl!+Æ (k) ffl!,Æ generates H!+Æ (G=G) (l)ffl!,Æ = dÆ0-Æ0ffl!0,Æ0if ! + Æ = !0+ Æ0 (m) ffl-1Æ~! = ffl-1Æ0~!0 if ! - Æ = !0- Æ0 (n) ~!0(ffl-1Æ~!) = ffl-1Æ~!+!08 >~!+Æ0-Æ if|Æ - Æ0|= 0 :pffl!+Æ0-Æif|Æ - Æ0|> 0 (p) (ffl-1Æ~!)(ffl-1Æ0~!0) = p(ffl-1Æ+Æ0~!+!0) (q) ~! Æ = !+Æ (r) ffl! Æ = !+Æ if |! + Æ|> 0 (s) ,! Æ = d !+Æ for some integer d prime to p if |(! + Æ)G| -3 (t) (ffl-1Æ~!) Æ0= 0 In the statements of these relations, the subscripts indicating the dimensions * *of the elements are implicitly assumed to be in the allowed range of dimensions for th* *at type of element. 1.4. Modules over H* In this section, we introduce the category H*-Mod of modules over the RO(G)- graded Mackey functor ring H*. Only those properties of H*-Mod needed to state our main result and to outline its proof are covered here. Most of what we need* * is related to the behavior of free H*-modules. The more sophisticated aspects of t* *he category H*-Mod , like its symmetric monoidal closed structure, are discussed l* *ater in Chapter 10. An H* -module M may be described as an RO(G)-graded collection M! of Mackey functors together with action maps HÆ M! __//_MÆ+! , for Æ, ! 2 RO(G), which make the obvious diagrams commute. In making use of the module structure on M, we often view these action maps in a slightly differ* *ent way. Definition 1.11. Assume that Æ 2 RO(G) and x 2 HÆ (G=G). By Lemma 1.2(a), there is a unique map ~x: A __//_HÆwhich takes the standard generator ~* * of A(G=G) to x. For any ! 2 RO(G), the composite M! ~=A M! _~x1_//_HÆ M! __//_MÆ+! is referred to as the multiplication by x map on M. 1.4. MODULES OVER H* 19 The appropriate definition of a free module in H*-Mod is not quite as obvio* *us as one would expect. Thus, our first objective is to assign a precise meaning t* *o that notion. To accomplish that goal, we must first describe a natural set of projec* *tive generators for the category H*-Mod . For any H*-module M and any ! 2 RO(G), let !M denote the H*-module specified by ( !M)Æ = MÆ-! . We refer to such a module as a dimension-shifted copy of M. Since H* is a graded ring, a set of projective generators for H*-Mod obviously ought to include dimension-shift* *ed copies of H*. If we were working with abelian groups rather than Mackey functor* *s, this would suffice. However, a somewhat larger set of generators is needed in t* *he context of Mackey functors. The source of this need can be seen even at the lev* *el of ungraded Mackey functors. If S is a Mackey functor ring and C is a module over * *S, then C need not be a quotient of a direct sum of copies of S because the elemen* *ts of C(G=e) are not clearly seen by S. These elements can only be seen properly by SG=e. Thus, for a typical Mackey functor ring S, the obvious set of projecti* *ve generators for S-Mod is {S, SG=e}. By analogy with the category of modules over* * a graded ring, one can think of SG=e as a copy of S shifted by "dimension" G=e. T* *he analog of SG=e for the category H*-Mod is the homology HG*(G=e ; A)of a single * *free orbit G=e. We denote this H*-module by (H* )G=e. Of course, we need to include dimension-shifted copies of (H* )G=e in our set of projective generators of H*-* *Mod . However, if Æ, ! 2 RO(G) and |Æ|= |! |, then the obvious G-homeomorphism between the G-spaces Æ(G=e)+ and !(G=e)+ induces an isomorphism between Æ(H* )G=e and !(H* )G=e. Thus, we include only the modules m (H* )G=e, for m 2 Z. The following result suffices to prove that the H* -modules !H* and m (H* )G=e together form a set of projective generators for H*-Mod . Lemma 1.12. Let M be a module over H* and ! be an element of RO(G). (a) The set of H*-module maps from !H* to M is isomorphic to the abelian group (M!)(G=G). (b) The set of H* -module maps from |!|(H* )G=e to M is isomorphic to the abelian group (M!)(G=e). This lemma implies that any direct sum _ ! _ ! M M M P = !iH* mj(H* )G=e i2I j2J of dimension-shifted copies of H* and (H* )G=e is projective as an H*-module. S* *uch a direct sum is, however, much better behaved than an arbitrary projective modu* *le in that Lemma 1.12 provides very precise control over maps out of P . We can th* *ink of P as having one "generator" in dimension !ifor each i 2 I and one "generator" in dimension mj for each j 2 J . An H* -module map from P to any other H* - module M is determined by what happens on these "generators". Further, there are no "relationsö n P which constrain where these "generators" can be sent. T* *his control over maps is the essential characteristic of a free module over an ordi* *nary ring, and so motivates our definition of a free module. Definition 1.13. A free module P over the ring H* is a module isomorphic to a direct sum of the form ( i2I !iH*) j2J mj(H* )G=e . The individual summands !iH* and mj(H* )G=e of P should be thought of as the generators of P . It is important to distinguish between these two types of summands. Since 20 1. PRELIMINARIES those of the form !H* correspond more closely to the generators of a free modu* *le over an ordinary ring, we refer to them as the purely free generators of P . In* * terms of their behavior, summands of the form m (H* )G=e sit somewhere between free and projective modules over an ordinary ring. Thus, we refer to these summands as the projective generators of P . Unfortunately, the dimension of a purely free generator of a free H*-module0P is not as well-defined as one might like because the H*-modules !H* and ! H* can be isomorphic even if !0 6= ! in RO(G). The following result describes this uncertainity in the dimension of a purely free generator. Lemma 1.14. The H*-modules !H* and !0H* are isomorphic if and only if !0- ! 2 RO0(G) and d!0-! 1 mod p. Proof. First assume that !H* ~= !0H*. Then A = ( !0H*)!0~=( !H* )!0. However, Proposition 1.7 indicates that ( !H* )!0cannot be isomorphic to A unle* *ss !0- ! 2 RO0(G). If !0- ! 2 RO0(G), then ( !H* )!0 = A[d!0-!] by the same proposition, and Lemma 1.1 gives that d!0-! 1 mod p. Now assume that !0- ! 2 RO0(G) and d!0-! 1 mod p. Use Lemmas 1.1, 1.2(a), and 1.12(a)0to select an H*-module map f : !H* ___// !0H*such that f! : ( !H* )!___//( ! H*)!is an isomorphism. Let Æ 2 RO(G), and consider the commuting diagram 1 f! 0 HÆ-! ( !H* )!__~=_//HÆ-! ( ! H*)! | | | | fflffl| fÆ fflffl|0 ( !H* )Æ____________//_( ! H*)Æ which expresses the fact that f is an H* -module map. By Proposition 8.12, the vertical maps in this diagram are isomorphisms. Thus, f is an isomorphism. Remark 1.15. (a) One good way of presenting the purely free generators of a free module over H* is by plotting their dimensions in the plane. A single gene* *rator in dimension ! is denoted by a dot o at the point (|!G |, |!)|. If two or more generators share the same coordinates, then the number of generators, rather th* *an a o, is plotted at that point. Note that the uncertainty in dimension of a pure* *ly free generator described in Lemma 1.14 does not affect the point to which the genera* *tor plots. Some information may be lost0in this plot since distinct elements ! and * *!0 of RO(G) for which !H* and ! H* are not isomorphic can plot to the same point if p > 3. However, the information which matters most in our arguments is retained in such a plot. (b) There is some uncertainty in the dimensions of the purely free generators of a free H* -module beyond that described in Lemma 1.14. There are sequences Æ1, Æ2, . . . , Æn and !1, !2, . . . , !n of elements of RO(G) for which the fr* *ee H* - modules i ÆiH* and i !iH* are isomorphic for reasons having nothing to do with either merely reindexing the lists of generators or the isomorphisms coming from Lemma 1.14. Examples of this sort are described in [5] in the case n = 2, p > 3, (|ÆGi|, |Æi|) = (|!Gi|, |!i|) for i = 1, 2, and either |ÆG1|= |ÆG2|or |Æ* *1|= |Æ2|. (c) The value of !H* in dimension ! is the Burnside ring Mackey functor A. An H*-module map f : !H* ___//M is determined by the image f!(G=G)(~) 1.4. MODULES OVER H* 21 of the canonical element ~ of A(G=G) in M!(G=G). Thus, we could think of ~ as the generator of !H* . Similarly, the value of m (H* )G=e in dimension m * *is the Mackey functor AG . An H*-module map f : m (H* )G=e__//M is determined by the image fm (G=e)((1, 0, 0, . .,.0)) 2 Mm (G=e) of the element (1, 0, 0, . * *.,.0) of AG (G=e) introduced in Lemma 1.2(b). Hence, we could think of (1, 0, 0, . .,.0)* * as the generator of m (H* )G=e. However, we rarely work at the level of elements.* * It is usually far more productive to think of a generator of a free H* -module P as the inclusion !H* ___//P or m (H* )G=e__//P of the appropriate summand rather than as the image of ~ or (1, 0, 0, . .,.0) under this inclusion. The dimensions of the free modules of interest in this paper frequently sati* *sfy two special conditions. Definition 1.16. (a)An element ! of RO(G) is said to be even-dimensional if both |! |and |!G |are even. Note that, if p is odd, then these two integers * *have to be either both even or both odd. A purely free generator of a free H*-module is said to be even-dimensional if its dimension is an even-dimensional element * *of RO(G). A projective generator is said to be even-dimensional if its dimension m* * is an even integer. (b) An element ! of RO(G) is said to be space-like if |! | |!G | 0. A purely free generator of a free H*-module is said to be space-like if its dimen* *sion is space-like. A projective generator is said to be space-like if its dimension* * m is a nonnegative integer. Having introduced free H*-modules, we now turn to an investigation of maps between finitely generated free H*-modules. Definition 1.17. (a)Let ! and !0 be elements of RO(G).0The set of maps from the free H* -module !H*0 to the free H* -module ! H* can be identified with the abelian group ( ! H*)!(G=G). Unless |! |= |!0|and |!G |= |(!0)G|,0this group is one of the cyclic groups Z, Z=p, or 0. A map f : !H* __//_ ! H*is called a standard shift map if it is a generator of this cyclic group. Clearly,* * f is a standard shift map if and only if f! is onto. Since our interest in standard sh* *ift maps comes from the role they play in dimension-shifting long exact sequences, * *we have not defined the notion of a standard shift map in the case where |! |= |!0| and |!G |= |(!0)G|. (b) Assume that M is a finitely generated free H* -module with purely free generators in dimensions !1, !2, . . . , !m , and that N is a finitely generate* *d free H*-module with purely free generators in dimensions !01, !02, . .0. , !0n. A m* *ap f : M __//_N is determined by its components fi,j: !iH* ___// !jH*. The map f is said to be constructed from standard shift maps if fi,jis a standard shift map for every i and j. Note that, if there is a pair i, j such that |!i|= |!0j|* *and |!Gi|= |(!0j)G|, then there is no map from M to N constructed from standard shi* *ft maps. Remark 1.18. This notion of a standard shift map is somewhat different from that introduced in [5]. The change is forced by our discussion of more complex dimension-shifting long exact sequences than those discussed in [5]. Definition 1.19. Four types of standard shift maps f : !H* ___// !0H*are of special interest to us. 22 1. PRELIMINARIES (a) Assume0that |! |= |!0|, |!G |> |(!0)G|, and ! - !0 is even-dimensional so that ( ! H*)! ~=R. Then f is a standard shift map if and only if f!(~) = ,, where , is the standard generator of R. Such an f is called a horizontal shift * *map. (b) Assume0that |! |< |!0|, |!G |= |(!0)G|, and ! - !0 is even-dimensional so that ( ! H*)! ~=. Then f is a standard shift map if and only if f!(~) = ffl, where ffl is the standard generator of . Such an f is called a vertical shif* *t map. (c) Assume0that |! |< |!0|, |!G |> |(!0)G|, and ! - !0 is even-dimensional so that ( ! H*)! ~=. Then f is a standard shift map if and only if it is nonz* *ero. Such an f is called a diagonal shift map. (d) Assume that |! |- |!0|is an odd0positive integer and |!G |- |(!0)G|is an odd integer less than -1. Then ( ! H*)! ~=, and f is a standard shift map if and only if it is nonzero. Such an f is called a boundary shift map because the boundary maps in our dimension-shifting long exact sequences are constructed from maps of this type. Because of the critical role played by boundary shift maps in our dimension- shifting long exact sequences, it is important to know the dimensions in which * *they are nonzero. Lemma 1.20. Let !, !0, and Æ be elements of RO(G) such that |! |- |!0|is an odd positive integer0and |!G |- |(!0)G|is an odd integer less than -1. Also,* * let f : !H* ___// ! H*be a nonzero H*-module map. Then 0 fÆ : ( !H* )Æ__//_( ! H*)Æ is nonzero if and only if all of the following hold: (i)|ÆG |- |!G |is even, (ii)|!(||Æ|, 0| ifp = 2 or (iii) |Æ| |! |Æ|> |!0| ifp 6= 2, (iv)|!G | |ÆG | |(!0)G|- 3. Proof. Observe that, for any Æ,0fÆ can be computed from f! by using the H*-module structures on !H* and ! H*. We begin by showing that fÆ vanishes 0 unless Æ satisfies the listed conditions. By examining the plots of !H* and !* * H*, it is easy to see0that, for most Æ not satisfying these conditions, at least on* *e of ( !H* )Æ or ( ! H*)Æ is zero. The only exceptions to this occur on the three li* *nes given by the equations |Æ|= |!,||Æ|= |!0|, and |ÆG |= |!G.|On the line |Æ|= |!,| the exceptions occur when either |!G |> |ÆG |or |ÆG |= |(!0)G|. In these cases, fÆ must be zero because there are no nonzero maps of the forms L __//_or R- ___//. On the line |Æ|= |!0|, exceptions occur only if |ÆG |> |(!0)G|- 3 and p = 2. In this case, fÆ must be zero because there are no nonzero maps from to R- . On the line |ÆG |= |!G,|the exceptions occur only if |! |< |Æ|. He* *re, Proposition 1.10(t) implies that fÆ is zero. Now assume that Æ satisfies the listed conditions. If |! |= |Æ|, then HÆ-! ~* *=R is generated at G=G by ,Æ-! . Proposition 1.10(s) therefore implies that fÆ is non* *zero. If |! |> |Æ|, then select Æ0 2 RO(G) such that |(Æ0)G|= |ÆG |and |Æ0|= |! |. The map fÆ0 is nonzero by our earlier argument.0 Moreover,0multiplication by fflÆ-Æ02 HÆ-Æ0 gives a monomorphism from ( ! H*)Æ0to ( ! H*)Æ by Lemma 8.7. It follows trivially that fÆ is nonzero. 1.5. GENERALIZED G-CELL COMPLEXES 23 1.5. Generalized G-cell complexes Our initial definitions in this section apply to any compact Lie group G. A generalized G-cell complex X is a G-space X together with an increasing sequence of G-subspaces Xn of X such that X0 is a disjoint union of orbits, Xn+1 is form* *ed from Xn by attaching G-cells, X = [nXn, and X has the colimit (or weak) topology derived from the subspaces Xn. The G-cells allowed in the formation of X are of the form G xH DV , where H is a (closed) subgroup of G and DV is the unit disk of a finite dimensional H-representation V . Such a cell is attached to Xn in t* *he process of forming Xn+1 via an attaching G-map from G xH SV , where SV is the unit sphere of V , to Xn. The set of cells attached to Xn to form Xn+1 is denot* *ed J n+1. Note that no restrictions are imposed on the dimension of the cells atta* *ched to Xn in the formation of Xn+1. A cell G xH DV is said to be even-dimensional if the fixed point subset V K is even-dimensional over R for every subgroup K of H. In the case of interest within the rest of this paper, G = Z=p for some prime p. Since the only subgroups of G are G itself and the trivial group, the only t* *ypes of cells appearing in a generalized G-cell complex X are those of the form DV ,* * for some G-representation V , and those of the form G x Dm , for some integer m. In cells of the latter type, G acts trivially on the disk Dm . These two types of * *cells are represented algebraically by the two types of generators which occur in free H*-modules. Cell complexes of this form are of interest because they arise naturally from equivariant Morse theory (see, for example, [21]). Further, if G is a finite ab* *elian group, then the usual Schubert cell structure on Grassmannian manifolds gener- alizes in an obvious way to a generalized G-cell structure on the Grassmannian manifold G(V, k) of k-planes in some G-representation V (see Chapter 7). Here, the action of G on G(V, k) is the obvious one derived from the action of G on V* * . For any n 0, there is a cofibre sequence W V Xn+___//Xn+1+__//_ G+ ^H S GxHDV 2J n+1 associated to the attachment of the cells in J n+1to Xn. Here, SV is the one-po* *int compactification of V . The point at infinity in SV is given trivial G-action* * and taken as the basepoint of SV . Associated to this cofibre sequence, we have lo* *ng exact sequences . ._._//HG*(Xn ;_S)//_HG*(Xn+1 ; S) M __//_ eHG*(G+ ^H SV ; S)@//_HG*-1(Xn ;_S)//_. . . GxHDV 2J n+1 and . ._._//eHG*(Xn ;_S)//_eHG*(Xn+1 ; S) M __//_ eHG*(G+ ^H SV ; S)@//_eHG*-1(Xn_;_S)//. . . GxHDV 2J n+1 in equivariant ordinary homology with any Mackey functor S as coefficients. We refer to these sequences as the cell-attaching long exact sequences of X. An analysis of the boundary map @ in these long exact sequences lies at the * *very center of the main argument in this paper. That analysis is tricky enough when 24 1. PRELIMINARIES only one cell is added to Xn in the formation of Xn+1, and can become hopelessly complicated when more than one cell is added. To get around this difficulty, we produce an alternative filtration of X in which cells are added one at a time. * *Since we have not assumed that X has only countably many cells, this filtration has t* *o be indexed on some ordinal J which may be larger than the set of natural numbers. The simpliest way to form J is to well order each of the sets J nand then take * *their union with the ordering which makes each element of J m less than any element of J nif m < n. The result is a well ordered set, and so may be thought of as an ordinal number. Notationally, however, it is convenient to think of J as an abs* *tract ordinal. Associated to each ff 2 J , there is a closed G-subspace Xffof X. Each ff 2 J has an immediate successor in J which is denoted ff + 1. The subspace Xff+1is formed from Xffby adding a single cell, which is denoted G xHffDVff. If fi is a limit point in J , then Xfi= [ff |!n|and |V G|< |!G1|. In this case, it is possible for the boun* *dary map @ : eHG*(SV ; A)//_eHG*-1(B; A) in the cell-attaching long exact sequence to hit each of the generators of eHG** *(B; A) in the sense that its composite with the projection onto the summand spanned by any one generator is nonzero. If this occurs, then eHG*(X; A)is free over H* wi* *th generators in dimensions !01, !02, . . . , !0n+1such that |!0i|= |!i|, fori n; |!0n+1|= |V;| |(!0i)G|= |!Gi-1|, fori 2; and |(!01)G|= |V G|. The relations among these various dimensions are best understood via the plot in Figure 2.1. Note that, in this case, none of the generators of eHG*(X; A)are* * in 25 26 2. THE MAIN FREENESS THEOREM (THEOREM 2.5) ||ff| | | |6 | 0 | V !n+1 | | | !0n !n | | | !n-1 | | | . | .. | | | | | | !0 ! | 2 2 | | !01 !1 | ___________________________________-||ffG| | | Figure 2.1. The generators of eHG*(B; A)and eHG*(X; A) the expected dimensions. The first n generators of HeG*(X; A)are, in a suitable sense, derived from the generators of eHG*(B; A). However, their dimensions plo* *t to the left of dimensions of the corresponding generators of B. The last generator* * is derived from the cell added to B to form X, but its dimension plots to the righ* *t of where it might be expected to lie. In Remark 1.15(b), we noted that two free modules over H* might be isomorphic even if their generators were in not in the same dimensions. In order to preven* *t that remark from causing some confusion here, it is important to note that eHG*(X; A) is obviously not isomorphic to the free H*-module with generators in dimensions !1, !2, . . . , !n, and V . Thus, this dimension-shifting is real and not mere* *ly a failure to notice that two free H*-modules with generators in different dimensi* *ons are nevertheless isomorphic. Whenever an even-dimensional cell DV is added to a space B whose homology is free over H* with even-dimensional generators and the boundary map @ : eHG*(SV ; A)//_HG*-1(B; A) in the cell-attaching long exact sequence is nonzero, some shifting of the dime* *nsions of the generators occurs. However, if the dimensions of the generators of HG*(B* *; A) do not plot in a nice "stairstep" pattern like that in Example 2.2, it can be d* *ifficult to predict exactly which shifts occur. Hence nothing is said in Theorem 2.1 abo* *ut the dimensions of the generators of HG*(X; A). All that can be said in general about this shifting is that old generators coming from HG*(B; A)may remain in the same dimension or move to the left in dimension; whereas the new generator coming from SV must move to the right of its expected dimension whenever @ is nonzero. 2. THE MAIN FREENESS THEOREM (THEOREM 2.5) 27 This shifting of dimensions means that, even if every finite complex of an i* *nfi- nite generalized G-cell complex X has homology which is free over H* with even- dimensional generators, it is not at all obvious that the homology of X must be free over H*. The following example illustrates what can go wrong. Example 2.3. Let G = Z=p, and let j be a nontrivial irreducible complex G-representation. We want to form a generalized G-cell complex X containing one 2-cell on which G acts trivially and one cell of the form D(mj) for each m 2. Let X1 be S2 with trivial action, and form X2 from X1 by attaching a cell of the form D(2j) in such a way that the boundary map @ of the cell-attaching long exa* *ct sequence is nonzero. Examples of linear actions of G on CP 4having a cell struc* *ture of this form can are given in [11]. Since @ 6= 0, the generators of eHG*(X2 ; A* *)must plot at the coordinates (0, 2) and (2, 4). One might assume that their dimensio* *ns were j and 2 + j. However, due to the way in which dimensions shift, the actual dimensions might involve irreducibles other than j. We would like to form X3 from X2 by attaching a cell of the form D(3j) in such a way that the boundary map @ of the cell-attaching long exact sequence is nonzero on the generator of eHG*(X2 ; A)which plots to (2, 4). Unfortunately, it is not obvious that the de* *sired attaching map can be constructed. However, the algebraic machinery presented in Chapter 6 allows us to construct a purely algebraic dimension-shifting long exa* *ct sequence which reflects what must happen in homology if the appropriate cell can be attached. It follows that, if X3 can be constructed, then it must have homol* *ogy generators which plot at the coordinates (0, 2), (0, 4), and (2, 6). In general, Xm should have homology generators which plot at the coordinates (0, 2), (0, 4), . . . , (0, 2m - 2), and (2, 2m). The next stage Xm+1 should be* * formed from Xm by adding a cell of the form D((m+1)j) in such a way that the boundary map @ is nonzero on the homology generator of Xm which plots at (2, 2m). Again, even though it is not obvious that the desired attaching map exist geometricall* *y, the results in Chapter 6 allow us to construct a purely algebraic long exact se- quence which reflects what must happen in homology if the appropriate cell can * *be attached. This algebra allows us to see that, if the space X = [m Xm exists, then eHG*(X; A)contains a free summand with a generator plotting at (0, 2m), for all m 1, and another, nonfree (and even nonprojective) summand which may be thought of as the "ghostö f the generators plotting at (2, 2m) in the various eHG*(Xm ; A). An easily seen part of this nonfree summand is a copy of at location (2, 2m) for every integer m. Strictly speaking, this example does not * *show that there are infinite generalized G-cell complexes formed from even-dimension* *al cells which have nonfree homology. However, it does suggest a general way in wh* *ich passing to colimits in homology could destroy freeness. Thus, to prove that eve* *ry generalized G-cell complex formed from even-dimensional cells had free homology, it would be necessary to show that a vast number of potential attaching maps do not exist. Our third example illustrates another kind of difficulty which can arise in * *prov- ing that the homology of an infinite complex is free over H* . The homology of the space constructed in this example is, in fact, free over H*. However, the p* *roof that it is involves an ad hoc argument which does not seem to have any reasonab* *le generalization. 28 2. THE MAIN FREENESS THEOREM (THEOREM 2.5) Example 2.4. Recall that, in the first stage of the example above, we attach* *ed a cell of the form D(2j) to a 2-sphere via an attaching map f : S(2j)___//S2that is known to exist. Here, we wish to work with the double suspension of f. Since the boundary S(2 + 2j) of D(2 + 2j) is a suspension, rather than mapping it to a single 4-sphere, we can map it to a wedge S4 _ S4 of two 4-spheres by taking one copy of the suspension of f going into each of the two 4-spheres. Now consider * *an infinite wedge _m2Z S4 of 4-spheres indexed on the integers. Form a new space X by adjoining a Z-indexed collection of cells of the form D(2 + 2j) to this infi* *nite wedge. The mthcell should be adjoined to the infinite wedge by attaching it to * *the spheres indexed on m and m + 1 via our sum map into S4 _ S4. The equivariant ordinary homology of the wedge of spheres is certainly free * *over H*. However, if one uses the standard generating set for this free module coming from the individual copies of S4, then the dimension shifting which occurs in t* *he passage to the homology of X makes it very hard to see that the homology of X is also free over H* . Nevertheless, by replacing the standard generating set w* *ith one consisting of exactly one of the standard generators plus the sum of the mth and (m + 1)ststandard generators, for every integer m, one can use an elementary variant of the proof of our main freeness theorem to show that eHG*(X; A)is free over H*. It has one generator plotting at the point (4, 4), a Z-indexed collect* *ion of generators plotting at the point (2, 4), and a Z-indexed collection of gener* *ators plotting at the point (4, 6). Basis changes similar to one used here are needed in the proof of our general freeness theorem. However, those basis changes, accomplished in Proposition 4.5, are much less precisely tuned to the geometry of the space than the one used for X. The difference is easily seen by noting that our general result, Theorem 2.* *5, applies to any subcomplex Y of X which contains only finitely many of the cells* * of the form D(2 + 2j). Looking over the proof of that result to see how it applies* * to Y , one can see that the finiteness condition imposed on Y allows us to concoct* * an appropriate change of basis in a rather naive way. In fact, it would not be unf* *air to say that we find this change of basis by stumbling around in the dark until we * *trip over it. Certainly this is a far less elegant approach than beginning the argum* *ent, as we did for eHG*(X; A), by picking a change of basis ideally suited to the ge* *ometry of the space. In part, this lack of elegance seems inherent in the ö ne cell at* * a time" approach used to prove our main result. However, for an arbitrary generalized G- cell complex, it is not at all obvious that there are changes of basis as well * *suited to a freeness argument as the one we suggest here for X. Examples 2.3 and 2.4 display the two distinct difficulties motivating the so* *me- what curious assumption about cell dimensions appearing in our main freeness theorem. It might be possible to weaken this assumption, but it seems unlikely that it can be removed entirely. Theorem 2.5 (Main Freeness Theorem). Let G = Z=p, and let X be a gener- alized G-cell complex formed from only even-dimensional cells of the types DV a* *nd G x Dk. Assume that, for each m 1 and each cell DV in J m, there is only a finite number of cells DW in the collections J n, for n > m, such that |W |> |V* * | and |W G |< |V G|. Then the RO(G)-graded Mackey functor-valued equivariant or- dinary homology of X with Burnside ring coefficients is free over H* . Moreover, there is a one-to-one correspondence between the generators of HG*(X; A)and the cells of X. 2. THE MAIN FREENESS THEOREM (THEOREM 2.5) 29 Remark 2.6. (a) In Chapter 7, we show that the complex Grassmann mani- fold G(V, k) of complex k-dimensional subspaces of a complex G-representation V is a generalized G-cell complex satisfying the hypotheses of this theorem. Thu* *s, HG*(G(V, k); A)is free over H* (see Corollary 7.2). (b) A reasonable notion of finite type for a generalized Z=p-cell complex X w* *ould be that, for each non-negative integer m, X has only finitely many cells of the* * form G x Dn with n m and only finitely many cells of the form DV with |V G| m. Clearly, Theorem 2.5 applies to such a finite type generalized Z=p-cell complex. CHAPTER 3 An outline of the proof of the main freeness result (Theorem 2.5) Throughout this chapter, we assume that G = Z=p and that X is a generalized G-cell complex formed from even-dimensional cells. We work with the ö ne cell a* *t a time" filtration {Xff}ff2Jof X in which Xff+1is formed from Xffby adding a sing* *le cell, which is either of the form DVfffor some even-dimensional G-representation Vffor of the form G x Dmfffor some even integer mff. The compactness axiom for equivariant ordinary homology provides the following result. Lemma 3.1. For any Mackey functor S, the canonical map colimff2JHG*(Xff;_S)//HG*(X; S) is an isomorphism. The proof of Theorem 2.5 therefore consists of two parts. In the first part, we show that, if the homology HG*(B; A) of a G-space B is free over H* with even-dimensional generators and the G-space Y is formed from B by adding a single even-dimensional cell of the form DV or G x Dm , then HG*(Y ; A)is also free over H* with even-dimensional generators. In the second part, we assume that the homology HG*(Xff; A)of each of the Xffis free over H*, and argue that HG*(X; A)~=colimff2JHG*(Xff; A)is also free over H*. Since the indexing set J is an ordinal which may be larger than the ordinal of natural numbers, this sec* *ond step is actually an inductive argument in which we show that, if fi is a limit * *point of J , then HG*(Xfi; A)~=colimff ff such that !ff,fi0= !ff,flfor all fl > fi0. Deno* *te this terminal value of the dimensions associated to ff by !ff. This element !ffof RO* *(G) should be thought of as the ultimate dimension of the generators in our diagram indexed on ff. In the diagrams of interest to us, all of the dimensions !ff,fi* *are space-like. Thus, if ff is an element of J associated to a purely free generat* *or, then for fl > fi > ff, |!Gff,fi| |!Gff,fl| 0. Since the integers |!Gff,fi|dec* *rease with increasing fi and are bounded below by 0, the fi0 2 J needed for convergence mu* *st exist. If ff 2 J is associated to a projective generator, then !ff,fi= !ff,flf* *or all fl > fi > ff, and we take !ffto be !ff,fifor any fi > ff. Remark 3.5. The fifth condition in Definition 3.4(a) may seem a bit strange. To understand it better, assume that ff < fi < fl in J , that ff is associated * *to a purely free generator, and that !ff,fi6= !ff,fl. The image of the ff-indexed ge* *nerator of Cfiunder the map ~fi,fl: Cfi__//_Cflcould be a multiple of the ff-indexed ge* *nerator of Cflby an element of H* in the appropriate dimension. However, it is more lik* *ely to be a linear combination of several generators of Cfl. The generators appeari* *ng in this linear combination are likely to depend on fl. Condition (v)provides a * *finite uniform bound on the collections of the generators which appear in these linear combinations. The misbehavior of the colimit in Example 2.3 arises precisely because no su* *ch bound is available. To see this, note that the positive dimensional generators * *ap- pearing in that example are indexed on the positive integers. The mth generator appears first in HG*(Xm ; A), where it has a dimension plotting to the point (2* *, 2m). However, at the very next stage of the filtration, it shifts to a dimension plo* *tting to (0, 2m) and remains at that dimension throughout the remainder of the diagram. For n > m, this generator of HG*(Xm ; A)maps to a nontrivial linear combination of the generators of HG*(Xn ; A)plotting to the points (0, 2m) and (2, 2n). Thu* *s, the only candidate for a set Jm satisfying condition (v)in this example would be the infinite set of positive integers greater than or equal to m. The failure o* *f this set to be finite is the source of the failure of the colimit to be free. For the proof of our algebraic freeness theorem, the subsets Jffof J introdu* *ced in condition (v)of Definition 3.4(a) need not satisfy any conditions beyond tho* *se given there. However, in the application of our algebraic result to the proof * *of Theorem 2.5, it is essential that these sets have an additional property. Definition 3.6. The sets Jffof an ordinal-indexed diagram of free H*-modules with a consistent set of generators are said to be well positioned if, for each* * ff02 Jff and each fi 2 J such that fi > ff0, |!ff0,fi| |!ff,ff+1|and |!Gff0,fi| |!Gff,* *ff+1| Our freeness result for diagrams of free H*-modules is precisely what one wo* *uld expect based on Definition 3.4. Proposition 3.7. Let J be an ordinal and ~ff,fi: Cff__//_Cfibe a J -indexed * *di- agram of free H*-modules having a consistent set of generators which is converg* *ent. Then C = colimfi2JCfi is a free H*-module whose generators are indexed on J . The generator of C inde* *xed on ff 2 J is of the same type (that is; purely free or projective) as the ff-in* *dexed 3.2. COLIMITS OF DIAGRAMS OF FREE H*-MODULES 35 generators appearing in the diagram and its dimension is the ultimate dimension !ffof the ff-indexed generators of the diagram. Moreover, if ff < fi and !ff,fi* *= !ff, then the diagram !ff,fiH*= !ffH* 'fffikkkkkk SSSS'ff SSS C uukkkkk SSS))S// fi____________~fi____________C commutes. Here, ~fi: Cfi___//C and 'ff: !ffH* __//_C are the canonical map into the colimit and the inclusion of the summand of C spanned by its ff-indexed generator, respectively. Proof. If J is not a limit ordinal, then it has a maximal element ff1 . In this case, C = Cff1, and there is nothing to prove. Thus, we assume that J is a limit ordinal. For each fi 2 J , let Dfibe the summand of Cfispanned by the generators indexed on those ff < fi such that !ff,fi= !ff. If fl > fi, then con* *dition (iv)of Definition 3.4 implies that the restriction of ~fi,fl: Cfi__//_Cflto Dfi* *factors through the inclusion of Dflinto Cfl. Moreover, the resulting map Dfi___//Dflis just the inclusion of a direct summand, basically because the only generators of Cfiallowed to appear in Dfiare those which undergo no dimension shifting in the part of the diagram beyond fi. It follows trivially that D = colimfi2JDfiis a f* *ree H*-module with generators indexed on J . Moreover, the types and dimensions of these generators are exactly those asserted in the proposition for the generato* *rs of C. The inclusions Dfi Cfiinduce a monomorphism OE : D __//_C. Thus, to show that C is free and has the appropriate generators, it suffices to show that OE * *is an epimorphism. For each ff < fi in J , consider the composite 'fffi ~fi !ff,fiH*__//Cfi____//_C. Taken together, the images of all these maps generate C. Thus, it suffices to s* *how that, for each ff < fi in J , the composite above factors through OE. Select fl* * 2 J satisfying: (i)fi < fl, (ii)!ff,fl= !ff, and (iii)for each ff02 Jff, ff0< fl and !ff0,fl= !ff0. The finiteness of the set Jffensures that such a fl exists. Our restrictions on* * fl imply that, for each ff02 Jff, the generator of Cflindexed on ff0is also a generator * *of Dfl. The desired factorization then follows from the commutativity of the diagram ~~fffi,flL !ff,fiH*___//_ff0 !ff0,flH* | | 'fflffl| 'fffi|| Dfl_________//_D | |OE fflffl|~fi,fl fflffl|~fl |fflffl Cfi____________//Cfl________//_C. The commutativity of the diagram in the proposition follows easily from the fact that 'fffiand 'fffactor through Dfiand D, respectively. 36 3. AN OUTLINE OF THE PROOF OF THE MAIN FREENESS RESULT 3.3. Completing the proof of the main freeness theorem Here, Proposition 3.7 is employed to complete the proof of Theorem 2.5. As- sume that X is a generalized G-cell complex satisfying the hypotheses of the th* *eo- rem. Recall the two filtrations on X discussed in Section 1.5. In the first fil* *tration {Xn}n 0, Xn+1 is formed from Xn by attaching the collection of cells J n+1. This filtration is used in the definition of a generalized G-cell complex and in the* * state- ment of Theorem 2.5. The second filtration {Xff}ff2Jis the ö ne cell at a time" filtration indexed on an ordinal J . This second filtration is assumed to satis* *fy the condition that, if ff, fi 2 J and the cells indexed on ff and fi are in the set* *s J m and J n, respectively, then ff < fi whenever m < n. For technical reasons, an addit* *ional condition must be imposed on this filtration. Assume that ff and fi are element* *s of J whose associated cells are of the form DVffand DVfiand that these two cells l* *ie in the same set J n. We require that ff < fi if |VfGf|< |VfGi|. Since |VfGl| 0* * for all fl 2 J , it is easy enough to arrange the order of the attachment of cells in t* *he ö ne cell at a time" filtration so that this extra condition is met. As in the proof of Proposition 3.7, we may as well assume that J is a limit ordinal. There are two main obstacles to completing the proof of the theorem by applying the proposition to the J -indexed diagram {HG*(Xff; A)} of H*-modules. The first is that, if ffi is a limit ordinal in J , then we do not know that HG* **(Xffi; A) is a free H*-module. This is resolved by using the proposition, together with t* *he freeness results from Section 3.1, in a transfinite induction argument on ffi. * * The second problem is that of establishing the existence of the finite sets Jffrequ* *ired by condition (v) in Definition 3.4(a). We construct these sets as a part of our induction argument. Throughout our inductive argument on ffi, we are going to work with a number of objects which must be indexed on the elements of J . In introducing these indexed objects, we often use an index like ffi +1 when the index ffi might see* *m more natural. The source of this notational clumsiness is that J typically contains * *both successor ordinals and limit ordinals. Our inductive argument has to deal with both of these types of ordinals. There is one perfectly reasonable indexing sch* *eme which works very well for the discussion of the successor ordinals, and another completely incompatible scheme which seems quite natural for the discussion of * *the limit ordinals. Thus, we have selected a third indexing scheme which is uniform* *ly a bit clumsy rather than one of the two that works well for half of the argument but disastrously for the other half. Our inductive assumption on ffi 2 J is that, for every ~ ffi + 1, the port* *ion of the diagram {HG*(Xff; A)} of H*-modules indexed on J (~) = {ff 2 J : ff < ~} is a diagram of free H*-modules with a consistent set of even-dimensional space-li* *ke generators. In the context of this assumption, we denote the finite subsets of* * J arising in condition (v)of Definition 3.4(a) by Jff(~) rather than Jff. We assu* *me that these finite sets are well positioned in the sense of Definition 3.6 and t* *hat, for ~ < ~0 ffi+1, Jff(~) Jff(~0). Observe that, if the portion of the diagram in* *dexed on J (~) satisfies (v)for ff with respect to the finite set Jff(~), then it sat* *isfies that condition with respect to any finite set containing Jff(~). This is important b* *ecause ultimately we define Jffto be [~Jff(~). One further technical assumption is needed about the dimension !ff,ff+1of a purely free generator at the point in our diagram where it first appears. Intui* *tively, this assumption says either that no dimension shifting occurs when the cell DVf* *fis 3.3. COMPLETING THE PROOF OF THE MAIN FREENESS THEOREM 37 attached or that the dimension shifting which does occur is tied to at least on* *e cell in a lower filtration J kthan the filtration J m of DVff. Our precise assumptio* *n is that, for each ff 2 J associated to a purely free generator, there is an elemen* *t b(ff) of J , also associated to a purely free generator, satisfying the conditions: (i)|!ff,ff+1| |Vb(ff)|and |!Gff,ff+1| |VbG(ff)|, (ii)either b(ff) = ff or the cells DVb(ff)and DVffare in filtrations J kand J m, respectively, such that k < m. We refer to this condition as our bounding assumption on !ff,ff+1. The element b(ff) is ff when the attachment of the cell DVffcauses no dimension shifting. I* *n this case, !ff,ff+1= Vff, so the two inequalities in condition (i)are trivially sati* *sfied. Looking back at Example 2.3 may provide some intuition for this bounding assumption. In that example, the problem with the colimit is not our bounding assumption, which is satisfied, but rather a problem with the finiteness requir* *ement in condition (v)of Definition 3.4(a). Recall that the positive dimensional gene* *rators appearing in that example are indexed on the positive integers. For each m 1, the mth generator first appears in a dimension plotting to the point (2, 2m). F* *or m > 1, this location for the first appearance of the mthgenerator is the result* * of a sequence of dimension shifts that begins with a shift involving the first gener* *ator at (2, 2). Thus, that bottom cell is the ultimate source of all our dimension shif* *ting, and b(m) = 1 for every m 1. We begin our inductive argument by looking at the transition from an element ffi of J to its successor ffi + 1. As a part of our inductive assumption, we kn* *ow that HG*(Xffi; A)is a free H*-module. If the cell added to construct Xffi+1is of the* * form G x Dmffi, for some integer mffi, or the boundary map @ in the cell-attaching l* *ong exact sequence associated to the formation of Xffi+1is zero, then Proposition 3* *.2 applies, and indicates that HG*(Xffi+1;iA)s a free H*-module. This result also * *allows us to take !ff,ffi+1to be !ff,ffifor all ff < ffi and !ffi,ffi+1to be the dimen* *sion (mffior Vffi) of the cell used to form Xffi+1from Xffi. For each ff which is associated to a * *purely free generator and is less than ffi, we can take Jff(ffi + 2) to be Jff(ffi + 1* *). If the new generator in HG*(Xffi+1; A)is purely free, then we begin the process of constru* *cting the set Jffiby defining Jffi(ffi + 2) to be {ffi}. Given these definitions, it * *is easy to see that Proposition 3.2 implies that the J (ffi + 2)-indexed diagram of H*-modules* * is a diagram of free H*-modules with a consistent set of even-dimensional space-li* *ke generators. Our inductive assumptions imply that the finite subsets Jff(ffi + 2* *) of J (ffi + 2) are well-positioned. Note that, if the generator associated to ffi * *is purely free, then !ffi,ffi+1= Vffiso our bounding assumption for !ffi,ffi+1is satisfie* *d. This condition must hold for ff < ffi by our inductive assumptions. The case in which the cell attached to construct Xffi+1is of the form DVffia* *nd the cell-attaching boundary map is nonzero must still be considered. Here, we invoke Theorem 3.3, which asserts that HG*(Xffi+1; A)is a free H* -module, and specifi* *es the dimensions of its generators. Recall that there is a finite set F@ of purel* *y free generators of HG*(Xffi; A)which do not pass over to generators of HG*(Xffi+1; A) in the same dimension. Assume that the elements of F@ are the generators of HG*(Xffi; A)indexed on the elements ff1, ff2, . . . , ffn of J (ffi + 1). Recal* *l that these generators are in dimensions !1, !2, . . . , !n satisfying the conditions liste* *d in Theorem 3.3. Also recall that HG*(Xffi+1; A)has n+1 new generators in dimensions !01, !02, . . . , !0n+1satisfying further conditions listed in that theorem. If* * ff < ffi is not one of the ffi, take the ff-indexed generator of HG*(Xffi+1; A)to be the ob* *vious 38 3. AN OUTLINE OF THE PROOF OF THE MAIN FREENESS RESULT one associated by the theorem to the ff-indexed generator of HG*(Xffi;.A)It fol* *lows that !ff,ffi+1= !ff,ffi. Subject to minor adjustments noted below, take the gen* *erator of HG*(Xffi+1; A)indexed on ffi, for 1 i n, to be the new generator in dime* *nsion !0iso that !ffi,ffi+1= !0i. Similarly, provisionally take the ffi-indexed gene* *rator of HG*(Xffi+1; A)to be the new generator in dimension !0n+1. Since !ffi,ffi+1= !0n+1, the conditions |!ffi,ffi+1|> |!n|and |!Gffi,ffi+1|=* * |!Gn|are satisfied. But !n is !ffn,ffi, which satisfies the conditions |!ffn,ffi|= |!ffn* *,ffn+1|and |!Gffn,ffi| |!Gffn,ffn+1|. It follows that b(ffn) is an obvious choice for the* * bound b(ffi) of ffi. The only difficulty which might arise from this choice is that the lower f* *iltration condition in our bounding assumption might fail. If b(ff) 6= ff, it obviously d* *oesn't fail. In the case b(ff) = ff, we can use the technical assumption on the ö ne c* *ell at a time" filtration imposed at the beginning of this section to show that, si* *nce |VfGfi|< |!Gffi,ffi+1| |!Gffn,ffn+1|= |VfGfn|, DVffiis in a higher filtration * *J m than DVffn. It should now be easy to see that conditions (i), (ii), and (iii)of Definiti* *on 3.4(a) are satisfied for the diagram {HG*(Xff; A)} of H* -modules indexed on J (ffi + * *2). Moreover, the description of the map Ø : HG*(Xffi; A)_//HG*(Xffi+1;gA)iven in Theorem 3.3 implies that condition (iv)is also satisfied. To complete our proof that the collection {HG*(Xff; A)}ff ffi+1is a J (ffi + 2)-indexed diagram of fr* *ee H*- modules having a consistent set of space-like generators, we must construct the finite sets Jff(ffi + 2) and show that condition (v)of Definition 3.4(a) is sat* *isfied for the part of our homology diagram indexed on J (ffi + 2). The set {ffi} is a nat* *ural choice for Jffi(ffi + 2) and is obviously well positioned. If ff < ffi is associated to a purely free generator and none of the ffi are* * in Jff(ffi+1), then we can take Jff(ffi+2) to be Jff(ffi+1). By our inductive assu* *mption, this set is well positioned. Moreover, it follows easily from Theorem 3.3 that * *this set suffices to ensure that condition (v)is satisfied with respect to ff for th* *e part of the diagram indexed on the set J (ffi + 2). To define Jff(ffi + 2) for those ff such that Jff(ffi + 1) contains at least* * one of the ffi, we must examine the restriction of the map Ø : HG*(Xffi; A)//_HG*(Xffi+1; * *A) to the summand of HG*(Xffi; A)spanned by each generator indexed on one of the ffi in Jff(ffi + 1). This examination may indicate that we need to adjust the g* *ener- ators of HG*(Xffi+1; A)indexed on the ffi and ffi. It is important to note that* * these adjustments do not involve a change in the dimension. These adjustments are best described by adopting the notational convention that ffi is ffn+1. Consider the composite i 0 0 !iH* HG*(Xffi; A)Ø//_HG*(Xffi+1;ßA)//_ !iH* !i+1H* in which the first map is the inclusion of the summand of HG*(Xffi; A)spanned by the generator indexed on ffi and the last map is the projection onto the summand of HG*(Xffi+1; A)spanned by the generators indexed on ffi and ffi+1. Theorem 3.3 indicates that this composite is constructed from standard shift maps. Clearly,* * if ffi2 Jff(ffi+1), then ffi+1must be added to the set Jff(ffi+1) in the process o* *f forming Jff(ffi + 2). However, it is possible that even more indices must be added. T* *he possible adjustment in the generators of HG*(Xffi+1; A)mentioned above provides us with some control over which indices must be added. The composite !iH* HG*(Xffi; A)Ø//_HG*(Xffi+1;,A) 3.3. COMPLETING THE PROOF OF THE MAIN FREENESS THEOREM 39 which we denote by Øi, is completely determined by the image of the standard element ~ of A(G=G) = ( !iH*)!i(G=G). This image (Øi!i)(G=G)(~) must lie in a summand of HG*(Xffi+1; A)spanned by a finite number of generators. Pick a minimal set of generators whose span contains this element. Denote the indices * *of these generators by fi1, fi2, . . . , fim and the dimensions of these generator* *s by !001, !002, . . . , !00m. If we did not need to show that the set Jff(ffi + 2) is wel* *l positioned, then we could just add the indices fi1, fi2, . . . , fim into Jff(ffi + 1) in t* *he process of forming Jff(ffi + 2) whenever ffi 2 Jff(ffi + 1). Doing this for all ff and * *i would produce finite sets Jff(ffi + 2) satisfying condition (v)for the part of our ho* *mology diagram indexed on J (ffi + 2). However, in order to ensure that the set Jff(ff* *i + 2) is well positioned, we must be a bit more careful about what we add to Jff(ffi * *+ 1). From the description of H* given in Proposition 1.7, it is easy to see that,* * for each j, the dimension !00jmust satisfy one of the following three conditions: (i)|!00j| |!i|and |(!00j)G| |!Gi|, (ii)|!00j|= |!i|and |(!00j)G|> |!Gi|, or (iii)|!00j|< |!i|and |(!00j)G|= |!Gi|. Note that, since ffi is assumed to be in the well-positioned set Jff(ffi + 1), * *!i must satisfy the conditions |!i| |!ff,ff+1|and|!Gi| |!Gff,ff+1|. Thus, if !00jsatisfies the first of the three conditions above, then adding the* * associ- ated index to Jff(ffi+1) will not prevent the set Jff(ffi+2) from being well po* *sitioned. However, if !00jsatisfies either of the other two conditions, then we cannot af* *ford to add the associated index to Jff(ffi +1). At this point, it becomes important th* *at the composite ßiO Øiis constructed from standard shift maps. This implies that, if * *!00j satisfies the second of the conditions above, then by adding an appropriate mul* *ti- ple of the generator of HG*(Xffi+1; A)indexed on fij to the generator indexed o* *n ffi, we can eliminate the need to include the fij-indexed generator in the list of t* *hose required to span the minimal summand containing (Øi!i)(G=G)(~). The desired multiple is, of course, obtained by multiplying by some element of H!0i-!00j(G=* *G). Similarly, if !00jsatisfies the third condition, then the generator of HG*(Xffi* *+1; A) indexed on ffi+1can be adjusted by adding a multiple of the fij-indexed generat* *or to eliminate the need for that fij-indexed generator in the spanning set for th* *is minimal summand. The one difficulty which might arise in this process comes from the fact tha* *t the generator of HG*(Xffi+1;iA)ndexed on ffi+1must be adjusted to control the spann* *ing sets for the images of both the ffi- and ffi+1-indexed generators of HG*(Xffi;.* *A) However, the stairstep arrangement of the generators of HG*(Xffi; A)indexed on * *F@ ensures that the adjustments made for each of these two generators of HG*(Xffi;* * A) are completely invisible to the other generator. Thus, the desired adjustment * *to the basis for the free H*-module HG*(Xffi+1; A)can be made. This ensures that we need not add the index fij to Jff(ffi + 1) unless !00jsatisfies the first of ou* *r three conditions. It follows that there is a finite well positioned set Jff(ffi + 2) * *satisfying condition (v)for the part of our homology diagram indexed on J (ffi + 2). This completes the part of our inductive argument dealing with the transition from f* *fi to its successor ffi + 1. 40 3. AN OUTLINE OF THE PROOF OF THE MAIN FREENESS RESULT Now we must verify our induction assumptions for a limit ordinal ffi of J . * *The first step in verifying our assumptions for ffi is showing that the part of our* * homology diagram indexed on J (ffi) has a consistent set of even-dimensional space-like * *genera- tors. Given this, Proposition 3.7 indicates that HG*(Xffi; A)~=colimff |!ff0,fi|and|VfGi|< |!Gff0,fi|. Since Jff(fi + 1) is well positioned and contains ff0, the dimension !ff0,fimus* *t also satisfy the conditions |!ff0,fi| |!ff,ff+1|and|!Gff0,fi| |!Gff,ff+1|. Our bounding assumption for !ff,ff+1provides an element b(ff) of J associated to a purely free generator such that |!ff,ff+1| |Vb(ff)|and|!Gff,ff+1| |VbG(ff)|. Combining these inequalities, we see that |Vfi|> |Vb(ff)|and|VfGi|< |VbG(ff)|. We would like to use the finiteness assumption in the hypotheses of Theorem 2.5 to argue that there are only finitely many fi for which these last two inequali* *ties hold. This would imply that, in the formation of Jff(ffi), there are only finit* *ely many times when we can add elements. Since only finitely many elements can be added whenever elements are added, it would follow that Jff(ffi) is finite. To invoke the finiteness assumption in Theorem 2.5, we must show that the the filtration J kof DVb(ff)is lower than the filtration J nof DVfi. Note that * *the 3.3. COMPLETING THE PROOF OF THE MAIN FREENESS THEOREM 41 filtration J nof DVfiis at least as high as that of DVffsince DVfiis added after DVff. Moreover, by our bounding assumption, either the filtration J kof DVb(ff) is lower than the filtration J m of DVffor b(ff) = ff. Thus, unless b(ff) = ff,* * the required filtration condition holds. If b(ff) = ff, then |Vfi|> |Vff|and |VfGi|< |VfGf|. However, we ordered the cells of X in such a way that this condition cannot hold if DVfiand DVffare in the same filtration. Thus, even in this case, the filtrat* *ion J kof DVb(ff)= DVffis lower than the filtration of DVfi. This completes our proof that the portion of our homology diagram indexed on J (ffi) has a consistent set of even-dimensional space-like generators. From* * this, we conclude that HG*(Xffi; A)is a free H*-module with even-dimensional space-li* *ke generators. It follows easily from Proposition 3.7 that the portion of our homo* *logy diagram indexed on J (ffi+1) satisfies conditions (i)through (iv)of Definition * *3.4(a). By taking Jff(ffi + 1) to be Jff(ffi) for each ff associated to a purely free g* *enerator of HG*(Xffi;,A)it is easy to see that condition (v)is also satisfied. The set Jff(* *ffi + 1) is, of course, well positioned since Jff(ffi) is. Moreover, our bounding assump* *tion is satisfied for each ff indexing a purely free generator of HG*(Xffi;.A)Thus, * *our inductive assumptions are satisfied for the limit ordinal ffi. The last step in the proof of Theorem 2.5 is showing that our entire homology diagram has a consistent set of even-dimensional space-like generators. Conditi* *ons (i)through (iv)of Definition 3.4(a) are obviously satisfied since each instance* * of them only refers to a portion of the diagram indexed on some subset J (ffi) of * *J . For condition (v), we take Jffto be [fiJff(fi), where fi runs over the elements* * of J larger than ff. As in the part of our inductive argument dealing with a limit ordinal ffi of J , if we can show that Jffis finite, it then follows that condi* *tion (v) is satisfied. The argument for the finiteness of Jffis essentially identical t* *o the one given for a limit ordinal ffi, and so is not repeated. Proposition 3.7 can * *now be invoked to complete the proof of Theorem 2.5. CHAPTER 4 Proving the single-cell freeness results Throughout this chapter, B is assumed to be a G-space whose homology is free over H* with even-dimensional space-like generators. We also assume that t* *he G-space Y is formed from B by adding a single even-dimensional cell of the form DV or G x Dm . Associated to this attachment we have a homology cell-attaching long exact sequence of the form . ._._//HG*(B; A)Ø_//_HG*(Y ;_A)_//_eHG*(SV_;@A)//_HG*-1(B;/A)/_. . . or the form . ._._//HG*(B; A)Ø_//HG*(Y ;_A)_//_eHG*(G+ ^ Sm_;@A)//_HG*-1(B;_A)//_... . Our goal is to prove Proposition 3.2 and Theorem 3.3. The proposition follows trivially from the following two results. Lemma 4.1. Assume that the homology HG*(B; A)of B is free over H* with even-dimensional space-like generators. If the cell attached to B is of the fo* *rm G x Dm , then the boundary map @ in the cell-attaching long exact sequence is z* *ero. Lemma 4.2. If the boundary map @ in the cell-attaching long exact sequence is zero, then either HG*(Y ; A)~=HG*(B; A) eHG*(SV ; A) or HG*(Y ; A)~=HG*(B; A) eHG*(G+ ^ Sm ;,A) depending on which type of cell is added to B in the formation of Y . Moreover, under this isomorphism, the natural map Ø : HG*(B; A)__//_HG*(Y ;iA)s identified with the inclusion of HG*(B; A)into the direct sum as the first summand. Thus, * *if HG*(B; A)is free over H*, then HG*(Y ; A)is free over H* with generators consis* *ting of the generators of HG*(B; A)and one additional generator. The first of these lemmas follows directly from Lemma 1.12(b), which indicat* *es that there can be no nontrivial maps from eHG*(G+ ^ Sm ; A)to HG*-1(B; A). The second follows from the projectivity of the H*-modules eHG*(SV ; A)~= V H* and eHG*(G+ ^ Sm ; A)~= m (H* )G=e. The remainder of this chapter, and both of the next two chapters, are devoted to the proof of Theorem 3.3. Because of the length of this proof, the next sect* *ion provides a quick overview of the argument. Modulo the proofs of some key techni* *cal results, the details of that argument are then presented in the remaining three sections of this chapter. The proofs of those technical results are rather leng* *thy, and are therefore given separately in the next two chapters. 42 4.1. A PROOF OVERVIEW FOR THE DIMENSION-SHIFTING THEOREM 43 4.1.A proof overview for the dimension-shifting theorem (Theorem 3.3) For the remainder of this chapter, we assume that the cell attached to the G- space B to form the G-space Y is of the form DV and that the boundary map @ in the associated cell-attaching long exact sequence is nonzero. In this context, * *some dimension shifting must occur in the transition from HG*(B; A)to HG*(Y ; A). The role of the set F@ appearing in Theorem 3.3 is to keep track of that shifting. * *Let J be the summand of HG*(B; A)spanned by the generators in F@, and Z be the summand spanned by all the other generators (both projective and purely free) so that HG*(B; A)~=J Z. The set F@ can be chosen to ensure that the composite Z HG*(B; A)_Ø_//_HG*(Y ; A) is a monomorphism. Define the quotient Q of HG*(Y ; A)by the short exact sequence 0 __//_Z __//_HG*(Y ;_A)ß//_Q __//_0. An appropriate choice of the set F@ also allows us to construct a long exact se* *quence 0 _0 @0 . ._._//J _Ø__//Q ____//eHG*(SV ;_A)//_ J __//_. . . for Q from the original cell-attaching long exact sequence for HG*(Y ; A). This* * new long exact sequence is essentially identical to the cell-attaching long exact s* *equence associated to the special case discussed in Example 2.2. This new sequence has * *the advantage of being considerably simpler than the one from which it is construct* *ed _ enough so that we can actually compute Q in some critical dimensions. Observe that, if Q is a free H*-module, then its defining short exact sequence splits, * *yielding an isomorphism HG*(Y ; A)~=Z Q. From this, the freeness of HG*(Y ; A)follows immediately. By looking a bit more carefully at this isomorphism, we can also verify the claims of the theorem abo* *ut the natural map Ø : HG*(B; A)__//_HG*(Y ;.A) We have now reduced the proof of Theorem 3.3 to showing that Q is free over H* on an appropriate set of generators. Recall that the generators of HG*(B; A)* *in F@ should be in dimensions !1, !2, . . . , !n satisfying certain restrictions g* *iven in Theorem 3.3, and the generators of Q ought to be in dimensions !01, !02, . . . * *, !0n+1 satisfying further restrictions given in that theorem. It follows easily from t* *he values of H* given in Proposition 1.7 that Q!0ishould be isomorphic to A for 1 i n* *+1. The first step in showing that Q is free is using the long exact sequence above* * to verify that, if the dimensions !0iare appropriately chosen, then Q is isomorphic to A in these dimensions. This is a nontrivial computation, the details of whi* *ch are summarized in Section 4.4 and then presented in Chapter 5. Computing Q in these dimensions allowsLus to construct0a map ` : J0___//Q comparing Q with a free H*-module J0 = 1 i n+1 !iH* having the appropriate generators. To show that ` is an isomorphism, we wish to insert J0 into a long exact sequence comparable to our long exact sequence for Q. This can be accomplished by lifting the map Ø0 : J __//_Q through the map ` : J0___//Q. Constructing the 44 4. PROVING THE SINGLE-CELL FREENESS RESULTS lifting ~Ø: J __//_J0which makes the diagram "J0?? "" ~Ø"""`|| "" fflffl| J _Ø0_//_Q commute requires the computation of the values of Q in some additional dimensio* *ns. These computations are also addressed in Section 4.4 and Chapter 5. The lifting ~Øallows us to construct the commuting diagram _~ ~@ . ._.___//J_~Ø_//J0___//_eHG*(SV ;_A)_// J____//. . . =| |` = |= fflffl|fflffl| fflffl| fflffl| . ._.___//J_Ø0_//Q__0_//_eHG*(SV ;_A)@0//__J__//. . . in which ~_is defined to be _0O `. If we knew that the top row of this diagram was a long exact sequence, it would follow immediately that ` is an isomorphism. Fortunately, it is possible to give fairly simple conditions for the exactness * *of a sequence of free H* -modules of this form. These exactness criteria are present* *ed in Section 4.3 and proven in Chapter 6. The precise existence results for the m* *aps ` and ~Østated in Section 4.4 include the information needed to show that the t* *op row in the diagram above satisfies these exactness conditions. The next three sections are devoted to filling in the details of this quick * *sketch of the proof of Theorem 3.3. The first of these sections discusses the selectio* *n of the subset F@ of F and the construction of our long exact sequence characterizi* *ng the quotient Q of HG*(Y ; A). The second of these is devoted to the presentation of our exactness criteria for sequences of free H*-modules like the top row of * *the last diagram above. This is somewhat out of order with regard to the sketch giv* *en above. However, being aware of the precise criteria for exactness makes it easi* *er to appreciate the detailed results about the values of Q and behavior of the maps ` and Ø presented in the third section. That third section also contains a wrap-u* *p of the proof of Theorem 3.3. 4.2. Simplifying the cell-attaching long exact sequence In this section, we retain the assumptions about B and Y made in the previ- ous section. As we noted there, the assumption that the cell-attaching boundary map @ is nonzero forces some dimension shifting in the transition from HG*(B; A) to HG*(Y ; A). This shifting is extremely hard to understand if the generators* * of HG*(B; A)hit by the map @ do not plot in a simple stairstep pattern like that in Figure 2.1. Fortunately, a rather minimal adjustment of our set of generators f* *or HG*(B; A)ensures that the new generators hit by the map @ do plot in a simple stairstep pattern. Fundamentally, the function of the set F@ in Theorem 3.3 is * *to keep track of this change of basis for HG*(B; A). Once this change of basis has* * been made, it is easy to see that the composite Z HG*(B; A)_Ø_//_HG*(Y ; A) is a monomorphism and to construct our long exact sequence for the quotient Q of HG*(Y ; A). 4.2. SIMPLIFYING THE CELL-ATTACHING LONG EXACT SEQUENCE 45 The desired change of basis is best understood by looking at the following example, which illustrates how that change of basis is accomplished in the simp* *lest possible cases. Example 4.3. Assume that the boundary map @ : eHG*(SV ; A)//_HG*-1(B; A) factors through the summand of HG*(B; A)spanned by two generators in dimensions !1 and !2 satisfying |!1| |!2|< |V | and |!G1| |!G2|> |V G|. Since HeG*(SV ; A)is a free H* -module on one purely free generator in dimension V , the map @ is completely determined by its behavior in dimension V . The two generators in dimensions !1 and !2 each contribute a copy of to HGV -1(B; * *A), so the map @V has the form A _@V_//_ HGV -1(B; A), and is completely determined by the image of the standard generator ~ 2 A(G=G) of A. Let @V (~) = (x, y) 2 ( )(G=G), and assume that both x and y are nonzero. We want to define : HG*(B; A)__//_HG*(B; A)so that V -1O @V (~) = (x, 0); that is, so that pulls the boundary map off of the generator in dimen* *sion !2. Clearly, we want to define to be the identity on all the generators other tha* *n our two special ones, and to define it on those two generators so that, in HGV -1(B* *; A), it takes (a, b) 2 ( )(G=G) to (a, b-x-1ya) 2 ( )(G=G). This formula suggests that should also be the identity on the generator in dimensi* *on !2, and should take the generator in dimension !1 to some linear combination of itself and the generator in dimension !2. The generator in dimension !1 contributes a copy of A to HG!1(B; A). Denote the standard generating element of this copy of A by ~1. The generator in dimen* *sion !2 contributes one of the Mackey functors A[d], , R, or to HG!1(B; A), depending on the relative positions of !1 and !2 in the usual plot of elements * *of RO(G). This contribution contains a generating element of one of the forms ~2, * *ffl, ,, or ffl,. Define on the generator in dimension !1 by 8 >>>~1 + c~2if|!1|= |!2|and |!G1|= |!G2|, <~ + cffl if|! |< |! |and |!G |= |!G,| (~1) = > 1 1 2 1G 2G >>:~1 + c, if|!1|= |!2|and |!1 |> |!2,| ~1 + cffl,if|!1|< |!2|and |!G1|> |!G2|. Here, c is an integer which can be selected to ensure that behaves as desired* * in dimension V - 1. Obviously, is an isomorphism of H*-modules. This example shows that we can push @ off of one generator onto any other generator which plots to the same point or to a point below and/or to the right. Essentially, by pushing @ off of as many generators as possible, we can push it onto a finite set F@ of purely free generators of HG*(B; A)plotting in stairstep pattern. Our precise definition of F@ is easily understood when viewed in terms* * of this pushing off process. Definition 4.4. Let B be a G-space whose homology HG*(B; A)is free over H* with even-dimensional space-like generators. Assume that the G-space Y is 46 4. PROVING THE SINGLE-CELL FREENESS RESULTS formed from B by adding a single even-dimensional cell of the form DV , and that the boundary map @ : eHG*(SV ; A)//_HG*-1(B; A) in the associated cell-attaching long exact sequence is nonzero. This map is co* *m- pletely determined by @V , which has the form @V: A __//_ , where the direct sum is indexed on those purely free generators of HG*(B; A)lyi* *ng in a dimension ! satisfying |! |< |V | and |!G |> |V G|. Moreover, if ~ 2 A(G=G) is the standard generator of A, then @V is completely determined by @V (~), which has only finitely many nonzero coordinates in the direct sum ( )(G=G). Let F be the set of purely free generators of HG*(B; * *A) and F1 be the subset of F consisting of those generators corresponding to the nonzero coordinates of @V (~). Since the generators of HG*(B; A)are space-like, the dimension ! of any one of them satisfies 0 |!G | |! |. Thus, there is a minimum value for |! |among the dimensions ! of the generators in F1. Among all the generators in F1 with this minimum value for |!,|select one for which |* *!G | is maximal. This generator is the first element of F@; denote its dimension by * *!1. Note that |!1|< |V |and |V G|< |!G1|since @V is nonzero on the selected generat* *or. Now assume that the first i elements of F@ have been selected and that their dimensions !1, !2, . . . , !i satisfy |!1|< |!2|< . .<.|!i|< |V | and |V G|< |!G1|< |!G2|< . .<.|!Gi|. Let Fi+1 be the subset of F1 consisting of those generators having a dimension ! satisfying |!G |> |!Gi|. Our selection process will ensure that the dimension !* * of any generator in Fi+1also satisfies |! |> |!i|. If the set Fi+1is nonempty, the* *re is a minimum value for |! |among the dimensions ! of the generators in Fi+1. Among all the generators in Fi+1 with this minimum value for |!,|select one for which |!G |is maximal. This generator is the (i + 1)stelement of F@; denote its dimen* *sion by !i+1. Since the sets Fi are finite and decreasing in size, this inductive pr* *ocess eventually stops at an integer n for which the set Fn+1 is empty. The construction of the desired change of basis isomorphism for HG*(B; A) is now an obvious generalization of the process presented in Example 4.3. Every generator of HG*(B; A)not in F@ but hit by the boundary map @ lies above and/or to the left of a generator in F@. Thus, we can push the boundary map off of the generators not in F@. Proposition 4.5. Let B be a G-space whose homology HG*(B; A)is free over H* with even-dimensional space-like generators. Assume that the G-space Y is formed from B by adding a single even-dimensional cell of the form DV . Then there is a H*-module isomorphism : HG*(B; A)__//_HG*(B; A)such that: (i)the composite HeG*(SV ; A)@_//HG*-1(B; A)_//_HG*-1(B; A) 4.2. SIMPLIFYING THE CELL-ATTACHING LONG EXACT SEQUENCE 47 factors through the summand of HG*(B; A)spanned by the generators of HG*(B; A) in F@. (ii)the map O @ hits every generator in F@ in the sense that the composi* *te of this map with the projection of HG*(B; A)onto the summand generated by any element of F@ is nonzero. (iii) is the identity map on all the projective generators of HG*(B; A)and on those purely free generators of HG*(B; A)not in F@. Recall that J is the summand of HG*(B; A)spanned by the generators in F@, and Z is the summand spanned by all the other generators. Thus, HG*(B; A) decomposes as the direct sum J Z. Using , we can now write our cell-attaching long exact sequence in the form (@0,0) . ._.//_J Z _~Ø//_HG*(Y ;_A)_//_eHG*(SV_;_A)_//_ (J Z) __//_... . Here, ~Øis the composite -1 O Ø, and @0 is the composite O @ regarded as a map into J. This sequence is cluttered by the summand Z of HG*(B; A)and its image in HG*(Y ; A). The function of the quotient Q of HG*(Y ; A)introduced in the previous section is to eliminate this clutter. Note that, since the imag* *e of the adjusted boundary map @0 lies entirely inside the summand J, the composite Z HG*(B; A)_Ø_//_HG*(Y ; A)must be a monomorphism. Recall that Q is just the quotient of HG*(Y ; A)obtained by killing the image of this composite. The long exact sequence for Q introduced in Section 4.1 is a special case of a gene* *ral algebraic construction which reappears several times in the proof of Theorem 3.* *3. Thus, we include the following lemma describing that construction. Lemma 4.6. Let (@0,0) . ._._//J Z _~Ø//_M ___//_N ______//_ (J Z) __//_. . . be a long exact sequence of H*-modules. Define the H*-module Q by the short exa* *ct sequence 0 __//_Z __//_M _ß_//_Q __//_0. Observe that the map _ factors through the projection ß : M __//_Q to provide a map _0: Q __//_N. Also, let Ø0: J __//_Q be the composite of ß and the restrict* *ion of ~Øto J. Then 0 _0 @0 . ._._//J _Ø__//Q ____//N ____// J __//_. . . is a long exact sequence of H*-modules. Proof. This follows easily by chasing the diagram 0 0 fflffl|= fflffl| Z _______//Z '2fflffl|| fflffl|| (@0,0) . ._.___//J Z_~Ø_//M_____//N____// (J Z)____//. . . ß1fflffl|| fßflffl||=fflffl|| ß1fflffl|| . ._._____//J__Ø0__//Q__0__//N___0___// J_______//_. . . | _ @ fflffl| fflffl| 0 0 48 4. PROVING THE SINGLE-CELL FREENESS RESULTS in which the maps in the left column are the obvious inclusion and projection. By taking M to be HG*(Y ; A)and N to be eHG*(SV ; A)in the lemma above, we obtain our fundamental long exact sequence 0 _0 @0 . ._._//J _Ø__//Q ____//eHG*(SV ;_A)//_ J __//_. . . for Q. Hereafter, the free H*-module eHG*(SV ; A)is usually denoted N for notat* *ional compactness. Recall that it has a single purely free generator in dimension V . 4.3. Characterizing dimension-shifting long exact sequences In this section, we assume that !1, !2, . . . , !n, !01, !02, . . . , !0n+1,* * and V are even-dimensional space-like elements of RO(G) satisfying: |!1|< |!2|< . .<.|!n|< |V |= |!0n+1| |(!01)G|= |V G|< |!G1|< |!G2|< . .<.|!Gn| |!0i|= |!i|, fori n; and |(!0i)G|= |!Gi-1|, fori 2. We also assume that J is a free H*-module with purely free generators in dimens* *ions !1, !2, . . . , !n, that J0is a free H*-module with purely free generators in d* *imensions !01, !02, . . . , !0n+1, and that N is a free H*-module having one purely free * *generator in dimension V . Our goal here is to characterize the maps ~Ø: J __//_J0, ~_: J* *0___//N, and ~@: N __//_ Jfor which the sequence ~_ ~@ . ._.//_J _~Ø//_J0____//N ___//_ J __//_. . . is a long exact sequence. We refer to such a long exact sequence as a dimension- shifting long exact sequence because of the close connection between such seque* *nces and the cell-attaching long exact sequences arising in situations like Example * *2.2. Recall the notions of a standard shift map and of a map constructed from standa* *rd shift maps from Definition 1.17. Proposition 4.7. The sequence ~_ ~@ . ._.//_J _~Ø//_J0____//N ___//_ J __//_. . . is a long exact sequence if and only if the following four conditions are satis* *fied: (i)Ø~and ~_are constructed from standard shift maps, (ii)each of the components ~@i: N __//_ !i+1H*of the boundary map ~@is nonzero, (iii)for 1 i n, the composite Ø~!i 0 _~!i J!i_____//J!i_____//N!i is zero, and (iv)the composite ~@V ( ~Ø)V NV ____//_( J)V _______//( J0)V is zero. 4.3. CHARACTERIZING DIMENSION-SHIFTING LONG EXACT SEQUENCES 49 This result is proven in Chapter 6. However, by examining the putative long exact sequence in the dimensions of the generators of J, J0, and N, it is relat* *ively easy to verify that all four conditions are necessary. Remark 4.8. Since J is a free H*-module with one generator in dimension !i, for 1 i n, condition (iii)is equivalent to the assertion that ~_O ~Ø= 0. Si* *milarly, because N has one generator in dimension V , condition (iv)is equivalent to the assertion that ~ØO ~@= 0. It is natural to wonder how hard it is to find maps ~Ø, ~_, and ~@satisfying* * the conditions in this proposition. In the remainder of this section, we show that * *there is only one obstruction to their existence. The first two of the four condition* *s in the proposition are quite straightforward, and there are obviously maps ~Ø, ~_,* * and ~@satisfying them. The last two conditions are actually much simplier and more easily satisfied than their appearance suggests. Note that the map ~Øis complet* *ely determined by its behavior in the dimensions !i of the generators of J. The only two generators of J0 which make nonzero contributions to J0 in dimension !i are the two in dimensions !0iand !0i+1. Thus, if j 6= i, i + 1, then the component * *~Øi,j of ~Øassociated to the generators of J and J0 in dimensions !iand !0j, respecti* *vely, is zero. The composite in condition (iii)therefore has the form ~Ø!i ~_!i A ____//_R _____// with the R coming from the generator of J0 in dimension !i and the coming from the generator in dimension !0i+1. Since ~Øand ~_must be constructed from standard shift maps by condition (i)of the proposition, the composite ~_!iO ~Ø!* *iis easily computing by using the multiplicative structure of H* (see Proposition 1* *.10). The computation of ~_!iO ~Ø!i, which we carry out in Section 6.5, reveals th* *at, unless a nontrivial constraint on the dimensions of the generators of J, J0, and N is satisfied, there are no maps ~Øand ~_satisfying conditions (i)and (iii). * *To understand this constraint, recall the function d : RO0(G) __//_Z introduced in Definition 1.4, and note that X X V + !i- !0j 1 i n 1 j n+1 is in RO0(G). Proposition 4.9. There exist maps ~Ø: J __//_J0and ~_: J0___//N, constructed from standard shift maps, such that ~_O ~Ø= 0 if and only if d(V +P !i-P !0j) 1 mod p. It is easier to obtain maps satisfying condition (iv)of Proposition 4.7. Lemma 4.10. Let ~Ø: J __//_J0be a map constructed from standard shift maps. Then there exist nonzero maps ~@: N __//_ Jsuch that the composite ~@ ~Ø 0 N ___//_ J ____//_ J is zero. Moreover, each component of any such map ~@is nonzero. 50 4. PROVING THE SINGLE-CELL FREENESS RESULTS Proof. The composite in condition (iv)of Proposition 4.7 has the form ~@V M ( ~Ø)V M A ____//_ _______// . 1 i n 1to the left direct sum above, and each of the generators of J0 except those in dimensions !01and !0n+1contributes* * a copy of to the right direct sum. To verify that this composite is zero, it* * suffices to check that its composite with the projection onto each of the summands of J0V vanishes. The composite of ( ~Ø)V O ~@Vwith the projection onto the jth-summand has the form A __//_ __//_. Here, the two copies of in the middle come from the generators of J in dimensions !j-1 and !j. Both components of the second map are nonzero since ~Ø is constructed from standard shift maps. It follows that the map ( ~Ø)V is surj* *ective, and so has kernel . Moreover, if x is a nonzero element of this kernel, th* *en all n of its coordinates are nonzero. By Lemma 1.12(a), there is a one-to-one correspondence between such nonzero elements and maps ~@: N __//_ Jsuch that each component of ~@is nonzero and ( ~Ø)V O ~@V= 0. Combining this lemma with Propositions 4.7 and 4.9 yields: Corollary 4.11. There are maps ~Ø, ~_, and ~@for which ~_ ~@ . ._.//_J _~Ø//_J0____//N ___//_ J __//_. . . is a long exact sequence if and only if d(V +P !i-P !0j) 1 mod p. 4.4. Constructing the comparison dimension-shifting sequence We return now to the assumptions about B, Y , and @ stated at the beginning of Section 4.1. In Section 4.2, the proof of Theorem 3.3 is reduced to analyzin* *g a long exact sequence of the form 0 _0 @0 . ._._//J _Ø__//Q ____//N ____// J __//_. . . (4.1) in which N = eHG*(SV ;,A)J is the summand of HG*(B; A)spanned by the gener- ators in the finite set F@, and Q is the quotient of HG*(Y ; A)by the image of * *the summand Z of HG*(B; A)spanned by the generators not in F@. Recall that the generators of HG*(B; A)in F@ lie in dimensions !1, !2, . . . , !n satisfying |!1|< |!2|< . .<.|!n|< |V | and |V G|< |!G1|< |!G2|< . .<.|!Gn|. To complete the proof of Theorem 3.3, we must show that Q is a free H*-module having n + 1 generators in dimensions !01, !02, . . . , !0n+1satisfying |!0i|= |!i|, fori n, |!0n+1|= |V,| |(!01)G|= |V G|, 4.4. CONSTRUCTING THE COMPARISON DIMENSION-SHIFTING SEQUENCE 51 and |(!0i)G|= |!Gi-1|, fori 2. Our first task is to verify that there are dimensions !0isatisfying these co* *nditions in which Q is sufficiently well-behaved to permitPthe construction0an appropria* *te map comparing it to the free H* -module J0 = n+1j=1 !iH*. In order to show that this map is an isomorphism, we must then describe the behavior of long exa* *ct sequence (4.1) in the dimensions of the generators of J, J0, and N precisely en* *ough to permit the construction of a comparison sequence whose exactness can be prov* *en via Proposition 4.7. These tasks are carried out in the next three propositions and their corollaries. The proofs of two of those propositions are lengthy and * *are therefore presented separately in Chapter 5. Proposition 4.12. There exist space-like even-dimensional elements !01, !02, . . . , !0n+1of RO(G) satisfying the equations above such that, for 1 i n +* * 1, Q!0i~=A. Moreover, if 1 < i < n + 1, then long exact sequence (4.1) reduces to * *the short exact sequence Ø0!0i _0!0i 0 __//_ L ____//_A ____//__//_0 in dimension !0i. For i = 1 or n + 1, this long exact sequence reduces to the s* *hort exact sequences Ø0!01 _0!01 0 __//_L _____//A _____//_//_0 and Ø0!0n+1 _0!0n+1 0 __//______//A ______//_R __//_0, respectively, in dimension !0i. In these short exact sequences, the copies of <* *Z>and L in the left-hand term are contributed by the generators of J in dimensions !i* *-1 and !i, respectively. L 0 Let J0 = 1 i n+1 !iH* be a free H*-module on generators in the dimensions !0iprovided by this proposition. We wish to construct a map ` : J0___//Q making a diagram of the form _~ ~@ . ._.___//J_~Ø_//J0___//_N____// J____//_. . . f=flffl||`fflffl|=fflffl||=fflffl|| (4.2) . ._.___//J_Ø0_//Q__0_//_N_@0_// J____//_. . . commute. Note that the bottom row of this diagram is long exact sequence (4.1). To construct `, it suffices to specify that map on each of the generators of J0* *. It is easy to see that J0, like Q, is isomorphic to A in the dimensions of those gene* *rators. Thus, we can define the desired comparision map ` by taking it to be the identi* *ty map of A in dimension !0i, for 1 i n + 1. To complete this diagram, we must select the maps ~Øand ~_. Since we wish to employ Proposition 4.7 to establish the exactness of the top row of this diagra* *m, these maps must be constructed from standard shift maps. The map ~_ought to be the composite _0O `. The short exact sequences in Proposition 4.12 imply tha* *t, if ~_is defined in this way, then it has the proper form. 52 4. PROVING THE SINGLE-CELL FREENESS RESULTS Corollary 4.13. The map ~_= _0O` is constructed from standard shift maps. The map ~Ø: J __//_J0is obtained by lifting Ø0 along `. To show that this lifting exists, we must analyze long exact sequence (4.1) in the dimensions of * *the generators of J. Proposition 4.14. For 1 i n, long exact sequence (4.1) reduces to the short exact sequence Ø0!i _0!i 0 __//_A ____//_R _____//_//_0 in dimension !i. Moreover, in these dimensions, the map ` : J0___//Q is an iso- ~= morphism `!i: R ___//_R . The copies of R and in the domain of `!i are contributed by the generators of J0 in dimensions !0iand !0i+1, respect* *ively. The desired lifting ~Ø: J __//_J0can be defined by assigning it the value `-* *1!iOØ0!i in dimension !i for 1 i n. Lemma 12.1, which characterizes short exact sequences of the form appearing in the proposition, indicates that Ø0!itakes the generator ~ of A(G=G) to ( ,, ffl), where , and ffl are the standard generator* *s of R(G=G) and (G=G), respectively. These observations suffice for the proof of * *the following corollary: Corollary 4.15. There is a map ~Ø: J __//_J0constructed from standard shift maps which makes the diagram "J0?? "" ~Ø"""`|| "" fflffl| J _Ø0_//_Q commute. In order to show that the top row of diagram (4.2) satisfies condition (iv)of Proposition 4.7, we need to understand the behavior of the map ` in dimension V - 1. Proposition 4.16. In dimension V , the diagram zJ0== zz ~Øzzz ||` zz fflffl| . ._.___//N_@0_//_ J_Ø0_// Q___0_// N_____//. . . has the form n-177 ppp ~ØVp1pppp `V|-1| ppp fflffl| . ._._____//A__@0___//n_0___//_n-1_______//0. V ØV -1 Thus, `V -1is an isomorphism. 4.4. CONSTRUCTING THE COMPARISON DIMENSION-SHIFTING SEQUENCE 53 Proof. The Mackey functors NV , ( J)V , ( J0)V , and ( N)V are easily com- puted from the information about H* contained in Proposition 1.7. The value of ( Q)V and the surjectivity of `V -1then follow from the exactness of the bottom row. Being a surjective map of finite dimensional vector spaces over Z=p of the* * same dimension, `V -1(G=G) must be an isomorphism. Since the range and domain of `V -1vanish at G=e, this completes the proof. Proof of Theorem 3.3. Proposition 4.12 enables us to construct a map ` comparing Q to the free H* -module J0 to which it should be isomorphic. That proposition and Corollary 4.15 allow us to construct the maps ~Øand ~_which make diagram (4.2) commute. We wish to use Proposition 4.7 to show that the top row of this diagram is a long exact sequence. Corollaries 4.13 and 4.15 indicate t* *hat condition (i)is satisfied. Assertion (ii)in Proposition 4.5 indicates that cond* *ition (ii)of Proposition 4.7 is satisfied. Condition (iii)follows immediately from t* *he exactness of the bottom row of our diagram, and condition (iv)follows from that exactness and Proposition 4.16. The exactness of the top row clearly implies th* *at ` is an isomorphism so that Q is a free H*-module with generators in the appropri* *ate dimensions. Since Q is a free H*-module, its defining short exact sequence 0 __//_Z __//_HG*(Y ;_A)ß//_Q __//_0. splits, giving an isomorphism HG*(Y ; A)~=Z Q. This implies that HG*(Y ; A)is a free H* -module with generators in the speci- fied dimensions. The assertion of the theorem about the behavior of the map Ø : HG*(B; A)__//_HG*(Y ;oA)n the generators of HG*(B; A)not in F@ follows from the fact that the isomorphism used to establish the freeness of HG*(Y ; A)is de* *rived from the inclusion of Z into HG*(Y ; A). To verify the claim of the theorem about the behavior of the map Ø on the generators of HG*(B; A)in F@, we would like to use the diagram Ø HG*(B; A)_____//HG*(Y ; A) ß0|| ß|| fflffl|Ø0 fflffl| J ___________//_Q in which ß0is the projection onto the summand J of HG*(B; A). However, it is not entirely obvious that this diagram commutes. If the map Ø were replaced by the map ~Ø= -1 O Ø, then the resulting diagram would certainly commute since it is a part of the appropriate special case of the diagram used to prove Lemma 4.* *6. Observe that the difference 1 - -1 between the identity map of HG*(B; A)and -1 factors through Z, basically because the difference between the identity and -1 arises from certain elements of Z which are used to adjust the generators of HG*(B; A)indexed on F@. It follows that ß OØ~= ß OØ, so the desired diagram doe* *s, in fact, commute. The claim about the behavior of the map Ø on the generators in F@ can now be checked by examining this diagram in dimension !i, for 1 i n, and applying Proposition 4.14. CHAPTER 5 Computing HG*(B [ DV ; A) in the key dimensions Throughout this chapter, B is a G-space whose homology HG*(B; A)is free over H* with even-dimensional space-like generators, and the G-space Y is formed from B by adding a single even-dimensional cell of the form DV . We assume that the boundary map in the associated cell-attaching long exact sequence is nonzero. O* *ur goal here is to prove Propositions 4.12 and 4.14, which describe the quotient l* *ong exact sequence (4.1) of this cell-attaching sequence in certain critical dimens* *ions. To prove these results, we work with the long exact sequences in homology coming from the short exact sequence f g 0____//L___//A____//__//_0. of coefficient Mackey functors. Coupling these long exact sequences with the cell-attaching long exact se- quences, we obtain the commuting diagram .. . . . . .. .. .. fflffl|ØL fflffl|_L fflffl| L |fflffl . ._.//_HG*(B; L)_//_HG*(Y ;_L)//_eHG*(SV ;@L)//_HG*-1(B;_L)//_. . . fflffl|Ø fflffl|_ fflffl| |fflffl . ._.//_HG*(B; A)_//_HG*(Y ;_A)//_eHG*(SV ;@A)//_HG*-1(B;_A)//_.(.5..1) fflffl|Ø fflffl|_ fflffl| |fflffl . ._.//_HG*(B; )//_HG*(Y ;_)//_eHG*(SV@;/)/_HG*-1(B;/)/_. . . fflffl| fflffl| fflffl| |fflffl .. . . . . .. .. .. with exact rows and columns. However, we do not want to work directly with this diagram. Instead, we want to work with a quotient of this diagram obtained by killing off everything associated to the summand Z of HG*(B; A). One row of this quotient diagram is the quotient long exact sequence (4.1) and the other t* *wo rows are the analogous long exact sequences for L and coefficients. Our fir* *st objective is to define this quotient diagram and show that it has exact rows and columns. This is done in the first section below. This quotient diagram is th* *en used to prove Propositions 4.12 and 4.14 in the second section. 5.1. Using the Universal Coefficient Theorem Recall the finite subset F@ of the set of purely free generators of HG*(B; A) selected in Definition 4.4. From F@ , we obtain the direct sum decompositon HG*(B; A)~=J Z in which J and Z are the summands spanned by the generators 54 5.1. USING THE UNIVERSAL COEFFICIENT THEOREM 55 in F@ and those not in F@, respectively. Recall also the map @0: eHG*(SV ; A)//* *_ J and automorphism of HG*(B; A)introduced in Proposition 4.5. Denote the com- posite of and the direct sum decomposition by : J Z __//_HG*(B;.A)The commuting diagram eHG*(SV ; A) N (@0,0)qqqqq NNNN@N qq NNNN xxqqqq N'' (J Z) ____________________// HG*(B; A) describes the connection between the boundary map @ of our cell-attaching long exact sequence and the maps @0 and . For any Mackey functor S, denote the RO(G)-graded homology of a point with S-coefficients by HS*. Let JS = J H* HS* and ZS = Z H* HS*. Since HG*(B; A) and eHG*(SV ; A)are free H*-modules, the edge homomorphisms oeSB: HG*(B; A) H* HS*___//HG*(B; S) and oeSV: eHG*(SV ; A)H*HS*___//eHG*(SV ; S) of the universal coefficient spectral sequence are isomorphisms (see Propositio* *n 11.1 for the cases which matter here). Let S be the composite isomorphism H*1 oeSB JS ZS ~=(J Z) H* HS*_______//_HG*(B; A)H*HS*_____//HG*(B; S). Also, let @0S: eHG*(SV ; S)//_ JSbe the composite S)-1 @0 H 1 HeG*(SV ; S)(oeV_//_eHG*(SV ; A)H*HS*____*_// J H* HS*= JS . The naturality of the edge homomorphism implies that the diagram HeG*(SV ; S) N (@0S,0)ppppp NNN@SNN pp NNNN wwpppp N'' (JS ZS )__________S_________// HG*(B; S) commutes. Using this diagram, we can write the homology cell-attaching long exa* *ct sequence for Y with S coefficients as S _S (@0S,0) . ._._//_JS ZS~Ø//_HG*(Y ;_S)//_eHG*(SV_;_S)//_ (JS ZS_)//_...(.5.2) Here, ~ØS=0Ø O S. This long0exact sequence is natural in S if we define the ma* *ps JS __//_JSand ZS ___//ZSassociated to a coefficient map S __//_S0in the obvious way. The long exact sequence above implies that the restriction ZS ___//HG*(Y ; S) of ~ØSto ZS is a monomorphism. Define QS by the short exact sequence S 0 __//_ZS ___//HG*(Y ;_S)ß//_QS __//_0. 56 5. COMPUTING HG*(B [ DV; A)IN THE KEY DIMENSIONS Note that is construction is natural in S since the map ~ØSis natural in S. By applying Lemma 4.6 to long exact sequence (5.2), we obtain the long exact seque* *nce Ø0S S _0S G V @0S S . ._._//JS ____//_Q ____//_eH*(S ;_S)_//_ J ___//. .,. which is also natural in S. Hereafter, we denote eHG*(SV ; S)by NS for consiste* *ncy with our other notation. The naturality of this sequence allows us to construct* * the desired quotient of diagram (5.1). However, we need one further result to ensure that the columns of the resulting diagram are exact. Lemma 5.1. There are maps QL ___//Q, Q __//_Q, and Q___// QL such that the diagram . ._._//_HG*(Y ;_L)//_HG*(Y ;_A)//_HG*(Y ; )//_ HG*(Y ;_L)//_. . . ßLfflffl|| ßfflffl|| fßflffl|| ffßLlffl|| . ._.____//QL_________//_Q_________//Q________// QL______//. . . commutes and has an exact bottom row. Proof. Consider the diagram 0 0 0 0 | | | | fflffl| fflffl| fflffl| fflffl| . ._.____//_ZL________//_Z_________//Z________//_ ZL______//. . . | | | | fflffl| fflffl| fflffl| fflffl| . ._._//_HG*(Y ;_L)//_HG*(Y ;_A)//_HG*(Y ; )//_ HG*(Y ;_L)//_. . . ßL fflffl|| ßfflffl|| fßflffl|| fßLflffl|| . .`.`` `//QL`` ` ` ``//Q` ` `` ` `//Q`` ` ` `// QL` ` ` `//. . . | | | | fflffl| fflffl| fflffl| fflffl| 0 0 0 0 in which the columns are exact. The top half of this diagram commutes because of the naturality of the inclusion ZS ___//HG*(Y ;wS)ith respect to S. Moreover, t* *he top row of this diagram is exact since it is obtained by taking the box product* * of the long exact sequence . ._._//HL*__//H*__//_H*//_ HL* __//_. . . (5.3) with the free H*-module Z. It follows that there are unique choices for the dot* *ted arrows on the bottom row which make the whole diagram commute. A straightfor- ward diagram chase then gives that the bottom row is exact. 5.2. CONSTRUCTING THE MAPS OF THE COMPARISON SEQUENCE 57 Proposition 5.2. The diagram .. . . . . .. .. .. fflffl|Ø0|fflffl_0 |fflffl@0 fflffl| . ._.___//_JL_____L//_QL___L//_NL____L//_ JL____//_. . . fflffl|Ø0|fflffl_0 |fflffl@0 fflffl| . ._.____//_J_____//_Q______//_N______//_ J_____//_. . .(5.4) fflffl|Ø0|fflffl|fflffl_0@0 . ._.__//_J___//Q____//N____// J____//.,. . fflffl| |fflffl |fflffl fflffl| .. . . . . .. .. .. obtained from diagram (5.1) by collapsing out everything associated to Z, commu* *tes and has exact rows and columns. Proof. The exactness of the rows in the diagram follows from Lemma 4.6, and the commutativity of the diagram follows from the naturality of the construction described in that lemma. The Q column of the diagram is exact by Lemma 5.1. The N column is just a long exact coefficient sequence for the space SV . The J column is exact because it is obtained by taking the box product over H* of long exact sequence (5.3) with a free H*-module. 5.2. Constructing the maps of the comparison sequence In this section, we prove Propositions 4.12 and 4.14, which describe the cel* *l- attaching long exact sequence for the G-space Y in certain critical dimensions. Our basic tool in these proofs is diagram (5.4) of Proposition 5.2. Perhaps the most delicate part of these proofs is selecting the elements !0iof RO(G). We ha* *ve already indicated where these elements ought to appear in our standard plot of elements of RO(G). However, for p 5, more than one element of RO(G) plots to each of these locations. Thus, in the early stages of our argument, we look at * *an arbitrary element ! of RO(G) which plots to one of these locations. Once we have learned enough about the appearance of diagram (5.4) in such a dimension !, we can then make the appropriate choice for each of the !0i. The first step in these proofs is analyzing the map @0L: NL ___// JLin certa* *in dimensions. Lemma 5.3. Let ! be an element of RO(G) such that either (i)|! |= |V |and |!G |= |!Gn| or (ii)|!=||!i|and |!G |= |!Gi-1|for some i such that 1 < i n. Then the map (@0L)!: NL!___//( JL )!is nonzero. Proof. Recall that NL and JL are obtained from free H*-modules by tak- ing a box product over H* with HL*. By Corollary 9.3, the H*-modules HL* and 2-,HR* are isomorphic for any nontrivial irreducible G-representation ,. Thus, NL ~= 2-,NR and JL ~= 3-,JR . Further, since the maps @0Land @0Rare ob- tained from @0 by taking a box product over H* with HL* and HR*, respectively, @0Lis identified with 2-,@0Runder these two isomorphisms. Thus, it suffices to prove that the map 2-,@0R : 2-,NR ___// 3-,JRis nonzero in the indicated 58 5. COMPUTING HG*(B [ DV; A)IN THE KEY DIMENSIONS dimensions. This task is simplified by the fact that HR*, NR , and JR are quo- tients of H*, N, and J, respectively. By assumption, each component of the map @0: N __//_J is nonzero in dimension V . It follows easily that each component * *of the map 2-,@0R: 2-,NR __//_ 3-,JRis nonzero in dimension V + 2 - ,. Now assume that ! 2 RO(G) satisfies one of the two conditions in the lemma. Then |! | |V + 2 - ,|and |!G | |(V + 2 - ,)G|. Further, the case |! |= |V + 2 - ,| and |!G |= |(V + 2 - ,)G|can occur only if n = 1. Consider first this special case of two equalities. In this case, 3-,JR is * * in dimensions V + 2 - , and !. The map 2-,@0Rmust then be surjective in dimension V + 2 - , since it is nonzero. From this and Corollary 8.13, it follo* *ws that the map 2-,@0Ris surjective, and therefore nonzero, in dimension !. Hereafter, we can assume that at least one of the two inequalities relating V + 2 - , and ! is strict. In this case, H!-(V +2-,)is one of the Mackey functo* *rs R, , or , and so is generated at G=G by an element of the form ,, ffl, * *or ffl,, respectively. Note that there is an integer j such that 1 j n and |!G |= |!* *Gj|. In dimension !, 3-,JR consists of a single copy of contributed by the genera* *tor of J in dimension !j. Multiplication by the generator of H!-(V +2-,)(G=G) induc* *es an isomorphism from ( 3-,JRj)V +2-,to ( 3-,JRj)! = ( 3-,JR )!. Since the jth component ( 2-,@0R)j: 2-,NR ___// 3-,JRjof the map 2-,@0Ris an H*-module map and is nonzero in dimension V + 2 - ,, it follows that this map is nonzero * *in dimension !. Observe that an element ! satisfying condition (i)in this lemma plots to the location at which the element !0n+1should plot. In fact, this lemma provides us with enough information about such an element ! to establish the existence of an element !0n+1of RO(G) with all of the appropriate properties. Proposition 5.4. There exists an element !0n+1of RO(G) such that (i)|!0n+1|= |V |and |!0n+1G|= |!nG,| (ii)Q!0n+1~=A, and (iii)in dimension !0n+1, the middle row in diagram (5.4) is a short exact sequence of the form Ø0!0n+1 _0!0n+1 0 __//______//A ______//_R __//_0. Proof. Let ! be any element of RO(G) such that |! |= |V |and |!G |= |!Gn|. It follows easily from the description of H*and HL*given in Propositions 9.1* *, 9.2, and 9.5 that diagram (5.4) has the form 0 (Ø0L)! fflffl|(_0L)!(@0L)! 0_______//_QL!_____//R_____// fflffl|Ø0!fflffl|_0!~=fflffl||fflffl|@0! 0______//_____//_Q!______//R_______//0 f~=flffl| f~fflflffl|| fflffl| 0______//(Ø0__//Q!0___//_0 )! (_)! in dimension !. Lemma 5.3 indicates that the map (@0L)! is nonzero. It follows from Lemma 12.3 that QL!~=L. 5.2. CONSTRUCTING THE MAPS OF THE COMPARISON SEQUENCE 59 A simple rank argument applied to the Q column of the diagram implies that the map ~fflis nonzero at G=G. Even though this map need not be onto, its image must then be a copy of . We can therefore derive a short exact sequence of t* *he form 0 ____//L___//Q!____//__//_0 from the Q column of the diagram. By Lemma 12.2(c), the only common solution to this short exact sequence and the short exact sequence in the middle row of * *the diagram above is a Mackey functor of the form A[d], for some integer d prime to* * p. Corollary 8.15 now indicates that we can select ! such that Q! ~=A. Taking !0n+1 to be this ! completes the proof. Selecting the elements !0i, for 1 i n, requires a bit more effort and mu* *st be done by starting with !0nand working inductively downward to !01. Observe that an element ! satisfying condition (ii)in Lemma 5.3 plots to the location at which the element !0ishould plot. Lemma 5.3 does not allow us to determine Q as completely in a dimension ! satisfying condition (ii)as it does for a dimens* *ion satisfying condition (i). However, we can significantly restrict the possible v* *alues of Q in a dimension satisfying condition (ii). Proposition 5.5. Let i be an integer such that 1 i n, and let ! be an element of RO(G) such that |! |= |!i|and ( G |!G |= |!i-1| ifi > 1, |V G| ifi = 1. Then Q! is either L or A[d], for some integer d prime to p. Moreover, if Q! = L, then Q!0= L for any other !02 RO(G) plotting to the same position as !. In dimension !, the middle row of diagram (5.4) is a short exact sequence of the form Ø0! _0! 0 __//_ L ____//_Q! ____//__//0ifi > 1 or Ø0! _0! 0 __//_L ____//_Q! ____//__//0 ifi = 1. Proof. For i = 1, it is easy to see that the middle row of diagram (5.4) mu* *st have the indicated form. This, together with Lemma 12.2(a), implies that Q! is either L or A[d]. The assertion about Q! being L implying that Q!0is also L follows from Corollary 8.13. For i > 1, observe that diagram (5.4) has the form 0 0 fflffl|(Ø0L)|fflffl!(_0L)! (@0L)! 0 ________//_L________//QL!_____//____//_ fi2flffl|Ø0 |fflffl0 ~=fflffl|0 fflffl| ! _! @! 0 ______// L______//Q!______//______//_0 fß1flffl| |~fflfflffl fflffl|| 0 ________//(Ø0___//Q!_0____//0 )! (_)! 60 5. COMPUTING HG*(B [ DV; A)IN THE KEY DIMENSIONS in dimension !. There are far too many solutions for the extension problem dis- played in the middle row of this diagram for this row to be of use in identifyi* *ng Q!. However, the map (@0L)! is nonzero by Lemma 5.3. This map is therefore an isomorphism, and QL!must be isomorphic to L. As in the proof of Proposition 5.4, we can argue that the image of the map ~fflis a copy of which may, or may no* *t, be all of Q!. Regardless, we can extract from the Q column of this diagram a short exact sequence of the form 0____//L____//Q!___//___//0. By Lemma 12.2(a), the only possible solutions to this extension problem are * * L and A[d], for some integer d prime to p. The claim in the proposition about Q! being L implying that Q!0 is also L follows, as in the case i = 1, fr* *om Corollary 8.13. We turn now to a pair of propositions which set the stage for an inductive p* *roof of Propositions 4.12 and 4.14. Proposition 5.6. Let i be an integer such that 1 i n, and assume that there is an element !0i+1of RO(G) such that ( (i)|!0i+1|= |!i+1| ifi < n |V | ifi = n, (ii)|(!0i+1)G|= |!Gi|, and (iii)Q!0i+1~=A. Then Q!i~=R , and the middle row of diagram (5.4) has the form Ø0!i _0!i 0 __//_A ____//_R _____//_//_0 in dimension !i. Proof. The middle row of diagram (5.4) clearly has the form Ø0!i _0!i 0 __//_A ____//_Q!i_____//__//0 in dimension !i. By Lemma 12.1, the only possible solutions of this extension problem are A , which occurs if the sequence splits, and R . Assume that this sequence splits so that Q!i~=A . The Mackey functor H!i-!0i+1is isomorphic to , and is generated at G=G by the element ffl!i-!0i+1. Multipli* *cation by ffl!i-!0i+1gives a map from the middle row of diagram (5.4) in dimension !0i* *+1 to that row in dimension !i. If i < n, this map of short exact sequences has the form Ø0!0i+1 _0!0i+1 0_____// L________//_A_______//___//0 ffl0|| |ffl| ~=ffl00|| fflffl|Ø0!i fflffl|_0!i fflffl| 0________//A______//_A ___//___//0. Let ~0, ~0, and ø0 be the usual elements of Q!0i+1~=A at G=G, and let ~ be the * *usual element of J!i ~=A at G=G. Denote by (1, 0) the generator of the first summand of J!0i+1~= L at G=G. Proposition 1.10(o) indicates that (ffl0(G=G))(1, 0) * *= ~. 5.2. CONSTRUCTING THE MAPS OF THE COMPARISON SEQUENCE 61 The exactness of the top row implies that (Ø0!0i+1(G=G))(1, 0) = ~0. Since we have assumed that the bottom row splits, it follows that (ffl (G=G))(~0) = ( ~,* * 0) in (A )(G=G). The map ffl vanishes at G=e, so (ffl (G=G))(ø0) = 0 and (ffl (G=G))(~0) = (a~, x) for some integer a and some x 2 Z=p. Recall however that ~0= p~0- ø0. From this we get the contradiction that ( ~, 0) = (pa~, 0) in (A )(G=G). Thus, the bottom row cannot split, and Q!i~=R . If i = n, then the map of short exact sequences given by multiplication by ffl!n-!0n+1has the form Ø0!0n+1 _0!0n+1 0_____//_______//_A________//_R______//0 ffl0|| |ffl| ffl00|| fflffl|Ø0!nfflffl|_0!n fflffl| 0______//A____//_A ___//___//_0. Taking ~, ~0, ~0, and ø0 to be as in the previous case, and looking at the image of the generator of J!0n+1~=at G=G under ffl0and Ø0!0n+1, we again obtain th* *at (ffl (G=G))(~0) = ( ~, 0) 2 (A )(G=G). The map ffl still vanishes at G=* *e, so (ffl (G=G))(ø0) = 0 and (ffl (G=G))(~0) = (a~, x) for some integer a and some x 2 Z=p. Thus, the equation ~0 = p~0- ø0 still gives us the contradiction that ( ~, 0) = (pa~, 0) in (A )(G=G). Again, it follows that the bottom row cannot split, so Q!n ~=R . Proposition 5.7. Let i be an integer such that 1 i n, and assume that Q!i~=R . Then there is an element !0iof RO(G) such that ( G (i)|(!0i)G|= |!i-1| ifi > 1 |V G| ifi = 1, (ii)|!0i|= |!i|, and (iii)Q!0i~=A. Proof. Let ! be an element of RO(G) such that |! |= |!i|, and |!G |is |!Gi-* *1| or |V G|, depending on whether i > 1 or not. By Proposition 5.5, we know that Q! is either L or A[d], for some integer d prime to p. If Q! = A[d], then Coro* *llary 8.15 allows us to pick an element !0iof RO(G) satisfying the three conditions in the proposition. Thus, it suffices to eliminate the possibility that Q! = * *L. Assume, to the contrary, that Q! is L. Observe that H!i-! is the Mackey functor R, which is generated at G=G by the element ,!i-!. Multiplication by ,!i-! gives a map from the middle row of diagram (5.4) in dimension ! to that row in dimension !i. If i > 1, this map of short exact sequences has the form Ø0! _0! 0_____// L____//_ L____//___//_0 ,0|| |,| ~=,00|| fflffl|Ø0!i fflffl|_0!i fflffl| 0________//A______//_R ___//___//_0. There are no nonzero maps from to L or from L to . This, plus the fact th* *at the map Ø0!must be an isomorphism at G=e implies that, by picking orientations 62 5. COMPUTING HG*(B [ DV; A)IN THE KEY DIMENSIONS correctly, we can assume that the map Ø0!is p id, where p denotes the multi- plication by p map. It follows easily that the restriction of the composite ,00* *O _0! to the summand of its domain must be surjective. The map , is derived from multiplication by ,!i-!, and so is a composite of the form ~,1 L ~=A ( L) _____//R ( L) __//_R , where ~,takes ~ 2 A to ,!i-! 2 R. Table 1.1 gives that R ~= . But there is no p-torsion in R , so the map , must be zero on the summand of its domain. Since the right square in the diagram commutes, this contradicts the surjectivity of the restriction of ,00O _0!to that summand. It follows that Q! 6= L if i > 1. If i = 1, then the map of short exact sequences given by multiplication by ,!1-! has the form Ø0! _0! 0 _____//L____// L_____//_____//0 ,0|| ,|| |,00| fflffl|Ø0!fflffl|1_0!fflffl|1 0 _____//A____//R ___//____//0. An argument like that used for the i > 1 case implies that the map , vanishes on the summand of its domain. The map _0!is, however, the projection onto this summand of Q!. Moreover, Proposition 1.10(k) implies that the map ,00is an epimorphism. Thus the composite along the top and right edge of the right square is a epimorphism when restricted to the summand . However, the composite along the left edge and bottom of this same square is zero when restricted to <* *Z>. This contradiction implies that Q! 6= L if i = 1. Proposition 4.12 and most of Proposition 4.14 follow easily from these resul* *ts. Proofs of Propositions 4.12 and 4.14. Observe that sequence (4.1) men- tioned in these two propositions is just the middle row of diagram (5.4). Propo* *si- tion 5.4 establishes the existence of an element !0n+1of RO(G) with the propert* *ies claimed for it in Proposition 4.12. Given this element !0n+1, Proposition 5.6 c* *an be applied to establish the assertion of Proposition 4.14 about sequence (4.1) * *in dimension !n. Once this claim is verified, Proposition 5.7 can be applied to es* *tab- lish the existence of an element !0nof RO(G) with the properties claimed for it* * in Proposition 4.12. By continuing to apply Propositions 5.6 and 5.7 in an alterna* *ting fashion, we can establish the existence of the remaining !0irequired by Proposi* *tion 4.12 and verify the claims of Proposition 4.14 about sequence (4.1). This compl* *etes the proof of Proposition 4.12. The only part of Proposition 4.14 which remains unproven is its claim about the map `!i: R __//_R . It is easy to che* *ck that there are no nonzero maps from R to or from to R. Thus, the map `!i must be of the form f g for some maps f : R __//_R and g : ___//. We need to prove thatLeach of0f and g is id. Recall that the domain of ` is the * *free H*-module J0 = 1 i n+1 !iH*. It is easy to verify the claim of Proposition 4.14 that the R and in the domain of `!i come from the generators of J0 in t* *he dimensions !0iand !0i+1, respectively. Thus, to prove the claim about `!i, it s* *uf- fices to understand the restrictions of ` to the summands J0iand J0i+1coming fr* *om these two generators. Denote these restrictions by `i and `i+1, respectively. R* *ecall 5.2. CONSTRUCTING THE MAPS OF THE COMPARISON SEQUENCE 63 that `i was defined by requiring that `i!0ibe the identity map from (J0i)!0i= A* * to Q!0i= A. To see that f = id, consider the commuting square ^, (J0i)!0i__//(J0i)!i `i!0=i|| `i!i|| fflffl|, fflffl| Q!0i______//Q!i in which the horizontal maps come from multiplication by the generator ,!i-!0i of H!i-!0i(G=G). Note that f is the composite of `i!iand the projection of Q!i onto its R summand. The maps ^,and `i!0itake the generator ~0of (J0i)!0i= A to the generators ,0iof (J0i)!i = R and ~ of Q!0i= A, respectively. Thus, to show that the map f is id, it suffices to show that the map , in this square takes* * the generator ~ of Q!0i= A to ( ,i, 0), where ,i is the generator of the R summand of Q!i= R . This map , is, essentially, the middle vertical map in one of * *the two main diagrams occurring the proof of Proposition 5.7. The appropriate one of the two depends on whether i > 1 or i = 1. In that proof we were arguing by contradiction and so assuming that Q!0iwas L rather than A. Redrawing the first of those two diagrams (the one for i > 1) with the corre* *ct value for Q!0i, we obtain the diagram Ø0!0i _0!0i 0_____// L_______//_A______//___//_0 ,0|| |,| ~=,00|| fflffl|Ø0!i fflffl|_0!i fflffl| 0________//A______//_R ___//___//_0. At G=e, each of the four corners of the left square in this diagram is a copy of Z, and the two horizontal maps in that square must be isomorphisms at G=e by exactness. It is easy to check that the left vertical map is also an isomorphis* *m at G=e. From this it follows that (,(G=G))(~) = ( ,i, x) for some x 2 (G=G) = * *Z. However, the map , must factor through R by an argument like that used to show the vanishing of , on one summand in the proof of Proposition 5.7. Since there * *are no nonzero maps from R to , it follows that the component of , going into the summand of its range must be zero. Thus, x = 0. It follows that f = id if i > 1. The argument for the case i = 1 requires redrawing the other diagram in the proof of Proposition 5.7, but is essentially identical thereafter. To see that g = id, consider the commuting square (J0i+1)!0i+1^ffl//_(J0i+1)!i `i+1!=0i+1|| `i+1!i|| fflffl| fflffl| Q!0i+1___ffl__//Q!i in which the horizontal maps come from multiplication by the generator ffl!i-!0* *i+1of H!i-!0i+1(G=G). Note that g is the composite of `i+1!iand the projection of Q!i* *onto its summand. The maps ^ffland `i+1!0i+1take the generator ~0 of (J0i+1)!0i+1* *= A 64 5. COMPUTING HG*(B [ DV; A)IN THE KEY DIMENSIONS to the generators ffl0 of (J0i+1)!i = and ~ of Q!0i+1= A, respectively. Th* *us, to show that the map g is id, it suffices to show that the map ffl in this sq* *uare takes the generator ~ of Q!0i+1= A to (0, ffli), where ffli is the generator o* *f the summand of Q!i = R . This map ffl is the middle vertical map in one of the two main diagrams occurring the proof of Proposition 5.6. The appropriate one of the two depends on whether i < n or i = n. As in the proof of Proposition 5.7, these two diagrams were drawn with an incorrect assumption about one of the entries in order to prove that incorrectness. Correcting the first of these diagrams (which applies for i < n), we obtain * *the diagram Ø0!0i+1 _0!0i+1 0_____// L_______//_A______//___//_0 ffl0|| |ffl| ~=ffl00|| fflffl|Ø0!i fflffl|_0!i fflffl| 0________//A______//_R ___//___//_0. Since the map ffl vanishes at G=e, the component of this map going into the sum* *mand R of its range must be zero. Chasing the element (1, 0) of ( L)(G=G) around this diagram in much the same way that it was chased around the analogous diagr* *am in the proof of Proposition 5.6, one obtains fairly easily that (ffl(G=G))(~) =* * (0, ffli). Thus, g = id if i < n. For the case i = n, the other diagram in the proof of Proposition 5.6 must be redrawn correctly. Then chasing the generator 1 of (G=G) around this diagram, much as in the proof of Proposition 5.6, gives th* *at (ffl(G=G))(~) = (0, ffln). This completes the proof that g = id for all i. CHAPTER 6 Dimension-shifting long exact sequences In this chapter, n is a positive integer and !1, !2, . . . , !n, !01, !02, .* * . . , !0n+1, and V are even-dimensional space-like elements of RO(G) satisfying: |!1|< |!2|< . .<.|!n|< |V |= |!0n+1|, |(!01)G|= |V G|< |!G1|< |!G2|< . .<.|!Gn|, |!0i|= |!i|, fori n, and |(!0i)G|= |!Gi-1|, fori 2. Also, N, J, and J0 are free H*-modules having only purely free generators. The only generator of N is in dimension V . The generators of J are in dimensions !1, !2, . . . , !n. Those of J0 are in dimensions !01, !02, . . . , !0n+1. Ou* *r primary goal here is to prove Proposition 4.7, which characterizes those maps ~Ø: J __/* */_J0, ~_: J0___//N, and ~@: N __//_ Jfor which the sequence ~_ ~@ . ._.//_J _~Ø//_J0____//N ___//_ J __//_. . . (6.1) is a long exact sequence. Throughout this chapter, we refer to any sequence of * *the form (6.1), regardless of whether it is exact, as a candidate sequence. An exa* *ct sequence of this form is referred to as a dimension-shifting long exact sequenc* *e. Our proof of Proposition 4.7 is a three stage induction argument. The first stage of this argument, carried out in Section 6.2, is an induction on the numb* *er n of generators of J. This number is called the complexity of the sequence. The o* *ther two stages are on the two differences |!Gn- V G|and |V - !n|, which we refer to* * as the horizontal and vertical spreads of the sequence. These two stages are carri* *ed out in Section 6.4. Our induction arguments reduce the proof of the proposition to * *the special case of a candidate sequence with minimal complexity and minimal spread. This special case is handled in Section 6.3. In the final section of this chapt* *er, we prove Proposition 4.9, which describes the only constraint on the dimensions !1, !2, . . . , !n, !01, !02, . . . , !0n+1, and V which must be satisfied in order* * for there to be an associated dimension-shifting long exact sequence. Before going into these arguments, we describe a number of general properties of candidate sequences in Section 6.1. 6.1.Preliminary observations about dimension-shifting sequences The restrictions imposed on the dimensions of the generators of N, J, and J0 have a number of implications for the behavoir of sequences of the form (6.1* *). These implications are explored in this section. In particular, we consider wha* *t the exactness of a sequence of this form tells us about the maps ~Ø, ~_, and ~@. On* *e goal 65 66 6. DIMENSION-SHIFTING LONG EXACT SEQUENCES of this discussion is to establish the necessity of the four conditions for exa* *ctness given in Proposition 4.7. A second goal is to describe some properties of seque* *nces of complexity one (n = 1) which are used in the proof of the sufficiency of tho* *se four conditions. We begin with some observations about the vanishing of the composit* *es in a candidate sequence. Lemma 6.1. In any sequence of the form (6.1), (i)the composite ~_OØ~is zero if and only if the composites ~_!iOØ~!iare z* *ero for 1 i n. (ii)the composite ~@O ~_is zero. (iii)the composite ~ØO ~@is zero if and only if the composite ( ~Ø)V O ~@V* *is zero. (iv)the composite ~ØO ~@is zero if the complexity n is 1. For the remainder of this section, we consider only exact sequences of the f* *orm (6.1), and investigate the implications of that exactness for the maps ~Ø, ~_, * *and ~@. Proposition 6.2. In an exact sequence of the form (6.1), the maps ~Ø: J __//* *_J0 and ~_: J0___//N are constructed from standard shift maps. Further, each compo- nent ~@i: N __//_ !i+1H*of the map ~@: N __//_ Jis nonzero. Proof. To prove this result, we must examine a sequence of the form (6.1) in the dimensions of the generators of J, J0, and N. In the dimension !iof a gener* *ator of J, this sequence has the form ~Ø!i ~_!i 0____//A____//R __//___//0. Here, the A comes from the generator of J in dimension !i, and R and come from the generators of J0 in dimensions !0iand !0i+1, respectively. The remaini* *ng generators of J and J0 contribute nothing in this dimension. Lemma 12.1 implies that the components ~Øi,iand ~Øi,i+1of the map ~Øare standard shift maps. Since these components of ~Øare the only ones that can be nonzero, it follows that ~Ø* *is constructed from standard shift maps. In dimension !0i, for i 6= 1, n + 1, our sequence has the form ~Ø!0i ~_!0i 0 ___//_ L___//A___//___//0. For i = 1, n + 1, it has the forms ~Ø!01 ~_!01 0____//L____//A___//___//0 and ~Ø!0n+1~_!0n+1 0____//___//A___//R____//0 respectively. In these sequences, the and L terms are contributed by the gen- erators of J in dimensions !i-1 and !i, respectively. The A terms are contribut* *ed by the generator of J0 in dimension !0i. The remaining generators of J and J0 c* *on- tribute nothing in this dimension. From this description, it follows directly t* *hat all of the components of ~_which can be nonzero are surjective in the critical dime* *nsion and are therefore standard shift maps. 6.1. PRELIMINARY OBSERVATIONS ABOUT DIMENSION-SHIFTING SEQUENCES 67 In the dimension V of the generator of N, our sequence has the form . ._.__//A@~V//_L1 i n~ØV/-1/_L1__//_0. Clearly ~@Vmust be nonzero if this sequence is exact. Thus, by Lemma 4.10, each component of ~@is nonzero. For the remainder of this section,0we consider only sequences of complexity one. In this case, let J0i= !iH*, for i = 1, 2, so that J0 = J01 J02. Also,* * let ~Øi: J __//_J0iand ~_i: J0i__//N denote the components of the two maps ~Øand ~_. Proposition 6.3. Assume that n = 1 and that the sequence ~_ ~@ . ._.//_J _~Ø//_J0____//N ___//_ J __//_. . . is exact. Then any of the following changes to the maps ~Ø, ~_, and ~@yields an* *other long exact sequence: (i)replacing ~@with any other nonzero map N ___// J (ii)replacing any two of the maps ~Ø1, ~Ø2, ~_1, and ~_2with their negativ* *es (iii)if p = 2, replacing any one of the maps ~Ø1, ~Ø2, _~1, and _~2with its negative. Proof. For part (i), note that the map ~@must be nonzero by Proposition 6.2 and that the collection of maps from N to J is a cyclic group of order p by Le* *mma 1.12(a). Thus, any two nonzero maps from N to J are multiples of each other. Moreover, in any dimension ! where the map ~@is nonzero, its target is either <* *Z=p> or L- (which can occur only if p = 2). In either case, ~@!(G=G) is surjective, * *and ~@!(G=e) is zero. Replacing ~@by a nonzero multiple can therefore alter neither* * its image nor its kernel. For part (ii), observe that, if ~Ø1and ~_1are replaced by* * their negatives, then the new sequence can be compared to the old via the identity ma* *ps on J and N and the map -1 1: J01 J02_//_J01 .J02The exactness of the new sequence follows immediately from this comparison. Similarly, replacing ~Ø2and * *~_2 by their negatives also yields a new exact sequence. The new sequence obtained * *by replacing ~Øand ~@by their negatives must be exact because we can compare it to the original one via the identity maps on J0 and N and the map -1 on J. However, since ~@can be replaced by any nonzero map, it follows that replacing only ~Øby its negative also produces a long exact sequence. Analogously, replacing ~_by i* *ts negative produces a long exact sequence. By combining pairs of these allowed si* *gn changes, any other change of signs on exactly two of ~Ø1, ~Ø2, ~_1, and ~_2can * *be accomplished. Thus, any change of exactly two signs does not alter the exactness of the sequence. Now assume that p = 2. For any ff 2 RO(G), at least one of the four Mackey functors Jff, (J01)ff, (J02)ff, and Nffis either or 0. It is i* *mpossible to tell whether the sign has been changed on a map into or out of either o* *r 0. Thus, in any given dimension, a sequence obtained from our original sequence by changing the sign on exactly one of the maps ~Ø1, ~Ø2, ~_1, and ~_2is indisting* *uishable from some sequence obtained by changing the signs on exactly two of these four maps. This indistinguishability implies the desired exactness. Corollary 6.4. Assume n = 1, and let ~Ø: J __//_J0, _~ : J0 __//_N, and ~@: N __//_ Jbe maps satisfying the four conditions in Proposition 4.7. Assume 68 6. DIMENSION-SHIFTING LONG EXACT SEQUENCES further that there are maps bØ: J __//_J0, b_: J0___//N, and b@: N __//_ Jsuch * *that the sequence b_ b@ . ._.//_J _bØ//_J0____//N ___//_ J __//_. . . is exact. Then the sequence ~_ ~@ . ._.//_J _~Ø//_J0____//N ___//_ J __//_. . . is also exact. Proof. Part (i)of Proposition 6.3 allow us to assume that ~@= b@. By assump- tion, the maps ~Øi: J __//_J0iand ~_i: J0i__//N, for i = 1, 2, are standard shi* *ft maps. Proposition 6.2 indicates that the maps bØi: J __//_J0iand b_i: J0i__//N must a* *lso be standard shift maps. Thus, ~Øi= bØiand ~_i= _bi. If p = 2, then we are done by part (iii)of Proposition 6.3. Thus, we may assume that p is odd. If an even number of the maps ~Ø1, ~Ø2, ~_1, and ~_2are the negatives of the maps bØ1, bØ2* *, b_1, and b_2, then part (ii)of Proposition 6.3 implies the desired exactness. If an * *odd number of the four maps are negatives, then the composites bØ!1 _b!1 A ____//R ___// and ~Ø!1 _~!1 A ____//R ___// obtained by looking at our two long sequences in dimension !1 cannot both be ze* *ro. Since this contradicts our assumptions about the two sequences, an odd number of sign differences is not possible. The following special case of Proposition 4.9 can be coupled with the corol- lary above to simplify significantly the proof of Proposition 4.7 for sequences* * of complexity one. Lemma 6.5. Assume that n = 1. There are maps ~Ø: J __//_J0and ~_: J0___//N, constructed from standard shift maps, such that the composite ~Ø!1 0 ~_!1 J!1_____//J!1_____//N!1 is zero if and only if d(V +!1-!01-!02) 1 mod p. Proof. Note that J!1, (J01)!01, and (J02)!02are all copies of the Burnside * *ring A, and let ~, ~01, and ~02be the standard generators of these three Mackey functor* *s. If the maps ~Øand ~_are constructed from standard shift maps, then there are integ* *ers e1, e2, e01, and e02, each of which is 1, such that (~Ø1)!1(~) = e1,!1-!01 (~Ø2)!1(~) = e2ffl!1-!02 (_~1)!01(~01) = e01ffl!01-V (~Ø2)!1(~02) = e02,!02-V. Here ,!1-!01, ffl!1-!02, ffl!01-V, and ,!02-Vare the standard generators of (J0* *1)!1, (J02)!1, N!01, and N!02respectively. The map ~_!1O ~Ø!1is zero if and only if (_~!1O ~Ø!1)(~) = 0. 6.2. THE REDUCTION TO COMPLEXITY ONE DIMENSION-SHIFTING SEQUENCES 69 However, (_~!1O ~Ø!1)(~)= e1e01ffl!01-V,!1-!01+ e2e02ffl!1-!02,!02-V = (e1e01+ e2e02d(V +!1-!01-!02))ffl!01-V,!1-!01. Here, the second equality follows from Proposition 1.10(l). Since each of e1, * *e2, e01, and e02is 1, (_~!1O ~Ø!1)(~) can be zero if and only if d(V +!1-!01-!02) * * 1 mod p. Together this lemma and the corollary preceding it reduce the sufficiency pa* *rt of the proof of Proposition 4.7 for a sequence of complexity one to showing tha* *t, whenever !1, !01, !02, and V are even-dimensional space-like elements of RO(G) for which d(V +!1-!01-!02) 1 mod p, there is at least one choice for the maps ~Ø: J __//_J0, ~_: J0___//N, and ~@: N __//_ Jwhich makes sequence (6.1) exact. 6.2. The reduction to complexity one dimension-shifting sequences Here, we give an inductive argument which reduces the proof of Proposition 4.7 to the case of sequences of complexity one. Thus, throughout this section, we assume that the complexity n of our sequence is at least 2 and that the maps ~Ø: J __//_J0, _~: J0___//N, and ~@: N __//_ Jsatisfy the four conditions in the proposition. Our goal is to show that sequence (6.1) is exact. We do this by co* *m- paring this sequence to two other sequences of complexities 1 and n-1, respecti* *vely. These other two sequences satisfy the conditions of the proposition, and so we * *may assume that they are exact. 0 For 1 i n and 1 j n + 1, let Ji = !iH* and J0j= !jH* , so that J = iJi and J0 = jJ0j. Let Ø00: Jn___//J0n J0n+1be the map obtained from ~Ø by restriction to the summand Jn of its domain and projection onto the summand J0n J0n+1of its range. The map Ø00is constructed from standard shift maps since the maps ~Øand _~are assumed to satisfy condition (i)of Proposition 4.7. Let ~_j: J0j__//N be the jth component of ~_. Condition (i)also implies that the map ~_nis nonzero. This map is determined by its value in dimension !0n, which has * *the form (_~n)!0n: A __//_. From parts (k) and (l) of Proposition 1.10, it fol* *lows that there are elements ff, fi 2 RO(G) such that (_~n)!0n(~) = fflff,fi. These* * two elements of RO(G) satisfy the conditions |ff|= |!0n|- |V |and |ffG|= 0 |fi|= 0 and |fiG|= |(!0n)G|- |V G|. Let ~N= V +fiH*. Then ~_ncan be written as the composite _001 b_n J0n____//_~N__//_N in which the first map is multiplication by fflffand the second is multiplicati* *on by ,fi. Note that both of these maps are standard shift maps. It follows from Proposition 1.10(g) that the map ~_n+1: J0n+1__//N can be written as a composite of the form _002 b_n J0n+1____//_~N__//_N in which the first map is multiplication by ,V +fi-!0n+1, and so is a standard* * shift map. 70 6. DIMENSION-SHIFTING LONG EXACT SEQUENCES Together, the maps _001and _002give a map _00: J0n J0n+1__//~Nwhich is constructed from standard shift maps. For dimensional reasons, the composite N~ _b_n//_N _~@//_ J factors through the inclusion in : Jn __//_ Jvia a map @00: ~N___// Jn. Since ~@n: N __//_ Jnis nonzero by condition (ii)of Proposition 4.7, Proposition 1.10* *(s) implies that the map @00is also nonzero. The maps Ø00, _00, and @00fit into the commuting diagram 00 _00 @00 . ._.___//JnØ___//J0n J0n+1__//~N____// Jn____//_. . . |in |i |b_n |in (6.2) fflffl| fflffl| fflffl| fflffl| . ._.___//_J__~Ø___//_J0__~___//_N_~@_//_ J____//_... . We have already observed that the top row of this diagram satisfies conditions (i)and (ii)of Proposition 4.7. Part (iv)of Lemma 6.1 indicates that it satisfi* *es condition (iv). By looking at parts (g) and (k) of Proposition 1.10, we can arg* *ue that the map b_nis an isomorphism in dimension !n. From this, it follows that t* *he top row also satisfies condition (iii)of the proposition. Thus, by Proposition * *6.10 in Section 6.4, the top row is a long exact sequence. Let ~J= n-1i=1Jiand ~J0= n-1j=1J0j. By adding ~J0to the J0n J0n+1and ~Nter* *ms of the top row of diagram (6.2), we obtain the long exact sequence 00 (0,@00) . ._._//Jn _~Øn//_J0__1____//~J0 ~N______//_ Jn __//_. . . in which ~Ønis just the restriction of ~Øto Jn. Let bØ: ~J__//~J0 ~N be the composite of the map 1 _00: J0___//~J0 ~Nand the restriction of the m* *ap ~Ø: J __//_J0to the summand ~Jof J. Also, let _b: ~J0 ~N__//N be the map formed from the restriction of ~_: J0___//N to the summand ~J0of J0 and the map b_n: ~N___//N. Further, let b@: N __//_ J~ 6.2. THE REDUCTION TO COMPLEXITY ONE DIMENSION-SHIFTING SEQUENCES 71 be the composite of the map ~@: N __//_ Jand the projection ß : J __//_ J~onto the summand J~of J. Then we have the commuting diagram .. . | fflffl| ~J ~Ø|J~qqqq qqq |bØ . ._.____// xxqqq fflffl| Jn__~Øn_//J01__00//_~J0 _~N(0,@00)//__Jn//_. . . in| = | |b_ | in (6.3) fflffl| fflffl| fflffl| fflffl| . ._.____//_J__~Ø_//_J0__~___//N____~@__//_ J_____//_. . . |b@ |ß fflffl| fflffl| J~____=___// J~ | fflffl| .. . The vertical column in the center of this diagram is exactly the sort of seq* *uence to which Proposition 4.7 applies. Moreover, since it is a sequence of complexit* *y n-1, we may assume inductively that the proposition is valid. It follows easily from* * the definitions of the maps in the vertical column that conditions (i)and (ii)of the proposition are satisfied by the column. The composite ~JbØ//_~J0 ~N_b_//_N is just the restriction of the composite ~_ J _~Ø//_J0____//N to the summand J~of J. Our assumption that the composite ~_O ~Øvanishes in dimension !i, for 1 i n, therefore implies that b_O bØvanishes in dimension* * !i for 1 i n - 1. Thus, condition (iii)of the proposition is satisfied. To see* * that the vertical column satisfies (iv)of the proposition, consider the diagram ~@ ~Ø N _____//AA_J_____//_ J0 AAA ||(1 _00) b@AA__A fflffl| bØ 0 J~ ____//_ (J~ ~N). The fact that the summand Jn is present in J but missing from J~might suggest that this diagram does not commute. However, for dimensional reasons, the portion ~@n: N __//_ Jnof the map ~@: N __//_ Jcontributes nothing to the composite along the top and right-hand side of this diagram. It follows easily * *that the diagram does commute. By assumption, the composite ~ØO ~@vanishes in dimension V . Thus, bØO b@also vanishes in dimension V , and condition (iv)of * *the proposition is satisfied. We now know that the top row and the central vertical column of diagram (6.3) are exact, and must prove that the bottom row is exact. We have assumed that the bottom row satisfies conditions (iii)and (iv)of Proposition 4.7. Lemma 72 6. DIMENSION-SHIFTING LONG EXACT SEQUENCES 6.1 indicates that this implies the vanishing of the composites ~_O ~Øand ~ØO * *~@. That lemma also asserts that the composite ~@O ~_is zero. Thus, to complete the proof that the bottom row of diagram (6.3) is exact, it suffices to show that t* *he kernel of each map is contained in the image of the previous map. We do this by chasing elements around the diagram. Even though this diagram is a diagram of Mackey functors, we may treat it as a diagram of abelian groups for the purpose* * of chasing elements _ basically because the diagram is exact if and only if it is * *exact when evaluated at G=G and G=e. Exactness at N follows from an utterly routine diagram chase. The key to establishing exactness at J and J0is the observation that J = ~J * *Jn and that, under this identification, the map ~Ø: J __//_J0is just the sum of th* *e maps ~Ø|J~: ~J__//J0and ~Øn: Jn___//J0. It is easy to see that, if y 2 J0such that ~* *_(y) = 0, then there are elements ~x2 ~Jand xn 2 Jn such that y = (~Ø|J~)(~x) + ~Øn(xn). * *It follows that, if we regard the pair (~x, xn) as an element of J, then ~Ø(~x, xn* *) = y. Similarly, given x 2 J such that ~Ø(x) = 0, regard x as a pair (~x, xn) with ~x* *2 ~J and xn 2 Jn. The assertion that ~Ø(x) = 0 is equivalent to the statement that (~Ø|J~)(~x) = -~Øn(xn). The exactness of the top row gives that ((1 _00)O~Øn)(x* *n) = 0. Thus, bØ(~x) = 0. The exactness of the vertical column then gives an element z * *of -1N such that ( -1@b)(z) = ~x. It is easy to see that there is an element x0n of Jn such that x - ( -1@~)(z) = in(x0n). Since the composite ~ØO -1@~is zero, ~Ø(x - ( -1@~)(z)) = 0, and so ~Øn(x0n) = 0. The exactness of the top row then gives an element w of -1(J~0 ~N) such that ( -1(0, @00))(w) = x0n. It follows that x = ( -1@~)(z + ( -1_b)(w)). This completes our reduction of the proof of Proposition 4.7 to the case in which the complexity n is 1. 6.3. Sequences with minimal complexity and spread The induction arguments presented in the previous section and the next secti* *on reduce the proof of the sufficiency part of Proposition 4.7 down to proving the following proposition and corollary. Proposition 6.6. Assume that n = 1 and that |!G1- V G|= |V - !1|= 2. If dV +!1-!01-!02 1 mod p, then there exist maps bØ: J __//_J0, b_: J0___//N, a* *nd b@: N __//_ Jsuch that the sequence b_ b@ . ._.//_J _bØ//_J0____//N ___//_ J __//_. . . is a long exact sequence. Corollary 6.7. Assume that n = 1 and that |!G1- V G|= |V - !1|= 2. If (i)Ø~and ~_are constructed from standard shift maps, (ii)the map ~@: N __//_ Jis nonzero, and (iii)the composite ~Ø!1 0 ~_!1 J!1_____//J!1_____//N!1 is zero, then the sequence ~_ ~@ . ._.//_J _~Ø//_J0____//N ___//_ J __//_. . . is a long exact sequence. 6.3. SEQUENCES WITH MINIMAL COMPLEXITY AND SPREAD 73 Proof. We have assumed that the maps ~Ø, ~_, and ~@satisfy the first three conditions of Proposition 4.7. Part (iv)of Lemma 6.1 indicates that the fourth condition in that proposition is also satisfied. By Lemma 6.5, dV +!1-!01-!02 * * 1 mod p. Thus, by Proposition 6.6, there are maps bØ: J __//_J0, b_: J0___//N, and b@: N __//_ Jsuch that the sequence b_ b@ . ._.//_J _bØ//_J0____//N ___//_ J __//_. . . is exact. Corollary 6.4 now implies the asserted exactness. There are two possible approaches to proving Proposition 6.6. The most direct approach, which was taken in [5], is to select the maps bØ, b_, and b@appropria* *tely, and then prove the exactness of the sequence simply by examining it in all poss* *ible dimensions. However, this approach is quite tedious and requires intimate famil- iarity with both the additive and multiplicative structure of H*. A shorter pro* *of can be obtained by applying the main freeness result from [11] to appropriately selected stunted complex projective spaces. Certain cell-attaching long exact * *se- quences for these spaces are sequences of exactly the desired form. The remaind* *er of this section is devoted to proving the proposition via this second approach. The first step in this approach is to note that we can reduce the proof of t* *he proposition to a special case in which we have replaced the relatively unrestri* *cted quadruple of elements V , !1, !01, and !02of RO(G) by a much more carefully selected quadruple. In particular, by desuspending the desired sequence by !01,* * we can reduce the proof of the proposition to the special case in which the quadru* *ple V , !1, !01, and !02has been replaced by the quadruple V - !01, !1- !01, 0, and !02* *- !01. By Lemma 1.5, we can select nontrivial irreducible complex G-representations j and ~ such that dj-(V -!01) d2-~-(!1-!01) 1 mod p. Lemma 1.14 then provides isomorphisms 0 j ! -!0 2-~ V -!1H* ~= H* and 1 1H* ~= H* of H*-modules which allow us to replace V -!01and !1-!01by j and 2-~, respec- tively. The congruences determining j and ~ can be coupled with the congruence dj-1-j 1 mod p of Lemma 1.5 and the congruence dV +!1-!01-!02 1 mod p assumed in the proposition to obtain the congruence d2-~+j-1-(!02-!01) 1 mod p. Applying Lemma 1.14 again gives us the isomorphism 0-!0 2-~+j-1 !2 1H* ~= H* of H*-modules which allows us to replace !02- !01by 2 - ~ + j-1. Thus, it suffi* *ces to prove the special case of Proposition 6.6 in which the quadruple of elements* * of RO(G) is j, 2 - ~, 0, and 2 - ~ + j-1. Via suspension by ~, this special case is equivalent to the special case of the quadruple j + ~, 2, ~, and 2 + j-1. In the special case where j = ~, applying the results of [11] to a copy of C* *P 2 with a linear action provides the desired long exact sequence. 74 6. DIMENSION-SHIFTING LONG EXACT SEQUENCES Lemma 6.8. Let j be a nontrivial irreducible complex G-representation. Then, there are maps -1 bØ: 2H* ___// jH* 2+j H*, _b: jH* 2+j-1H* ___// 2jH*, and b@: 2jH* __//_ 3H* such that the sequence -1 b_ 2j b@ 3 . ._._// 2H* _bØ//_ jH* 2+j H* ___//_ H* ___//_ H* ___//. . . is exact. Proof. For any complex G-representation W , denote the associated complex projective space with a linear G-action by P (W ). Also, denote the trivial com* *plex G-representation with complex dimension n by nC. By Proposition 3.1 of [11], the reduced homology of P (2C+j-1) is a free H*-module with generators in dimensions j and 2 + j-1. This description of eHG*(P (2C + j-1);iA)s obtained by viewing t* *his space as being obtained from P (1C + j-1) by attaching the obvious 4-cell. If, instead, we view this space as being obtained from P (2C) by attaching a differ* *ent 4-cell, then the associated cell-attaching long exact sequence has the form . ._._// 2H* ___//eHG*(P (2C + j-1);_A)// 2jH* _@_//_ 3H* ___//. ... Replacing eHG*(P (2C + j-1);bA)y the isomorphic H* -module jH* 2+j-1H* gives the desired long exact sequence. Note that the boundary map @ in this long exact sequence has to be nonzero since the free H* -modules 2H* 2jH* and jH* 2+j-1H* are obviously not isomorphic. If j 6= ~, then we must use a stunted projective space with a linear G-action to produce the desired long exact sequence. Observe that, since j and ~ are both nontrivial irreducible complex G-representations, there is an integer k such th* *at ~ = jk and 1 < k < p. Let W be the complex G-representation 1C + j + . .+.jk-1. Note that neither ~ nor j-1 = jp-1 is contained in W . Applying the results of * *[11] to the stunted projective space P (W + j-1 + 1C)=P (W ) provides the desired lo* *ng exact sequence. Lemma 6.9. Let j be a nontrivial irreducible complex G-representation, and k be an integer such that 1 < k < p. Then, there are maps k 2+j-1 bØ: 2H* ___// j H* H*, b_: jkH* 2+j-1H* __//_ j+jkH*, and b@: j+jkH* __//_ 3H* such that the sequence k 2+j-1 b_ j+jk b@ 3 . ._._// 2H* _bØ//_ j H* H* ___//_ H* ___//_ H* ___//. . . is exact. 6.4. THE REDUCTION TO SEQUENCES OF MINIMAL SPREAD 75 Proof. Let X = P (W + j-1 + 1C)=P (W ) and ! = j + j2 + . .+.jk-1. By Proposition 3.1 of [11], the reduced homology of X is a free H*-module with generators in dimensions ! + jk and 2 + ! + j-1. This description of eHG*(X; A) is obtained by viewing this space as being obtained from P (W + j-1)=P (W ) by attaching the appropriate (2k + 2)-cell. If, instead, we view this space as be* *ing obtained from P (W + 1C)=P (W ) by attaching a different (2k + 2)-cell, then the associated cell-attaching long exact sequence has the form k+j @ 3+! . ._._// 2+!H* ___//eHG*(X; A)//_ !+j H* ___//_ H* ___//. ... Replacing eHG*(X; A)by the isomorphic H*-module !+jkH* 2+!+j-1H* and desuspending by ! gives the desired long exact sequence. Note that the mapk@ in this sequence has to be nonzero since the free H*-modules 2+!H* !+j +jH* and !+jkH* 2+!+j-1H* are obviously not isomorphic. Together Lemmas 6.8 and 6.9 provide a long exact sequence for every quadruple of the form j + ~, 2, ~, and 2 + j-1. Suspending by the appropriate element of RO(G) and applying the appropriate isomorphisms from Lemma 1.14 then provides a long exact sequence for each quadruple V , !1, !01, and !02satisfying the hyp* *otheses of Proposition 6.6. 6.4.The reduction to sequences of minimal spread Throughout this section, we assume that the complexity n of our sequences is one. Our goal in this section is to prove the following special case of Proposi* *tion 4.7. Proposition 6.10. Assume n = 1. The sequence ~_ ~@ . ._.//_J _~Ø//_J0____//N ___//_ J __//_. . . (6.4) is a long exact sequence if and only if the following three conditions are sati* *sfied: (i)Ø~and ~_are constructed from standard shift maps (ii)the map ~@: N __//_ Jis nonzero (iii)the composite ~Ø!1 0 ~_!1 J!1_____//J!1_____//N!1 is zero. The fourth condition which one might expect to see here is unnecessary by part (iv)of Lemma 6.1. Condition (iii)in this proposition is obviously necessary for exactness. Proposition 6.2 implies that the first two conditions are also n* *ecessary for exactness. The remainder of this section is devoted to proving that these t* *hree conditions are also sufficient. Thus, assume that ~Ø: J __//_J0, ~_: J0___//N,* * and ~@: N __//_ Jare maps satisfying the three conditions in the proposition. Note that, by Lemma 6.5, d(V +!1-!01-!02) 1 mod p. Our proof is a two stage induction argument based on Corollary 6.7. In the f* *irst stage, we retain the restriction from the corollary that |V - !1|= 2, but elimi* *nate the constraint on |!G1- V G|by an induction on the size of |!G1- V G|. Corollary 6.7 serves as both the base case of this induction and a key tool in proving the inductive step. In the second stage of the induction, we use the result from t* *he 76 6. DIMENSION-SHIFTING LONG EXACT SEQUENCES first stage to eliminate the constraint on |V - !1|by an induction on the size * *of |V - !1|. For the first stage of our induction, we work with a quadruple V , !1, !01, and !02of even-dimensional space-like elements RO(G) such that |V - !1|= 2. In order to apply our induction hypothesis, we wish to replace this quadruple of elements with two other quadruples having smaller horizonal spreads. Each of th* *ese new quadruples are formed by replacing a pair of the elements from the original quadruple by the elements !01+ 2 and 2!01- V + 2 of RO(G). Figure 6.1 illustrat* *es the relative positions of these six elements of RO(G). |ff| | |6 | 0 0 | V !1 + 2 !2 | | | !01 2!01- V + 2 !1 | _______________________________________-||ffG| | | Figure 6.1. The six elements of RO(G) used in stage one of the induction Let N~ = !01+2H*and J~= 2!01-V +2H*. Observe that the quadruple V , 2!01- V + 2, !01, and !01+ 2 (with 2!01- V + 2 and !01+2 taken as replacements * *for !1 and !02, respectively) satisfies the the hypotheses of Proposition 6.6. Thus* *, for appropriately chosen maps Ø0, _0, and @0, we have a dimension-shifting long exa* *ct sequence 0 _0 @0 . ._._//~JØ_//J01 ~N____//N ____// J~___//. ... (6.5) Consider also the quadruple !01+ 2, !1, 2!01- V + 2, and !02(with !01+ 2 taken * *as a replacement for V and 2!01- V + 2 taken as a replacement for !01). Since (!01+ 2) + !1 - (2!01- V + 2) - !02= V + !1 - !01- !02, Lemma 6.5 provides us with maps Ø00: J __//_~J J02and _00: ~J J02_//_~N, con- structed from standard shift maps, such that the composite _00!1O Ø00!1is zero. Moreover, because |!G1- (!01+ 2)G|< |!G1- V G|, our induction hypothesis allows us to assume that the sequence 00 _00 @00 . ._._//J _Ø__//_~J J02____//_~N__//_ J __//_. . .(6.6) is exact provided the map @00is nonzero. Proposition 6.3 implies that we have a certain amount of flexibility in the * *choice of the maps Ø0, _0, @0, Ø00, _00, and @00in these two long exact sequences. We * *want to use that flexibility to select those maps in such a way that we can derive t* *he exactness of sequence (6.4) from the exactness of these two sequences. Each of the maps ~Ø, _~, Ø0, _0, Ø00, and _00has two components, which we denote using 6.4. THE REDUCTION TO SEQUENCES OF MINIMAL SPREAD 77 subscripts (as in ~Ø1and ~Ø2). These six components fit into the diagram ~_2 ___________________________________________________* *______________________________________________________________@ _____________________________________________________* *______________________________________________________________@ ONOoo__0___N~uoo_00___J02u______OO | | 2 OOOO _2 OO|OO| | | | | | | ~_1||_01|| Ø02||_001||Ø002|~Ø2| | | | | |||| | | Ø01 | | Ø001 | | J01oo______J~oo_______J.ii______________________________* *______________________________________________________________@ ____________________________________________________* *______________________________________________________________@ Ø~1 Since all of the maps in this diagram are standard shift maps, the maps in e* *ach parallel pair are either equal or negatives of each other. Moreover, the compos* *ites Ø01O Ø001and _02O _002are ~Ø1and _~2, respectively. Using the flexibility giv* *en to us by Proposition 6.3, we can adjust the signs of the components of Ø0, _0, Ø00* *, and _00so that ~Ø1= Ø01O Ø001Ø~2= Ø002 _~2= _02O _002 _~1= _01. The condition ~_O ~Ø= 0, which is equivalent to condition (iii)of Proposition 6.10, can be restated as the assertion that the exterior of the diagram above a* *n- ticommutes (that is, commutes up to a minus sign). Similarly, the exactness of sequences (6.5) and (6.6) implies that the primed and double primed squares in * *the diagram above anticommute. It follows that, after all our sign changes have been made, _001= -Ø02. Proposition 6.3 indicates that we can take the maps @0 and @00in sequences (6.5) and (6.6) to be any nonzero maps. The composites Ø001O ~@and ~@O _02are easily seen to be nonzero, so we take these as our choices for @0and @00, respe* *ctively. Define maps fl : ~J__//J01 ~J J02 s : J0 = J01 J02//_J01 ~J J02 ` : J01 ~J J02_//J0 by the formulae fl(x)= (Ø01(x), -x, 0) s(u, v)= (u, 0, v) `(a, b, c)= (a + Ø01(b), c). Of course, these are maps between RO(G)-graded Mackey functors, so these for- mulae must be interpreted as applying for each ff 2 RO(G) and each of the orbits G=G and G=e. 78 6. DIMENSION-SHIFTING LONG EXACT SEQUENCES Now consider the diagram .. . | fflffl| m ~J flmmmm mmm |Ø0 (0,Ø00) vvmmmm fflffl|(0,@00) . ._.____//J_____//J01II~J J021__00//_J01__~N_// J_____//. . . _______ = | s_|`_____________|_0________|=______________________ fflffl| _fflffl|________fflffl|____|fflffl___ . ._.____//J___Ø~____//J0_____~_____//N___~@___// J_____//. . . pp f@0flffl||pØ00pppp xxpp 1 J~ | fflffl| .. . in which the vertical column is just long exact sequence (6.5). The top full ro* *w of this diagram is obtained from long exact sequence (6.6) by adding J01to the ~J * * J02 and ~Nterms in that sequence. Thus, the top full row is exact. The bottom row is the sequence whose exactness is to be proven. It is fairly easy to see that, if* * s is removed from the diagram, then the remainder of the diagram commutes. Clearly, ` O s = id, and ` O fl = 0. In fact, the two maps fl and ` form a split short e* *xact sequence. The two composites ~@O_~and ~ØO@~are zero by Lemma 6.1, and the composite ~_O ~Øis assumed to be zero. Thus, to show that the bottom row is exact, it suf* *fices to show that the kernel of each map is contained in the image of the previous m* *ap. At N, this follows from a perfectly straightforward diagram chase. If y is an e* *lement of J0!, for some ! 2 RO(G), and ~_!(y) = 0, then chasing s(y) around the diagram easily gives that y is in the image of ~Ø. Now assume that x 2 J!, for some ! 2 RO(G), and ~Ø!(x) = 0. If (0, Ø00)!(x) is zero, then it follows easily from the exactness of the top row of the diagra* *m that x is in the image of -1@~: -1N ___//J. On the other hand, if (0, Ø00)!(x) 6= 0,* * then there is a nonzero element z of ~J!such that fl!(z) = (0, Ø00)!(x). Since Ø0!(z* *) = 0, z must be in the image of the map ( -1@0)!: ( -1N)! ___//~J!. The map -1@0is the composite of -1@~and Ø001. Thus, ( -1@~)! must be nonzero. In any dimension where this map is nonzero, its target is either or L- . If the target is <* *Z=p>, then x is in the image of ( -1@~)! since any nonzero map into is surjectiv* *e. If the target of ( -1@~)! is L- , then p = 2 and the domain of this map is . Since ( -1@~)! is nonzero, it is surjective at G=G, and any x 2 J!(G=G) is in i* *ts image. The map ~Ø!is injective at G=e since ~Øis constructed from standard shift maps. Thus, if x 2 J!(G=e), then x = 0 2 Im( -1@~)!. This completes the proof of the first stage of our induction. For the second stage of the induction, we remove the restriction |V - !1|= 2 from the quadruple V , !1, !01, and !02. Again, in order to apply our induction hypothesis, we want to replace this single quadruple by two others. However, th* *is time we want these two to have smaller vertical, rather than horizontal, spreads 6.4. THE REDUCTION TO SEQUENCES OF MINIMAL SPREAD 79 than that of the original quadruple. Select a nontrivial complex irreducible G- representation j. Each of our new quadruples is formed by replacing a pair of the elements from the original quadruple by0the elements !1 + j and !01+ j of RO(G). Thus, let J^= !1+jH* and N^ = !1+jH* . Consider the quadruple !01+ j, !1, !01, and !1 + j in which !01+ j replaces V and !1 + j replaces !02. Since (!01+ j) + !1 - !01- (!1 + j) = 0, Lemma 6.5 indicates that there are maps Ø0: J __//_J01 ^Jand _0 : J01 ^J__//^N, constructed from standard shift maps, such that the composite _0!1O Ø0!1is zero. Moreover, because |(!01+ j) - !1|= 2, we can conclude from the first stage of our induction that the sequence 0 _0 @0 . ._._//J _Ø__//J01 ^J___//^N___// J __//_. ... (6.7) is exact provided the map @0 is nonzero. Consider also the quadruple V , !1+ j, !01+ j, and !02in which !1+ j replaces !1 and !01+ j replaces !01. Note that V + (!1 + j) - (!01+ j) - !02= V + !1 - !01- !02. Thus, by Lemma 6.5, there are maps Ø00: ^J__//_^N J02and _00: N^ J02__//N, constructed from standard shift maps, such that the composite _00!1+jO Ø00!1+jis zero. Since |V - (!1 + j)|< |V - !1|, our induction hypothesis for the second stage allows us to assume that the sequ* *ence 00 _00 @00 . ._._//^JØ_//_^N J02____//_N ____//_ J^___//. . .(6.8) is exact provided the map @00is nonzero. As in the first stage of the induction, Proposition 6.3 gives us a certain a* *mount of flexibility in the choice of the maps Ø0, _0, Ø00, and _00in these two long * *exact sequences. We want to use that flexibility to select those maps in such a way t* *hat we can derive the exactness of sequence (6.4) from the exactness of sequences (* *6.7) and (6.8). The six components of the maps ~Ø, ~_, Ø0, _0, Ø00, and _00fit into * *the diagram oo_~_2__ NOOoo______J02II______ _||____002__OO|____________________________________* *_UU__________________________________________ ___________________________________________________* *__________ __|_001|__Ø002||____________________________________* *___________________________________________________________ __|__________|______________________________________* *______________________________________ __|_oo_Ø001__|______________________________________* *_______________________________________ ~_1N^_oo______J^______________________________________* *________~Ø2______________________________________________ __OO|___02__________________________________________* *_OO|_______________________________________________ ________________________|___________________________* *_____________ __|_01|_____________________________________________* *_________Ø02||________________________________________ _|________________________________________|________* *_________________________________ _|__________________|Ø01oo_________________________* *__ J01oo______J. ~Ø1 As before, all of the maps in this diagram are standard shift maps, and so t* *he maps in each parallel pair are either equal or negatives of each other. Moreove* *r, the composites Ø002O Ø02and _001O _01are ~Ø2and _~1, respectively. Using the flex* *ibility given to us by Proposition 6.3, we can adjust the signs of the components of Ø0* *, _0, 80 6. DIMENSION-SHIFTING LONG EXACT SEQUENCES Ø00, and _00so that Ø~2= Ø002O Ø02 ~Ø1= Ø01 _~1= _001O _01 _~2= _002. The assumed vanishing of the composites ~_O ~Ø, _0O Ø0, and _00O Ø00implies that the exterior of this diagram and the primed and double primed squares in t* *his diagram anticommute. It follows that, after all our sign adjustments have been made, Ø001= -_02. The only condition which the maps @0 and @00must satisfy is that they must be nonzero. It is easy to check that the composites ~@O _001and Ø02O ~@are non* *zero, so we take these two composites to be @0 and @00, respectively. Consider the diagram .. . | fflffl| m ^J flmmmm mmm |Ø00 (Ø0,0) vvmmmm fflffl|(@0,0) . ._.____//J_____//J01JJ^J J02_0_1//^N J02___// J_____//. . . _______ = | s_|`_____________|_00_______|=_______________________ fflffl| _fflffl|________fflffl|____|fflffl___ . ._.____//J___Ø~____//J0_____~_____//N___~@___// J_____//. . . pp f@00flffl||pØ0pppp xxpp 2 J^ | fflffl| .. . in which the maps fl : ^J__//J01 ^J J02 s : J0 = J01 J02//_J01 ^J J02 ` : J01 ^J J02_//J0 are defined by the formulae fl(x)= (0, -x, Ø002(x)) s(u, v)= (u, 0, v) `(a, b, c)= (a, Ø002(b) + c). The vertical column in this diagram is just long exact sequence (6.8). The top * *full row of this diagram is obtained from long exact sequence (6.7) by adding J02to * *the J01 ^Jand N^terms in that sequence. Thus, the top full row is exact. As in the first stage of the induction, the bottom row is the sequence whose exactness is* * to be proven. Also, if s is removed from the diagram, then the remainder of the diagr* *am commutes. Moreover, the two maps fl and ` form a short exact sequence which is split by s. From this point on, the argument for the exactness of the bottom row of this diagram follows exactly the same pattern as the one presented in the fi* *rst stage of the induction. Thus, the proof of Proposition 6.10 is complete. 6.5. THE CONGRUENCE CONDITION ON diV +P !i-P !0jj 81 6.5. The congruence condition on d(V +P !i-P !0j) In this section, we return to the context presented at the beginning of this chapter in which the complexity n of our sequences is an arbitrary positive int* *eger. Our goal is to prove Proposition 4.9. This result describes the congruence cond* *ition on dV +P !i-P !0jthat is the sole obstruction to the existence of a dimension-s* *hifting long exact sequence associated to the elements !i, !0j, and V of RO(G). The information on the multiplicative structure of H* provided by Proposition 1.10 * *is needed for the explicit computations required in the proof of this result. For these computations, it is useful to define elements fij of RO(G) by j-1X fij = V + !i- !0i, i=1 for 1 < j n + 1. Observe that |fij|= |V |and |fiGj|= |(!0j)G|for 1 < j n + * *1. Denote the canonical generator of J!i= A by ~ifor 1 i n. The key to the proof of the assertion about ~Øin Proposition 6.2 is the observation that, for * *any map ~Ø: J __//_J0, Ø~!i(~i) = ei,!i-!0i+ e0iffl!i-!0i+1 for some integers ei and e0i. The map ~Øis constructed from standard shift maps* * if and only if these integers are 1 for 1 i n. Similarly, denote the standard generator of J0!0j= A by ~0jfor 1 j n + 1. From the proof of the assertion about ~_in Proposition 6.2, it follows that, fo* *r any map ~_: J0___//N, there are integers e00jsuch that 8 >:j j j j e00n+1,!0n+1-V forj = n + 1. Note that, for 1 < j n, the integer e00jis only determined mod p. Also observe that the map ~_is constructed from standard shift maps if and only if e001and e* *00n+1 are 1 and the e00jare relatively prime to p for all j. These observations suff* *ice for the computations needed in our proof. Proof of Proposition 4.9. We begin with the ö nly if" part of the proof. Assume that ~Ø: J __//_J0and ~_: J0___//N are constructed from standard shift maps. From the formulae above, we obtain that (_~!1O ~Ø!1)(~1)= e1e001ffl!01-V,!1-!01+ e01e002ffl!1-!02ffl!02-fi2,fi2* *-V = (e1e001+ e01e002)ffl!01-V,!1-!01. Thus, (_~!1O ~Ø!1)(~1) = 0 if and only if e002 -e1e01e001mod p. Since each of e1, e01, and e001is 1, it follows that e002 1 mod p if (_~!1O * *~Ø!1)(~1) is zero. Similarly, for 1 < i < n, (_~!iO ~Ø!i)(~i)=eie00iffl!0i-fii,!i-!0i,fii-V+ e0ie00i+1ffl!i-!0i+1ffl!0i+1-* *fii+1,fii+1-V = (eie00i+ e0ie00i+1)ffl!0i-fii,fii+1-V. 82 6. DIMENSION-SHIFTING LONG EXACT SEQUENCES It follows that (_~!iO ~Ø!i)(~i) = 0 if and only if e00i+1 -eie0ie00imod p. Inductively, this allows us to argue that e00i+1 1 mod p if (_~!jO ~Ø!j)(~j) * *= 0 for j i. Finally, (_~!nO ~Ø!n)(~n)= ene00nffl!0n-fin,!n-!0n,fin-V+ e0ne00n+1ffl!n-!0n+1,!0n+1* *-V = (ene00n+ dfin+1-!0n+1e0ne00n+1)ffl!n-fin+1,fin+1-V. This gives that (_~!nO ~Ø!n)(~n) = 0 if and only if dfin+1-!0n+1e00n+1 -ene0ne00nmod p. Since e00n+1= 1, we can conclude that dfin+1-!0n+1 1 mod p if (_~!iO ~Ø!i)(~* *i) is zero for 1 i n. The observation that X X fin+1 - !0n+1= V + !i- !0j, 1 i n 1 j n+1 completes the ö nly if" part of the proof of the proposition. For the "if" part, assume that integers ei and e0i, for 1 i n, and an in* *teger e001have been chosen so that each is 1. If dfin+1-!0n+1 1 mod p, then we can select integers e00i= 1, for 1 < i n+1, which satisfy the appropriate congru* *ences noted above. This collection of integers specifies maps ~Ø: J __//_J0and ~_: J0* *___//N such that ~_!iO ~Ø!i= 0 for all i. CHAPTER 7 Complex Grassmannian Manifolds If V is a complex G-representation and k is a positive integer, then the Gra* *ss- mannian manifold G(V, k) of complex k-dimensional subspaces of V carries an ob- vious G-action derived from the action of G on V . Nonequivariantly, G(V, k) is* * a CW-complex whose cells are the Schubert cells (see, for example, [7, 8, 19]). H* *ere, we show that, if G is a finite abelian group, then there is an equivariant vers* *ion of the Schubert cell structure of G(V, k) which provides this G-space with the str* *uc- ture of a generalized G-cell complex. We also show that, for G = Z=p, this cell structure on G(V, k) satisfies the hypotheses of Theorem 2.5 so that the equiva* *riant RO(G)-graded ordinary homology HG*(G(V, k); A)is free over H*. To describe the generalized G-cell structure on G(V, k), we assume initiallyL that V is a finite dimensional complex G-representation and express V = ms=1O* *Es as a sum of complex irreducibles. Since we are assuming that G is abelian, these irreducible representations OEs have complex dimension one. From this descripti* *on of V in terms of an ordered collection of irreducible representations, we obtai* *n a flag of subspaces 0 < V1 < V2 < . .<.Vm = V L t of V in which Vt = s=1OEs for each 1 t m. In terms of this fixed flag, the standard Schubert cells can now be described as usual. Here, we follow the nota* *tion used in [8]. Given a sequence of integers 0 a1 . . .ak m - k, define the * *cell by = {X 2 G(V, k) : dimC(X \ Vai+i) = i for 1 i k}. This cell is usually represented as a matrix OE1. . .OEa1OEa1+1 . . . OEa2+2 . . .OEak+k . . .OEm 2 3 66* . . .* 1 0 . . . 7 66* . . .* 0 * . . .* 1 0 . . . 77 66.. .. 777 4 . . 5 * . . .* 0 * . . .* 0 * . . . 1 0 . . .0 written in standard form. Here, * is used to denote an arbitrary complex number, and there are ai *'s in the ith row, for each 1 i k. The G-action on the ce* *ll comes from letting G act on each entry of the matrix as it acts o* *n the irreducible representation above its column. The cell is the inte* *rior of 83 84 7. COMPLEX GRASSMANNIAN MANIFOLDS the representation cell DW where Mk ai+i-1M W = OE-1ai+iOEj. i=1 j=1 j62{a1+1,...,ai-1+i-1} Here, OE-1ai+iOEj denotes the tensor product over C of OEj and the conjugate of* * OEai+i. Observe that the (real) dimension of the cell is |W | = 2 ki=1ai. The space G(V, k) is built from these cells by beginning with the 0-cell <0,* * . .,.0> as the 0-filtration X0. The cell can be attached to any subcomple* *x of G(V, k) containing all the cells such that bi ai, for 1 i k* *, and bi< aifor at least one i. The usual filtration of G(V, k) is specified by the e* *quation X Xn - Xn-1 = {| ai= n}. i However, there are other useful filtrations. If V V 0, then clearly G(V, k)* * is a sub-G-cell complex of G(V 0, k). Hence, G(V, k) carries an obvious G-cell struc* *ture if V is a countably infinite dimensional complex representation. If G = Z=p and V is finite dimensional, then Theorem 2.1 obviously implies t* *hat the RO(G)-graded homology HG*(G(V, k); A)of G(V, k) is free over H*. However, if V is countably infinite dimensional, then Theorem 2.5 must be used to show t* *hat the homology is free. In order to apply that result, we need to find a lower bo* *und for the fixed dimension of the representation W associated to the cell . Note that W contains one copy of the irreducible trivial complex G-representati* *on for each pair of positive integers i and j such that i k, j ai+i-1, j 6= at* *+t for any t < i, and OEj is isomorphic to OEai+ias a complex G-representation. In ord* *er to obtain a useful lower bound for the number of such pairs, we need to choose the ordering on the irreducible summands OEs of V carefully. For each positive integer r, let Br be a set of representatives of the isomorphism classes of irr* *educible complex G-representations that appear in the decomposition of V at least r time* *s. Note that, if the dimension of V is countable, then the disjoint union [rBr can be identified with the set of irreducible summands of V . Assign each set Br so* *me ordering, and then order the set [rBr as the concatenation B1, B2, . . . , Br, * *. . . of the ordered sets Br. Each of the Br is referred to as a block, and an ordering * *of the irreducible summands of V constructed in this fashion is referred to as a stand* *ard block ordering of the cells. For each positive integer n, let Vn be the sum of * *the first n irreducible summands of V in a standard block ordering of these summand* *s. The sequence of inclusions * = G(Vk, k) G(Vk+1, k) . . .G(Vn, k) . . . provides a filtration of G(V, k) by finite sub-G-cell complexes. The control ov* *er the fixed dimensions of the cells of G(V, k) needed to apply Theorem 2.5 is provide* *d by the following result: Proposition 7.1. Let G = Z=p, and let V be a countably infinite dimensional complex G-representation. Assume that the irreducible complex summands of V have been given a standard block ordering. Then, for any positive integer m, ev- ery cell of G(V, k) having fixed dimension at most 2m is contained in the finite subcomplex G(V(m+k)p, k) of G(V, k). 7. COMPLEX GRASSMANNIAN MANIFOLDS 85 Proof. Observe that the cell of G(V, k) is contained in the s* *ub- complex G(Vn, k) of G(V, k) if and only if ak + k n. Assume that the cell of G(V, k) is added after G(V(m+k)p, k) so that ak + k > (m + k)p. Since there are only p distinct irreducible complex representations of G, irred* *ucible representations isomorphic to OEak+k must have appeared at least m + k times pr* *ior to OEak+k in our standard block ordering of the irreducible summands of V . At * *most k - 1 of these appearances can come from the set {OEai+i|1 i < k}. Hence, the irreducible trivial complex G-representation must appear at least m + 1 times in the portion of the sum Mk ai+i-1M W = OE-1ai+iOEj i=1 j=1 j62{a1+1,...,ai-1+i-1} coming from i = k. Thus, the fixed dimension of the representation W associated to the cell is at least 2(m + 1). Since, for any positive integer m, there are only finitely many cells of G(V* *, k) having fixed dimension below 2m, Theorem 2.5 applies to G(V, k). Corollary 7.2. Let G = Z=p. If V is a finite or countably infinite dimen- sional complex G-representation, then the RO(G)-graded Mackey functor-valued equivariant ordinary homology HG*(G(V, k); A)of G(V, k) is free over H*. Part 2 Observations about RO(G)-graded equivariant ordinary homology CHAPTER 8 The computation of HS* for arbitrary S Here, we compute the RO(G)-graded equivariant ordinary homology HS* of a point with an arbitrary Mackey functor S as coefficients. The approach taken is a variant of that employed in [5] and [11]. Our basic tools are derived from unpublished work of Stong. In order to describe HS*, we need to introduce some new maps and a collection of new Mackey functors. The restriction map æ : S(G=G) ___//S(G=e)and transfer map ø : S(G=e)___//S(G=G)of the Mackey functor S induce maps ~æ: S(G=G) ___//S(G=e)G and ~ø: S(G=e)=G __//_S(G=G). They also induce the maps of Mackey functors bæS: S __//_SG=e and øbS: SG=e___//S described in Section 1.1. The cokernel of bæSand the kernel of bøSplay an impor* *tant role in our computations, and so are denoted C(S) and K(S), respectively. For any abelian group BPcarrying a G-action, there is a trace map tr: B __//* *_B given on x 2 B by tr(x) = g2G gx. This map induces maps B=G __//_B and B __//_BG, which we also denote by tr. The Mackey functors L and R introduced in Section 1.1 are special cases of two general constructions which produce Mac* *key functors L(B) and R(B) from any Z[G]-module B. Diagramatically, these Mackey functors are given by: L(B) R(B) B=G_____ BG ____________________UU_________________________________* *________________________VV________________________________ tr_________________________________ß_____________________* *____________i__________________________________tr_____________@ _~~_____________________________~~________________ _B_YY_________________ B_YY_______________ _______________________________________________________* *_________________________________________________ _______________________________________________________* *________________________________________________________ ` ` Here, ß and i denote the projection onto the orbit module and the inclusion of the fixed point submodule, respectively, and ` denotes the action of G on B. In [14], these constructions are denoted LeB and JeB, respectively. There it is sh* *own that these functors are the left and right adjoint, respectively, to the functo* *r from the category M of Mackey functors to the category of Z[G]-modules which sends a Mackey functor S to S(G=e). In the remainder of this chapter, we often need to consider the Mackey funct* *ors L(S(G=e)) and R(S(G=e)) derived from a Mackey functor S. For compactness of notation, we denote these by L(S) and R(S), respectively. Observe that each of * *S, L(S), and R(S) has value S(G=e) at G=e. The counit and unit of the appropriate 88 8. THE COMPUTATION OF HS*FOR ARBITRARY S 89 adjunctions give us canonical maps ~fflS: L(S)___//S and ~jS: S __//_R(S). These two maps are the unique maps whose value at G=e is the identity map. If p = 2, then there is a sign action of G on Z which sends 1 to -1. Denote Z with this action by Z- . The Mackey functors L- and R- introduced in Section 1.1 are just L(Z- ) and B(Z- ), respectively. Just as L(B) and R(B) are generalizat* *ions of L and R, the Mackey functors L(B Z- ) and R(B Z- ), which we denote by L- (B) and R- (B), generalize L- and R- . For any Z=2-Mackey functor S, L-(S(G=e)) and R- (S(G=e)) are denoted L- (S) and R- (S), respectively. Typically, the Mackey functor HS! depends only on the integers |! |and |!G |. Thus, HS* is most easily visualized by plotting it out in the plane. The one ca* *se in which the integers |! |and |!G |don't suffice to determine HS! is that in wh* *ich ! 2 RO0(G); that is, when |! |= |!G |= 0. For such an !, the value of d! is the only additional bit of information needed to determine HS!. Theorem 8.1. Let ! 2 RO(G). Then, (i)if |!G |= 0, 8 > 0, HS! = >A[d!] S if |! |= 0, :Coker bøS: S ___// G=e S if |! |< 0. (ii)if |!G |> 0 and |! |= 0, 8 >R- (S) if |!G | 3 and odd, :K(S) if |!G |= 1. (iii)if |!G |< 0 and |! |= 0, 8 >L- (S) if|!G | -3 and odd, :C(S) if|!G |= -1. (iv)if |!G |and |! |are both positive or both negative, then HS!= 0. (v)if |!G |> 0 and |! |< 0, 8 >Ker ~jSO ~fflS: L(S)___//R(S)if|!G | 3 and odd, :Ker ~fflS: L(S)___//S if |!G |= 1. (vi)if |!G |< 0 and |! |> 0, 8 >Coker ~jSO ~fflS: L(S)___//R(S)if|!G |-3 and odd, :Coker ~jS: S __//_R(S) if |!G |= -1. Remark 8.2. (a) If |! |6= 0, then HS!vanishes at G=e because the nonequivari- ant spectrum underlying the equivariant Eilenberg-Mac Lane spectrum representing homology with S-coefficients is the nonequivariant Eilenberg-Mac Lane spectrum associated to S(G=e). 90 8. THE COMPUTATION OF HS*FOR ARBITRARY S (b) In parts (ii)and (iii)of Theorem 8.1, the second and third cases occur on* *ly if p = 2. (c) In the proof of this theorem, we show that Ker j~SO ~fflS: L(S)___//R(S) can also be described as Ker bæL(S): L(S)___//L(S)G=e. Similarly, the two maps ~jSO ~fflS: L(S)___//R(S)and bøR(S): R(S)G=e___//R(S)have isomorphic cokernels. (d) For p = 2, alternative descriptions for the values of HS! described in pa* *rts (v)and (vi)of the theorem are given in Proposition 8.9 and Remark 8.10. It is fairly easy to see that these alternative descriptions actually yield the same * *Mackey functors. Six of the values of HS*appearing in the theorem above vanish for some common choices of S. Lemma 8.3. (a) If the restriction map æ : S(G=G) ___//S(G=e)is a monomor- phism, then HS!= 0 for those ! 2 RO(G) such that |!G |= 0 and |! |> 0. (b) If the transfer ø : S(G=e)___//S(G=G)is an epimorphism, then HS!= 0 for those ! 2 RO(G) such that |!G |= 0 and |! |< 0. (c) If the transfer ø : S(G=e)___//S(G=G)is a monomorphism, then HS!= 0 for those ! 2 RO(G) such that |!G |= 1 and |! |< 0. (d) If the map ~æ: S(G=G) ___//S(G=e)Gis an epimorphism, then HS! = 0 for those ! 2 RO(G) such that |!G |= -1 and |! |> 0. (e) If the restriction map æ : L(S)(G=G) ___//L(S)(G=e)is a monomorphism, then HS!= 0 for those ! 2 RO(G) such that either |!G |is odd and |! |< 0 or |!G* * | is even and |! |> 0. Proof. Parts (a) and (b) follow easily from the fact that, at G=G, the maps bæS: S __//_SG=eand bøS: SG=e __//_S can be identified with the restriction map æ : S(G=G) ___//S(G=e)and the transfer map ø : S(G=e)___//S(G=G), respectively. Part (c) follows from the fact that, at G=G, the map ~fflS: L(S)___//S is ju* *st the map ~ø: S(G=e)=G __//_S(G=G). If the map ø : S(G=e)___//S(G=G)is a monomor- phism, then G must act trivially on S(G=e). Thus, S(G=e)=G = S(G=e), and ~ø= ø. It follows that Ker ~fflS: L(S)___//S= 0. Part (d) follows directly from the fa* *ct that, at G=G, the map ~jS: S __//_R(S)is just the map ~æ: S(G=G) ___//S(G=e)G. If ! 2 RO(G) satisfies the conditions indicated in part (e), then either HS!* *= 0 by part (iv)of Theorem 8.1 or HS! is the kernel of a map of Mackey functors whose domain is L(S) and whose value at G=e is the identity map of S(G=e). The asserted vanishing therefore follows from the observation that a map f : M __//* *_N of Mackey functors is a monomorphism if the maps æ : M(G=G) __//_M(G=e)and f(G=e): M(G=e) ___//N(G=e)are both monomorphisms. The remainder of this chapter is devoted to the proof of Theorem 8.1. For p 6= 2, our argument is essentially that given in [5]. For p = 2, it is an exte* *nsion of the computation of H* given in [11]. Throughout our proof, , denotes a nontrivi* *al irreducible complex representation of G, and i denotes the real one-dimensional* * sign representation of Z=2. The key tools for our computation of HS*are the equivari* *ant cofibre sequences ', ß0, (G=e)+ ____//S,+ ____// (G=e)+ , ffl, , ß, S0 ___//_S ____// S,+ , 8. THE COMPUTATION OF HS*FOR ARBITRARY S 91 and ffli i ßi S0 ___//_S ____// (G=e)+ , the last of which applies only for p = 2. Remark 8.4. Several observations about these cofibre sequences are needed in our computations. (a) The next map on the right in our first cofibre sequence is of the form (G=e)+ _1-g_//_ (G=e)+ for some generator g of G. (b) The next map on the right in our second cofibre sequence is the suspension of the collapse map OE : S,+ __//_S0which sends all of S, to the non-basepoint * *of S0. (c) The composite ', OE 0 (G=e)+ ____//S,+ ___//_S is just the geometric restriction map (G=e)+ __//_S0which collapses all of G=e * *to the non-basepoint of S0. In cohomology, this map induces the restriction map. In homology it induces the transfer since its Spanier-Whitehead dual is the geomet* *ric transfer map (see Corollary III.5.2 of [17]). (d) The composite ß, ß0, 2 S, ____// S,+ _____// (G=e)+ is related to the geometric transfer map ø,: S, __//_ ,(G=e)+by the commuting diagram ß, S, __________// S,+ ø,|| |ß0,| fflffl|~ fflffl| ,(G=e)+ _____// 2(G=e)+ . Here, ~ is a special case of the G-homeomorphism ~ : (G=e)+ ^ X___//(G=e)+ ^ X, available for any G-space X, which relates (G=e)+ ^ X with G acting diagonally * *to (G=e)+ ^X with G acting only on (G=e)+ . This map is given by ~(g, x) = (g, g-1* *x). (e) The commuting diagram øi i Si _____//H (G=e)+ HH HHH |~ ßiHHH$$Hfflffl|| (G=e)+ identifies the map ßi of our third cofibre sequence with the geometric transfer* * map øi: Si___// i(G=e)+. Our next two results generalize Lemma A.1 of [11]. Proposition 8.5. Let ! be an element of RO(G). Then 92 8. THE COMPUTATION OF HS*FOR ARBITRARY S (a) 8 >: !-1 0 otherwise. Moreover, if |! |= 0, then the diagram ~= L(HS!)_____//eHG!(S,+ ; S) LLL LLL |OE* ~fflHS!LLLL| L%%fflffl| HS! commutes. (b) 8 >: G 1-! 0 otherwise. Moreover, if |! |= 0, then the diagram He!G(S0; S) O *| OOOO~jO OE | OO fflffl|~ OOO'' eH!G(S,+ ;_S)=_//R(He!G(S0; S)) commutes. Proof. Geometrically, the key to this proof is the observation in Remark 8.4(a) about the next map on the right in our first cofibre sequence. Algebraic* *ally, the key is the fact that, since G is cyclic, the Mackey functors L(M) and R(M) and the canonical maps fflM: L(M) ___//M and jM : M __//_R(M)are specified by the commuting diagram M G | GGGbæMG jM | GG fflfflGG##| 1-g 0 ____//R(M)____//MG=e____//MG=e___//_L(M)___//_0 (8.1) GG GGG |fflM bøM GG##Gfflffl|| M in which the long row is exact. The middle map in this row is the difference of* * the self maps of MG=e induced by the identity map and the multiplication by g map on G=e. The homology and cohomology long exact sequences associated to our first cofibre sequence can be compared to the row in this diagram via the isomorphisms eHG!((G=e)+; S)~=(HS!)G=e and eH!G((G=e)+; S)~=(He!G(S0; S))G=e. This compari- son yields the short exact sequences 0 __//_L(HS!) __//_eHG!(S,+ ;_S)//_R(HS!-1) __//_0 8. THE COMPUTATION OF HS*FOR ARBITRARY S 93 and 0 __//_L(HS1-!) __//_eH!G(S,+ ;_S)//_R(HS-!) __//_0. For any Mackey functor M, L(M) and R(M) are completely determined by the G- module M(G=e). Since HS!vanishes at G=e unless |! |= 0, it follows immediately that eHG!(S,+ ; S)and eH!G(S,+ ; S)vanish unless |! |is 0 or 1. If |! |is 0 or * *1, the asserted values of eHG!(S,+ ; S)and eH!G(S,+ ; S)follow immediately from the sh* *ort exact sequences. The commutativity of the two diagrams follows directly from the characterization of the maps bæMand bøMgiven by diagram (8.1). Explicit values for HeG!(S,+ ; S)and He!G(S,+ ; S)can be obtained by looking more closely at HS!(G=e). Corollary 8.6. For ! 2 RO(G), 8 >>>L(S) if |! |= 0 and |!G |is even, >>> G >> >>>R- (S)if |! |= 1 and |!G |is even, :0 otherwise. and 8 >>>R(S) if |! |= 0 and |!G |is even, >>> G >> >>>L- (S)if |! |= 1 and |!G |is even, :0 otherwise. Proof. If |! |= 0, then the abelian group HS!(G=e) is just S(G=e). However, the actions of G on HS!(G=e) and S(G=e) need not be the same due to the action * *of G on the sphere S! . If |!G |is even, then the two G-actions agree since the ac* *tion of G on S! is homologically trivial. However, if |!G |is odd, then p = 2 and ! con* *tains an odd number of copies of the real sign representation i. In this case, HS!(G=* *e) is isomorphic as a G-module to S(G=e) Z- . This description of the G-action on HS!(G=e), together with the descriptions of eHG!(S,+ ; S)and eH!G(S,+ ; S)given* * in Proposition 8.5, suffices to complete the proof. The proposition above, coupled with our second and third cofibre sequences, yields the following variant of Lemma A.2 of [11]. Here, and elsewhere, the hom* *ol- ogy and cohomology maps induced by the maps ffl,: S0___//S,and ffli: S0___//Si are also denoted by ffl, and ffli. Lemma 8.7. Let ! be an element of RO(G). (a) The map ffl,: HS!+,___//HS!is 8 >:mono if|! |6= -1, -2, iso if|! |6= 0, -1, -2. 94 8. THE COMPUTATION OF HS*FOR ARBITRARY S (b) If p = 2, then the map ffli: HS!+i___//HS!is 8 >:mono if |! |6= -1, iso if |! |6= 0, -1. (c) If S(G=e) = 0, then the map ffl,: HS!+,___//HS!and, for p = 2, the map ffli: HS!+i___//HS!are isomorphisms for all ! 2 RO(G). This result plus the dimension axiom gives the vanishing of HS!for any ! whi* *ch plots in either the first or third quadrant, but not on the coordinate axis (se* *e also Lemma A.3 of [11]). Lemma 8.8. Let ! be an element of RO(G). If |! |and |!G |are either both positive or both negative, then HS!= 0. Proof. Consider the special case in which ! = n+,1-,2, where n is a positive integer and ,1, ,2 are irreducible complex representations of G. Then HSn= 0 by the dimension axiom. The pair of isomorphisms ffl,2 ffl,1 HSn+,1-,2o~=oHSn+,1_~=//_HSn= 0 then gives the vanishing of HS!. The general case follows by the obvious extens* *ion of this argument. Lemma 8.7 indicates that all of HS*can be determined from its values at those ! 2 RO(G) such that -2 |! | 2. The following result, which generalizes Lemmas A.4, A.8, and A.9 of [11], further reduces the computation of HS*down to that of its values at those ! 2 RO(G) such that |! |= 0. Proposition 8.9. Let ! be an element of RO(G). (a) If |! |= -2, then i j HS! = Coker bø: HS!+,G=e___//HS!+,. (b) If |! |= 2, then i j HS! = Ker bæ: HS!-,___//HS!-,G=e. (c) If |! |= -1 and |!G |> 0, then i j HS! = Ker ~fflHS!+,-1: L(HS!+,-1)__//_HS!+,-1. (d) If |! |= 1 and |!G |< 0, then i j HS! = Coker ~jHS!-,+1: HS!-,+1__//_R(HS!-,+1). (e) If p = 2 and |! |= -1, then i j HS! = Coker bø: HS!+iG=e __//_HS!+i. (f) If p = 2 and |! |= 1, then i j HS! = Ker bæ: HS!-i___//HS!-iG=e. 8. THE COMPUTATION OF HS*FOR ARBITRARY S 95 Proof. For part (a), assume |! |= -2 and consider the diagram eHG!+,((G=e)+; S) PPPP (',)*|| PbøPPPP fflffl| PP'' HeG!+,(S,+ ; S)OE*//_eHG!+,(S0;_S)//_eHG!+,(S,;_S)//_eHG!+,( S,+=;0S) in which the exact row comes from our second cofibre sequence. The Mackey functor on the right vanishes by Proposition 8.5 since |! + , - 1|= -1. Thus, HS! ~=HeG!+,(S,; S)is the cokernel of OE*. However, (',)* is the surjective map displaying eHG!+,(S,+ ;aS)s the quotient L(HS!+,) of eHG!+,((G=e)+;.S)Thus, HS! is also the cokernel of bø. For part (b), assume |! |= 2 and consider the diagram OE* ,-! 0 = eH,-!G( S,+ ; S)_//_eH,-!G(S,;_S)//_eH,-!G(S0;_S)//_PeHG(S,+ ; S) PPPPbæP | * PPP (',)| PPP'' fflffl| He,-!G((G=e)+; S) in which the exact row comes from our second cofibre sequence. Here, the left- most Mackey functor vanishes by Proposition 8.5 so that HS!~=He,-!G(S,; S)is the kernel of OE*. In this case, (',)* is the injective map displaying eH,-!G(S,+ ;* *aS)s the subobject R(HS!-,) of eH,-!G((G=e)+; S). Thus, HS!is the kernel of bæ. For part (c), assume |! |= -1 and |!G |> 0, and consider the exact sequence ( OE)* 0 = eHG!+,(S0;_S)_//_eHG!+,(S,;_S)//_eHG!+,( S,+_;_S)//_eHG!+,(S1; S) derived from our second cofibre sequence. Here, the left-most Mackey functor va* *n- ishes by Lemma 8.8. Thus, HS!~=HeG!+,(S,; S)is the kernel of ( OE)*. Proposition 8.5(a) provides the identification of ( OE)* with ~fflHS!+,-1: L(HS!+,-1)__//_H* *S!+,-1. Part (d) is handled in much the same way as part (c) by examining the exact sequence * eH,-!G(S1; S)(_OE)//eH,-!G( S,+_;/S)/_eH,-!G(S,;_S)//eH,-!G(S0;=S)0 and using Proposition 8.5(b) to identify the map ( OE)*. Now assume that p = 2 and that |! |= -1. Part (e) follows immediately from the exact sequence eH-!G( i(G=e)+ ; S)bø//_eH-!G(Si;_S)//_eH-!G(S0;_S)//_eH-!G((G=e)+;=S)0 derived from our third cofibre sequence. Notice that we have used the identific* *ation of the map ßi in that sequence with the transfer. Part (f) follows similarly fr* *om the exact sequence bæ i-! 0 = eHi-!G( (G=e)+ ;_S)//_eHi-!G(Si;_S)//_eHi-!G(S0;/S)/_eHG((G=e)+; S) and the isomorphism HS!~=Hei-!G(Si; S). 96 8. THE COMPUTATION OF HS*FOR ARBITRARY S Remark 8.10. If p = 2, then any one of the three pairs (a) and (b), (c) and (d), or (e) and (f) of parts of this proposition can be used to derive all the * *values of HS!from those for which |! |= 0. However, if p is odd, then all four of part* *s (a), (b), (c) and (d) are needed since, for any ! 2 RO(G), |! |and |!G |are either b* *oth even or both odd. Our next two results, which generalize Lemmas A.5 and A.6 of [11], give the values of HS!for those ! 2 RO(G) such that |! |= 0. Proposition 8.11. Let ! be an element of RO(G) such that |! |= 0. (a) If |!G |< -1, then HS!~=HeG!(S,+ ; S). (b) If |!G |> 1, then HS!~=He-!G(S,+ ; S). (c) If p = 2 and |!G |= -1, then HS!~=C(S). (d) If p = 2 and |!G |= 1, then HS!~=K(S). Proof. For part (a), consider the exact sequence eHG!(S,-1; S)__//eHG!(S,+ ;_S)//_eHG!(S0;_S)//_eHG!(S,; S) derived from our second cofibre sequence. The first and last terms in this sequ* *ence are isomorphic to HS!-,+1and HS!-,, respectively. These vanish by Lemma 8.8. For part (b), consider the analogous cohomology exact sequence He-!G(S,; S)___//eH-!G(S0;_S)_//eH-!G(S,+ ;_S)//_eH-!G(S,-1;.S) Here, the first and last terms are isomorphic to HS!+, and HS!+,-1, respectivel* *y. These also vanish by Lemma 8.8. In part (c), ! must be i - 1. Our third cofibre sequence yields the cohomolo* *gy exact sequence He0G(S0; S)__//_eH0G((G=e)+;_S)//_eH1G(Si;_S)//_eH1G(S0;.S) By the dimension axiom, the last term in this sequence is zero and the first two are S and SG=e, respectively. The third term, which is isomorphic to HSi-1, must therefore also be isomorphic to C(S). In part (d), ! must be 1 - i. Our third cofibre sequence yields the homology exact sequence eHG1(S0; S)__//eHG1(Si;_S)_//eHG0((G=e)+;_S)_//eHG0(S0;.S) By the dimension axiom, the first term in this sequence is zero and the last two are SG=e and S, respectively. The second term, which is isomorphic to HS1-i, mu* *st therefore also be isomorphic to K(S). Proposition 8.12. Let X be a G-space, Æ 2 RO(G), and ! 2 RO0(G). Then the product map H! eHGÆ(X; S)_//_eHG!+Æ(X; S) is an isomorphism. In particular, eHG!(X; S)~=H! HeG0(X; S). Proof. Apply Theorem 3.2 of [4] to the G-spectrum S-! , which is obviously invertible. Our assumption that |! |= |!G |= 0 is equivalent to the assertion t* *hat the function d of that theorem (which is not our function d) vanishes on S-! . 8. THE COMPUTATION OF HS*FOR ARBITRARY S 97 Thus, by that theorem, S-! is a Künneth object. Proposition 1.2 of [4] therefore indicates that S-! is a retract of a wedge of copies of S0. The multiplication map eHG0(S0; A)HeGÆ(X; S)__//eHGÆ(S0 ^ X;~S)=eHGÆ(X; S) is just the unit isomorphism for the action of H* on eHG*(X; S). It follows eas* *ily that the map HeG0(S0; A) eHGÆ(X; S)_//_eHGÆ(S0 ^ X; S) remains an isomorphism if S0 is replaced by either a wedge of copies of S0 or a retract of a wedge of copies of S0. The obvious identification of eHG0(S-! ; A)* *with H! suffices to complete the proof. In the proof of our main freeness theorem, we need to know that an analog of this proposition holds for any module over H* rather than just those modules arising in homology. Corollary 8.13. Let N be a module over H*, Æ 2 RO(G), and ! 2 RO0(G). Then the action map !,Æ: H! NÆ __//_N!+Æ is an isomorphism. Proof. In the commuting diagram H! H-! N!+Æ ~~1=//_H0 N!+Æ 1 -!,!+Æ|| ~=|0,!+Æ| fflffl| fflffl| H! NÆ _____!,Æ____//N!+Æ the top horizontal map is an isomorphism by the proposition, and the right vert* *ical map is the unit isomorphism for the action of H* on N. Thus, the map 1 -!,!+Æ is a monomorphism. Similarly, the top horizontal and right vertical maps in the commuting diagram H-! H! NÆ ~~1=//_H0 NÆ 1 !,Æ|| ~=|0,Æ| fflffl| fflffl| H-! N!+Æ __-!,!+Æ_//NÆ are isomorphisms. Thus, the map -!,!+Æ is an epimorphism. But then the right exactness of box products implies that the map 1 -!,!+Æ: H! H-! N!+Æ ___//H! NÆ must also be an epimorphism. It is therefore an isomorphism. The first commuting square then gives that !,Æis an isomorphism. Remark 8.14. To use Proposition 8.12 in the proof of Theorem 8.1, we need to identify H! with A[d!]. This is done in detail in Lemma A.12 of [11]. Howeve* *r, a quick summary of that argument is easily given. The portions . ._.__//eHG!(S,+ ;_A)//_eHG!(S0;fA)fl,//_eHG!(S,;/A)/_. . . 98 8. THE COMPUTATION OF HS*FOR ARBITRARY S and . ._._//eH-!G(S,;_A)_//eH-!G(S0; A)_//_eH-!G(S,+ ;_A)//_. . . of the homology and cohomology long exact sequences coming from our second cofibre sequence reduce to the short exact sequences ffl, 0 ____//L___//H!____//__//_0 and 0____//___//H!____//R___//0, respectively. By Lemma 12.2(c), the only common solutions to these two extension problems are of the form A[d], for some integer d prime to p. The appropriate d* * is determined by the fact that there is an element of A[d](G=G) which restricts to* * d in A[d](G=e) and maps under ffl, to a generator of (G=G) in the first short * *exact sequence above. Now consider the special case in which ! = ,-,0, where ,0is another irreduci* *ble complex G-representation. Both , and ,0are copies of the complex plane on which G acts by multiplication by pthroots of unity. Thus, there is a integer m prime* * to p such that the complex power map taking z to zm is an equivariant0map from , to * *,0. This map extends to an equivariant map f from S, to S, , which may be regarded as an element of H! (G=G) via the Hurewicz map. It is easy to see that ffl,(f) * *is a generator of . Clearly the restriction of f to H! (G=e) is just its degre* *e m, which is d! by our definition of d!. Since the element ø of A[d](G=G) restricts* * to p in A[d](G=e) and maps to zero under ffl,, this argument only determines d modulo p. Moreover, since we haven't worried about the orientation of , we have rea* *lly determined d only up to sign. However, A[d] ~=A[d0] if and only if d d0 mod * *p, so this level of imprecision is irrelevant. The argument for a general element * *! of RO0(G) is just the obvious extension of this argument. The following consequence of Corollary 8.13 is not used in our computation of HS*, but is needed for the proof of our main freeness theorem. Corollary 8.15. Let N be a module over H* . If there is an element Æ of RO(G) and an integer d prime to p such that NÆ ~=A[d], then there is an element Æ0 of RO(G) such that NÆ0~=A, |Æ0|= |Æ|, and |(Æ0)G|= |ÆG.| Proof. Select an element ! of RO0(G) such that d!d 1 mod p. The map !,Æ: H! NÆ __//_N!+Æ is an isomorphism by Corollary 8.13. By Remark 8.14, H! ~=A[d!]. By Table 1.1, A[d!] A[d] ~=A[d!d]; and, by Lemma 1.1, A[d!d] ~=A[1] = A. Thus, Æ0= ! + Æ is the desired element of RO(G). Completing the proof of this chapter's main theorem requires nothing more than tying the preceding results together properly. Proof of Theorem 8.1. First note that parts (ii)and (iii)follow immedi- ately from Proposition 8.11 and Corollary 8.6. Also note that part (iv)is just* * a restatement of Lemma 8.8. Similarly, the middle of the three values described in part (i)comes directly from Proposition 8.12 and Remark 8.14. 8. THE COMPUTATION OF HS*FOR ARBITRARY S 99 To verify the other two values given in part (i), first observe that HS-,= Coker bøS: SG=e___//S and HS,= Ker bæS: S __//_SG=e by parts (a) and (b) of Proposition 8.9. If p = 2, then HS-i= Coker bøS: SG=e___//S and HSi= Ker bæS: S __//_SG=e by parts (e) and (f) of the same proposition. For p = 2, Lemma 8.7 obviously su* *ffices to complete the proof since the only elements ! of RO(G) satisfying |!G |= 0 and |! |6= 0 are of the form mi for some integer m. If p 6= 2, such an ! is a line* *ar combination ,1+ ,2+ . .+.,m - j1- j2- . .-.jn of nontrivial irreducible complex G-representations in which m 6= n. For the special case in which m = 2 and n = * *1, the pair of isomorphisms fflj1 ffl,2 HS,1+,2-j1~=oHS,1+,2o_~=//_HS,1 suffices to complete the proof. Longer chains of isomorphisms of the same sort suffice to handle any case in which m > n. A very similar chain of isomorphisms argument handles the cases for which m < n. For part (v), observe that Proposition 8.9(a) and part (ii)of Theorem 8.1 indicate that the first value should be Coker bøR(S): R(S)G=e___//R(S). The nat- urality of the maps bøimplies that the diagram bøL(S) L(S)G=e _____//L(S) (~jSO~fflS)G=e|| |~jSO~fflS| fflffl|bøR(fflffl|S) R(S)G=e ____//_R(S) commutes. The left vertical map in this diagram is an isomorphism since ~jSO ~f* *flS is an isomorphism at G=e. The top horizontal map in the diagram is an epimor- phism since the transfer map of L(S) is an epimorphism. Thus, the maps bøR(S) and ~jSO ~fflShave isomorphic cokernels. The second value in part should (v)be Ker ~fflR(S): L(R(S) )__//R(S)by Proposition 8.9(c) and part (ii)of this theo- rem. Note, however, that L(R(S) ) ~=L(S) since R(S)(G=e) and S(G=e) are equal. Moreover, under this isomorphism, the map ~fflR(S)is identified with the compos* *ite ~jSO ~fflSsince each of these maps is uniquely determined by the fact that it i* *s the identity map at G=e. For the third value in part (v), first consider the special case in which ! * *= 1-,. Then, from Proposition 8.9(c), we have that HS1-,= Ker ~fflS: L(S)___//S. If p = 2, then Lemma 8.7 suffices to complete the proof since the only elements ! satisfying the appropriate conditions are of the form 1 - mi, where m 2. Here, we use the fact that , = 2i in RO(G). For p 6= 2, ! must be of the form 100 8. THE COMPUTATION OF HS*FOR ARBITRARY S 1 + j1+ j2+ . .+.jm - ,1- ,2- . .-.,n, with 0 m < n. For the special case in which m = 1 and n = 2, the pair of isomorphisms fflj1 ffl,2 HS1+j1-,1-,2~=_//HS1-,1-,2~=HS1-,1oo_ completes the proof. Longer chains of isomorphisms of the same sort suffice to handle the general case. For part (vi), note that Proposition 8.9(b) and part (iii)of this theorem gi* *ve that the first value should be Ker bæL(S): L(S)___//L(S)G=e. The naturality of * *the maps bæindicates that the diagram bæL(S) L(S) ____//_L(S)G=e ~jSO~fflS|| |(~jSO~fflS)G=e| fflffl|bæR(fflffl|S) R(S) ____//_R(S)G=e commutes. The right vertical map in this diagram is an isomorphism since ~jSO~f* *flSis an isomorphism at G=e. Moreover, the bottom horizontal map is a monomorphism since the restriction map of R(S) is a monomorphism. Thus, the maps bæL(S)and ~jSO ~fflShave isomorphic kernels. Proposition 8.9(d) and part (ii)of this theo* *rem indicate that the second value should be Coker ~jL(S): L(S)___//R(L(S)). How- ever, R(L(S)) and R(S) are isomorphic since L(S)(G=e) and S(G=e) are equal. As in part (v), this isomorphism identifies the map ~jL(S)with the composite ~jSO * *~fflS. For the third value, look first at the special case in which ! = , - 1. Proposi* *tion 8.9(d) asserts that HS,-1= Coker ~jS: S __//_R(S). The general case follows from this special case by a chain of isomorphisms argu* *ment essentially identical to that used for the third value in part (v). CHAPTER 9 Examples of HS* In this chapter, the results of the previous chapter are applied to the spec* *ial cases S = , R, and L needed in our computations. By comparing HR*and HL*, we establish a connection between these two H*-modules which plays an important role in the computations carried out in Chapter 5. We begin with since this * *is especially simple. Proposition 9.1. Let B be an abelian group. Then ( G H!= if |! |= 0, 0 otherwise. Proof. The vanishing of H!for |!G |6= 0 follows from the fact that, for * *any Mackey functor S, every value of HS* off of the vertical axis is constructed fr* *om S(G=e) via some additive functor. The values on the vertical axis follow from T* *able 1.1 and part (i)of Theorem 8.1. The values of HR* and HL*depend critically on whether p = 2. The case p 6= 2 is simpler, and so is treated first. Proposition 9.2. Assume that p 6= 2. Then 8 >>> R if |! |= 0 and |!G | 0, >>> G >>< L if8|! |= 0 and |! |< 0, ><|! |< 0, |!G | 0, and |!G |is even HR! = > >>>if> or >>> : |! |> 0, |!G | -3, and |!G |is odd, >: 0 otherwise. and 8 >>> R if |! |= 0 and |!G |> 0, >>> G >>< L if8|! |= 0 and |! | 0, ><|! |< 0, |!G |> 0, and |!G |is even HL! = > >>>if> or >>> : |! |> 0, |!G | -1, and |!G |is odd, >: 0 otherwise. Proof. The values of HR* and HL* at the origin follow from Table 1.1. Both R and L satisfy the hypotheses of parts (a), (c) and (e) of Lemma 8.3. Moreover, L satisfies the hypotheses of part (b) of that lemma, and R satisfies the hypot* *heses of part (d). These observations give a number of vanishing results for HR*and H* *L*. Most of the still undetermined values of HR* and HL* follow from the observation 101 102 9. EXAMPLES OF HS* that, if S is either R or L, then the map ~jSO ~fflS: L(S)___//R(S)is just the * *canonical map : L __//_R. This map is a monomorphism with cokernel . The map ~jL: L __//_R(L)can also be identified with . This gives the one remaining val* *ue of HL*. Since R = R(A), Remark 8.2(c) implies that the map bøR: RG=e __//_R has the same cokernel as . The remaining uncomputed value of HR* follows from this. The values of HR*and HL*are best visualized by plotting them in the plane, as in Figures 9.1 and 9.2 below. Observe from these figures that the plot of HL* can be obtained simply by shifting the plot of HR* two units to the right. One way to say this is that, f* *or any nontrivial irreducible complex G-representation ,, HL*and 2-,HR* are isomorphi* *c, at least as RO(G)-graded Mackey functors. In fact, a much stronger result holds. |! | V .. . .. . 0 . .. . .<.Z=p> 0 . .<.Z=p> . . . L L R R R . .>.|!G | . . . . . . .. . . . .. .. Figure 9.1. HR* for p odd Corollary 9.3. Assume that p 6= 2. Let , be any nontrivial irreducible com- plex G-representation, and let HA, HL, and HR be the equivariant Eilenberg- Mac Lane spectra representing equivariant ordinary homology with A, L, and R coefficients, respectively. Then HL and 2-,HR are equivalent in the equivariant stable category as module spectra over HA. Thus, HL*and 2-,HR* are isomorphic H*-modules. 9. EXAMPLES OF HS* 103 Proof. Proposition 9.2 asserts that the equivariant stable homotopy "groups" ßGn(HL) and ßGn( 2-,HR) are isomorphic Mackey functors for any integer n. Thus, 2-,HR is an equivariant Eilenberg-Mac Lane spectrum with ßG0( 2-,HR) = L. The uniqueness of equivariant Eilenberg-Mac Lane spectra, as module spectra over HA, follows easily from Proposition 5.4 of [14]. The claim about HL*and 2-,HR* is an obvious consequence of this. |! | V .. . .. . 0 . .. . .<.Z=p> 0 . .<.Z=p> . . . L L L R R . .>.|!G | 0 . . . 0 . . . .. . . . .. .. Figure 9.2. HL* for p odd To compute HS*for p = 2, we must first compute the Mackey functors C(S) = Coker(bæS: S __//_SG=e) and K(S) = Ker(bøS: SG=e___//S) introduced at the beginning of Chapter 8. The following result identifies these Mackey functors in some important special cases. Lemma 9.4. Let p = 2. (a) If B is an abelian group with a Z=2-action, then there are natural isomor- phisms C(L(B) ) ~=L- (B) and K(R(B) ) ~=R- (B). (b) The canonical map ~jA: A __//_R induces an isomorphism C(A) __//_C(R). Moreover, C(R) ~=R- . 104 9. EXAMPLES OF HS* (c) The canonical maps ~fflA: L __//_A and ~jAO ~fflA: L __//_R induce isomor* *phisms K(L)___//K(A) and K(L) ___//K(R). Thus, K(A), K(L), and K(R) are all isomorphic to R- . Proof. We want to think of B as having two Z=2-actions _ its original action and an alternative action coming from the identification of B with B Z- , on wh* *ich Z=2 acts diagonally via the original action on B and the sign action on Z- . To reduce confusion, we denote B with this alternative action by B- . Observe that L- (B) and R- (B) are just L(B- ) and R(B- ). Denote the generaor of Z=2 by oe, and let Z=2 act on B B by oe(b, b0) = (oeb0, oeb). The usual diagonal and fol* *ding maps : B __//_B Band r : B B __//_B are Z=2-maps. Moreover, the "signed" diagonal and folding maps given by - (b) = (b, -b) and r- (b, b0) = b - b0. are Z=2-maps when regarded as maps - : B- ___//B Band r- : B B __//_B-. It is easy to check that the sequences r- 0____//B____//B B___//B-____//0 and - r 0____//B-____//B B___//B____//0 are short exact. The adjunctions defining L and R provide canonical maps L(B B) __//_L(B)G=e and R(B)G=e __//_R(B B). These two maps are isomorphisms, essentially because each is an isomorphism at G=e and all of the Mackey functors involved satisfy an appropriate form of indu* *ction with respect to the trivial subgroup. Under these isomorphisms, the maps L( ) a* *nd R(r) are identified with bæL(B): L(B) __//_L(B)G=eand bøR(B): R(B)G=e __//_R(B), respectively. Part (a) now follows from the two short exact sequences because L, being a left adjoint, preserves cokernels, and R, being a right adjoint, preser* *ves kernels. When evaluated at G=e, the exact sequence R _bæR//_RG=e___//C(R) __//_0 defining C(R) can be identified with the special case of the first of the two s* *hort exact sequences above in which B = Z. From this and the fact that the map bæRis an isomorphism at G=G, it follows easily that C(R) ~=R- . For the rest of part * *(b), consider the commuting diagram bæA A ____//_AG=e___//_C(A) | ~jA|| |(~jA)G=e||| fflffl|bæfflffl|R fflffl| R ____//_RG=e___//_C(R) in which the right vertical map is the one asserted to be an isomorphism. Its existence is ensured by the commutativity of the left square. Note that the le* *ft vertical arrow is an epimorphism and the middle vertical arrow is an isomorphis* *m. From this, it follows that the right vertical arrow is an isomorphism. 9. EXAMPLES OF HS* 105 For the first isomorphism in part (c), consider the commuting diagram K(L) _____//LG=ebøL_//L | | || | (~fflA)G=e| ~fflA| fflffl| fflffl|bøfflffl|A K(A) _____//AG=e____//A in which the left vertical map is the one asserted to be an isomorphism. Its ex* *istence is ensured by the commutativity of the right square. Again, the middle vertical arrow is an isomorphism. The left vertical arrow is therefore an isomorphism si* *nce the right vertical arrow is a monomorphism. The second isomorphism in part (c) follows from the analogous diagram in which A is replaced by R and the map ~ffl* *Ais replaced by the composite ~jAO ~fflA. The final assertion in part (c) follows f* *rom part (a) since R = R(Z) if Z is given trivial Z=2-action. The arguments used to prove Proposition 9.2 suffice to determine all the val* *ues of HR* and HL* for p = 2 except those for which |! |= 0 and |!G |= 1. Those missing values come from Theorem 8.1 and Lemma 9.4. Proposition 9.5. Assume that p = 2. Then 8 >>> R if |! |= 0, |!G | 0, and |!G |is even, >>> G G >>> R- if |! |= 0, |! | -1, and |! |is odd, >>> L if |! |= 0, |!G |< 0, and |!G |is even, >< G G HR! = > L- if |!8|= 0, |!G | -3, and |!G |is odd, >>> ><|!<|0, |! | 0, and |! |is even >>>if or >>> >: >>> |! |> 0, |!G | -3, and |!G |is odd, : 0 otherwise. and 8 >>> R if |! |= 0, |!G |> 0, and |!G |is even, >>> G G >>> R- if |! |= 0, |! |> 0, and |! |is odd, >>> L if |! |= 0, |!G | 0, and |!G |is even, >< G G HL!= > L- if |!8|= 0, |!G |< 0, and |!G |is odd, >>> ><|!<|0, |! |> 0, and |! |is even >>>if or >>> >: >>> |! |> 0, |!G | -1, and |!G |is odd, : 0 otherwise. As in the case where p 6= 2, The values of HR* and HL* for p = 2 are best visualized by plotting them in the plane. These plots are given in Figures 9.3 * *and 9.4 below. Note that, as in the p 6= 2 case, the plot for HL* can be obtained * *by shifting the plot of HR* two units to the right. This motivates the following r* *esult: Corollary 9.6. Assume that p = 2. Let i be the one-dimensional real sign representation of Z, and let HA, HL, H(L- ), HR, and H(R- ) be the equivariant Eilenberg-Mac Lane spectra representing equivariant ordinary homology with A, L, 106 9. EXAMPLES OF HS* L- , R, and R- coefficients, respectively. Then there are equivalences H(R- ) ' 1-iHR HL ' 2-2iHR H(L- ) ' 3-3iHR of module spectra over HA in the equivariant stable category. These equivalences yield isomorphisms HR-* ~= 1-iHR* HL* ~= 2-2iHR* HL-*~= 3-3iHR* of H*-modules. Proof. As in the case p 6= 2, these results are obtained by computing the stable homotopy Mackey functors ßGn( m-mi HR) for 1 m 3 and n 2 Z. From this, it follows that m-mi HR is an equivariant Eilenberg-Mac Lane spectrum of the indicated type if 1 m 3. 9. EXAMPLES OF HS* 107 |! | V .. . . . .. .. . .<.Z=2> 0 . .<.Z=2> 0 . . .L- L L- L R- R R- R R- R . .>.|!G | . . . . . . .. . . . .. .. Figure 9.3. HR* for p = 2 |! | V .. . . . .. .. . .<.Z=2> 0 . .<.Z=2> 0 . . .L- L L- L L- L R- R R- R . .>.|!G | 0 . . . 0 . . . .. . . . .. .. Figure 9.4. HL* for p = 2 CHAPTER 10 RO(G)-graded box products Here, we introduce the box product on the category of modules over H*. This construction lies at the heart of the weak form of the Universal Coefficient Th* *eorem invoked in Chapter 5. We begin by introducing the category M* of RO(G)-graded Mackey functors. This category is an obvious generalization of the category of * *Z- graded abelian groups. The basic properties of M* are therefore presented rath* *er tersely, with fuller commentary only at the points where its behavior is not wh* *at one might expect from the nonequivariant case. The homology H* of a point is a ring object in M* . Thus, from M* , we can construct the category H*-Mod of modules over H*. The bulk of this chapter is devoted to establishing the proper* *ties of this category which are needed for the proof of our weak Universal Coefficie* *nt Theorem. Since restricting to the case G = Z=p would save very little effort he* *re, G is assumed to be an arbitrary finite group throughout this chapter. Recall from [10], or Section V.9 of [17], that a Mackey functor C may be re- garded as an additive functor from a small additive category BG, called the Bur* *nside category, to the category Ab of abelian groups. The objects of the category BG * *are finite G-sets. Also recall that, if C is a Mackey functor and X is a finite G-* *set, then the Mackey functor CX is given on a G-set Y by CX (Y ) = C(X x Y ). This is a generalization of the MG=e construction introduced in Section 1.1. The category M of Mackey functors is a bicomplete symmetric monoidal closed abelian category which has enough projectives and injectives and satisfies AB5. Definition 10.1. (a)An RO(G)-graded Mackey functor M is a collection {Mff} of Mackey functors indexed on the set RO(G). A map f : M __//_N of RO(G)-graded Mackey functors is the obvious collection of maps of Mackey func- tors. The category of RO(G)-graded Mackey functors for the group G is denoted M* . (b) Let ff be an element of RO(G). The functor efffrom the category M* to the category M of Mackey functors sends the RO(G)-graded Mackey functor M to its value Mffat ff. The functor cfffrom M to M* sends a Mackey functor C to the RO(G)-graded Mackey functor whose values are C at ff and zero at all other elements of RO(G). (c) For ff an element of RO(G), the functor ff: M* ___//M* sends an RO(G)- graded Mackey functor M to the RO(G)-graded Mackey functor ffM given by ( ffM)fi= Mfi-ff, for fi 2 RO(G). 108 10. RO(G)-GRADED BOX PRODUCTS 109 (d) If M is an RO(G)-graded Mackey functor and X is a finite G-set, then the RO(G)-graded Mackey functor MX is given at ff 2 RO(G) by (MX )ff= (Mff)X. (e) If M and N are RO(G)-graded Mackey functors, then the RO(G)-graded Mackey functor M * N is given at fl 2 RO(G) by M (M * N) fl= Mff Nfi, ff+fi=fl where the box product on the right is in the category M of Mackey functors (see [10, 14]). Also, the RO(G)-graded Mackey functor *is given at fl 2 RO(G) by Y (*)fl= , ff where the construction on the right is the internal hom functor on the ca* *tegory M of Mackey functors (see [10, 14]). The basic properties of the category M* and the various constructions defin* *ed above are summarized in the next two propositions. Proposition 10.2. (a)M* is a bicomplete abelian category which has enough projectives and injectives and satisfies AB5. (b) M* is enriched over the category M. Moreover, it is tensored and cotenso* *red over M. (c) The functors ? *? and *provide M* with a symmetric monoidal closed structure which is consistent with its enrichment over M. The unit for the prod* *uct operation * on M* is c0(A), where A is the Burnside ring Mackey functor. Proof. Most of part (a) follows trivially from the observation that, if RO(* *G) is regarded as a discrete category, then M* is just the category of functors f* *rom RO(G) to M. For any ff 2 RO(G), the functors effand cffare related by two adjunctions described in Proposition 10.4(a) below. It follows easily from the* *se adjunctions that, if C is a projective (or injective) Mackey functor, then cff(* *C) is a projective (or injective) RO(G)-graded Mackey functor. These adjunctions also imply that objects of this form provide M* with enough projectives and injecti* *ves. If M and N are RO(G)-graded Mackey functors, then the Mackey functor- valued hom construction which enriches M* over M is just Y e0(*) = . ff It is easy to see that, if C is a Mackey functor and M is an RO(G)-graded Mackey functor, then c0(C) *M and *are the tensor and cotensor, respectivel* *y, of C with M. Of course, (c0(C) *M) ff= C Mff and (*)ff= . The unit and associativity isomorphisms for M* follow easily from those for M. The adjunction between the constructions ? *? and *is the obvious generalization of the corresponding adjunction for graded abelian groups. Thus, the only non-obvious part of the symmetric monoidal closed structure on M* is the commutativity isomorphism, which _ like the commutativity isomorphism used in homological algebra for the category of graded abelian groups _ involves sign 110 10. RO(G)-GRADED BOX PRODUCTS changes. Unfortunately, the sign changes needed here involve nontrivial units * *in the Burnside ring Mackey functor. These are discussed in Remark 10.3 below. The symmetric monoidal structure is consistent with the enrichment over M in the sense that the functors ? *? and *, and their adjunction, are enriched o* *ver M. This is actually a formal consequence of the definition of the enrichment, b* *ut is also easily checked directly. Remark 10.3. The commutativity isomorphism for the product on M* takes the summand Mff Nfiof (M * N)ff+fito the summand Nfi Mffof (N * M)ff+fi by the commutativity isomorphism for the box product on M composed with a "sign" change. Naively, one might expect the "sign" change to be given by multi- plication by (-1)|ff||fi|, where |ff|denotes the virtual dimension of ff. If th* *e order of G is odd, this naive approach to "sign" changes actually works because the Burn* *side ring of G contains no nontrivial units. However, for groups of even order (incl* *uding Z=2), the required "sign" is multiplication by a unit of the Burnside ring which need not be 1. The appropriate signs are given by a symmetric bilinear map sgn: RO(G) x RO(G) __//_A(G)x, where A(G)x is the group of units of the Burnside ring of G. To define sgn, it suffices to specify sgn(V, W ) when V and W are irreducible G-representations. * *If V and W are non-isomorphic irreducible representations, then sgn(V, W ) = 1. The element sgn(V, V ) of A(G)x is best described by thinking of it as an equivaria* *nt stable map S0 __//_S0. It is the map obtained by stabilizing the multiplication* * by -1 map from SV to itself. Proposition 10.4. Let C and D be Mackey functors, M and N be RO(G)- graded Mackey functors, ff and fi be elements of RO(G), and X and Y be finite G-sets. Then (a) The functors eff: M* ___//M and cff: M __//_M*are both enriched over M. Moreover, effis both left adjoint and right adjoint to cff, and these two adjun* *ctions are enriched over M. (b) There is a natural isomorphism cff(C) *cfi(D) ~=cff+fi(C D). Thus, c0 is a strict monoidal functor and e0 is both a lax monoidal and a lax comonoidal functor. (c) The functor ff: M* ___//M* is enriched over M, and is an enriched adjoint equvalence with inverse -ff. (d) There are natural isomorphisms fi ff+fi ( ffM) * N ~= (M * N) and < ffM , fiN>*~= fi-ff*. (e) The endofunctor of M* sending M to MX is a self-adjoint functor enriched over M. (f) There are natural isomorphisms (MX ) *(NY )~=(M * N)XxY and *~=(*)XxY . 10. RO(G)-GRADED BOX PRODUCTS 111 For our purposes, the basic connection between the category M* and equivari- ant homology is given by the following result. Proposition 10.5. Let T be a commutative ring spectrum, and Y be a module spectrum over T in the equivariant stable homotopy category. Then the homology T* of a point with respect T is a commutative ring object in M* , and the homol* *ogy Y* of a point with respect Y is a module over T* in M* . In order to prove our weak Universal Coefficient Theorem, we need several results about the category of modules over H* . However, everything we need to know about this category is true in the broader context of the category of modu* *les over any commutative ring object in M* . Thus, for the rest of this chapter, T** * is a commutative ring object in the category M* . A module over T* is an RO(G)- graded Mackey functor M together with a map i : T* *M __//_M for which the obvious diagrams commute. The category of T* modules is denoted T*-Mod. This category inherits all the good properties of the category M* . Definition 10.6. Let M and N be modules over T*, and let K be an RO(G)- graded Mackey functor. (a) The box product M T* N is given by the coequalizer diagram M * T* *N _____////_M *_N_//_M T* N in which the parallel arrows come from the actions of T* on M and N. (b) The internal hom functor T*is given by the equalizer diagram T*_____//*____////_**>* in which the parallel arrows come from the actions of T* on M and N. (c) The RO(G)-graded Mackey functors T* *K and *carry T*-module structures derived from the action of T* on itself. These two constructions are called the free and cofree T*-modules associated to K. Proposition 10.7. (a)The category T*-Mod is a bicomplete abelian category having enough projectives and injectives and satisfying AB5. (b) The category T*-Mod is enriched over the category M. Moreover, it is tensored and cotensored over M. (c) The functors ? T*? and T*provide T*-Mod with a symmetric monoidal closed structure which is consistent with its enrichment over M. The unit for t* *he product operation T* on T*-Mod is T*. (d) The functors sending an RO(G)-graded Mackey functor K to T* *K and *are left and right adjoint, respectively, to the forgetful functor from* * T*-Mod to M* . All three of these functors are enriched over M, and so are their assoc* *iated adjunctions. The free T*-module functor T* *? is strict monoidal, and the forg* *etful functor from T*-Mod to M* is lax monoidal. (e) The constructions ffM, for ff 2 RO(G), and MX , for a finite G-set X, restrict to endofunctors on T*-Mod. 112 10. RO(G)-GRADED BOX PRODUCTS (f) Let M and N be T*-modules, ff and fi be elements of RO(G), and X and Y be finite G-sets. Then there are natural isomorphisms fi ff+fi ( ffM) T* N ~= (M T* N), < ffM , fiN>T*~= fi-ffT*, (MX ) T* (NY )~=(M T* N)XxY , and T*~=(T*)XxY of T*-modules. Proof. The category T*-Mod inherits limits and colimits from M* . Sufficient projectives for T*-Mod are obtained by applying the free T*-module construction* * to the projectives in M* . It follows from the appropriate adjunction in part (d) * *that these objects are, in fact, projective in T*-Mod. Analogously, sufficient injec* *tives for T*-Mod are obtained by applying the cofree T*-module construction to injectives* * in M* . If M and N are T*-modules, then the Mackey functor-valued hom construction which enriches T*-Mod over M is just e0(T*). The tensors and cotensors for T*-Mod are just the obvious restrictions to T*-Mod of the analogous constructio* *ns on M* . The unit, associativity, and commutativity isomorphisms for T*-Mod are easily derived from the corresponding isomorphisms for M* . Similarly, the adjunction relating the functors ? T*? and T*follows immediately from the analogous adjunction for M* . The proof of part (d) is similar to a suitably formal proof of the correspon* *ding result for ordinary commutative rings. Note that the lax monoidal structure on * *the forgetful functor from T*-Mod to M* is given by the unit map u : c0(A)__//_T*a* *nd the canonical quotient map M * N ___//M T* N. Part (e) is rather obvious, and part (f) follows directly from Propositions 10.4(d) and 10.4(f). CHAPTER 11 A weak Universal Coefficient Theorem All that we need from the Universal Coefficient Theorem for the proof of our main results is the assertion that, if the RO(G)-graded ordinary homology HG*(X; A)of a G-space X with Burnside ring coefficients is free over H*, then, * *for certain Mackey functors M, the canonical map oeMX: HG*(X; A) H* HM* __//_HG*(X; M) is an isomorphism. Such a result, which we hereafter refer to as a weak Uni- versal Coefficient Theorem, would clearly follow from any reasonable Universal Coefficient Theorem for RO(G)-graded ordinary homology. Unfortunately, since RO(G)-graded ordinary homology cannot be described in terms of chain complexes, obtaining such a theorem is not trivial. It is possible to extend the Universal* * Co- efficient Theorems contained in [3] to G-spectra for any finite group G. Such an extension will appear in [15]. However, these results are applicable only to E1* * -ring G-spectra and their E1 -module G-spectra. It is widely acknowledged that equi- variant Eilenberg-MacLane spectra should carry such E1 -structures, but a proof for this has not yet been published. To bridge this gap, we provide here an ad * *hoc proof of the weak Universal Coefficient Theorem in the cases for which we need * *it. Recall the Z=p-Mackey functors L, R and introduced in Section 1.1. Proposition 11.1. Let G = Z=p, and let X be a G-space whose RO(G)-graded ordinary homology HG*(X; A)with Burnside ring coefficients is free over H*. Then the canonical maps oeLX: HG*(X; A) H* HL*___//HG*(X; L), oeRX: HG*(X; A) H* HR*___//HG*(X; R), and oeX: HG*(X; A) H* H*__//HG*(X; ) are isomorphisms. Our proof of this result uses spectrum-level arguments. It is therefore conv* *e- nient to break our usual convention and work with reduced, rather than unreduce* *d, homology. Throughout the argument, the analogous results for unreduced homol- ogy can be obtained by replacing the G-space X by X+ . To set the stage for the proof of this result, we begin with some observatio* *ns applicable to any finite group G, any commutative ring G-spectrum T , and any module G-spectrum Y over T . Motivated by Definition 1.13, we say that the reduced T -homology eT*X of a G-space X is free over T* if there is an isomorph* *ism M !i(T*)G=Hi ~=eT*X i 113 114 11. A WEAK UNIVERSAL COEFFICIENT THEOREM of T*-modules for some collection {Hi} of subgroups of G and some collection {!* *i} of elements of RO(G). Given such an isomorphism, there are canonical maps (G=Hi)+ ^ S!i___//X ^ T which may be thought of as the free generators of eT*X. Combining these generat* *ors, we obtain a map W f : (G=Hi)+ ^ S!i___//X ^ T. i Let ~fY: W(G=Hi)+ ^ S!i^ Y ___//X ^ Y i be the composite W ! f^1 1^i (G=Hi)+ ^ S i^ Y _____//X ^ T ^ Y _____//X ^ Y i in which i is the map giving the action of T on Y . It is easy to check that th* *e map ~fYis natural with respect to maps between T -modules. Denote the RO(G)-graded, Mackey-functor-valued stable homotopy "groupsö f a G-spectrum Z by ßG*Z. For any T -module Y , there is a canonical isomorphism M `W ' !i(Y*)G=Hi ~=ßG* (G=Hi)+ ^ S!i^ Y . i i The original isomorphism identifying eT*X as a free T*-module can be recovered * *as the composite M `W '(f~T)* !i(T*)G=Hi ~=ßG* (G=Hi)+ ^ S!i^ T ______//_ßG*(X ^ T~)=eT*X i i of this canonical isomorphism and the map in homotopy induced by ~fT. Thus, the map ~fTis a stable equivalence of G-spectra. The connection between the maps ~fYand the desired weak Universal Coefficient Theorem is described by the commuting diagram L !i ~= L !i i (T*)G=Hi T* Y*____________________// i (Y*)G=Hi ko ko ßG*(_i(G=Hi)+ ^ S!i^WTW)T*Y* ßG*(_i(G=Hi)+3^3S!i^gY ) | WWWWW++W ggggggg || (f~T)* T*1~=| ßG*(_i(G=Hi)+ ^ S!i^ T ^ Y ) |(f~Y)* | | fflffl| fflffl| eT*X T* Y*______________________________//eYX oeYX * in which the top isomorphism comes from Proposition 10.7(f). From the diagram, it follows that oeYX is an isomorphism if and only if ~fYis a stable equivalenc* *e of G-spectra. Unfortunately, the most we can prove about ~fYin general is the following: Lemma 11.2. Let T be a commutative ring G-spectrum, Y be a module G- spectrum over T , and X be a G-space whose reduced T -homology is free. Then the map ~fYis a split epimorphism in the equivariant stable category. 11. A WEAK UNIVERSAL COEFFICIENT THEOREM 115 Proof. Let {Hi} and {!i} be the collections of subgroups of G and elements of RO(G), respectively, such that M !i(T*)G=Hi ~=eT*X. i Then the commuting diagram _i(G=Hi)+ ^ S!i^ T ^ Y_1^1^i//__i(G=Hi)+ ^ S!i^ Y ~fT^1=f~T^Y~=|| || fflffl| |~ X ^ TO^OY VVV fY|| | VVVVVV1^iVV | 1^e^1| VVVVVV | | VVVV++ fflffl| X ^ S0 ^ Y_________~=________//X ^ Y displays the desired splitting. Here, e : S0___//T is the unit map for T . Since the composite Y ~=S0 ^ Y _e^1_//T ^ Y is not a map of T -modules, it is not possible to give an analogous argument sh* *owing that ~fYis a split monomorphism. We now specialize to the case in which T is the Eilenberg-Mac Lane spectrum HA associated to the Burnside ring Mackey functor for some finite group G. In proving Proposition 11.1, we make use of a special property of modules over HA. Assume that 0 __//_M0__j//_M _q_//M00__//_0 is a short exact sequence of Mackey functors. Then the equivariant Eilenberg- Mac Lane spectra HM0, HM, and HM00are HA-module spectra and the induced maps ~j: HM0 __//_HM and ~q: HM ___//HM00are HA-module maps in the equi- variant stable category (see Proposition 5.4 of [14]). Moreover, the sequence ~j ~q HM0___//HM ___//HM00 is a cofibre sequence in the equivariant stable category. Let @ : HM00 ___// HM0 be the boundary map associated to this cofibre sequence. Lemma 11.3. The boundary map @ : HM00 ___// HM0 is a map of HA-module spectra. Proof. As noted in the proof of Lemma 2.2 of [12], the unit map e : S0___//* *HA of HA is the inclusion of its 0-skeleton, and no 1-cells are needed to form HA * *from S0. Thus, the cofibre HA=S0 of e is 1-connected. We wish to show that the diagr* *am i00 HA ^ HM00 ______//HM00 1^@ || |@| fflffl|1^ i0 fflffl| HA ^ HM0 _____// HM0 116 11. A WEAK UNIVERSAL COEFFICIENT THEOREM commutes in the equivariant stable category. This can be rephrased as the asser- tion that a certain element of the equivariant homotopy set [HA ^ HM00, HM0]G vanishes. The image of this element under the map (e^1)* 0 00 0 [HA ^ HM00, HM0]G _______//[S ^ HM , HM ]G certainly vanishes. However, since [(HA=S0) ^ HM00, HM0]G is zero, (e ^ 1)* is* * a monomorphism. In the nonequivariant context, this result suffices to give a weak Universal Coefficient Theorem for ordinary homology because the category of abelian groups has homological dimension one. However, for a nontrivial finite group G, most Mackey functors have infinite homological dimension. Rather than attempt to deal with this homological difficulty, we now restrict our attention to the special case G = Z=p. It is easy to see that the Z=p-Mackey functor R is a Mackey functor ring. Thus, for any G-space X, eHG*(X; R)consists of R-modules. The following lemma allows us to exploit this fact. Lemma 11.4. Let M be a Mackey functor module over the Mackey functor ring R, and let C be an abelian group. Then any map h : M __//_factors through , where C0 is the subgroup of C consisting of elements annihilated by p. Th* *us, if there is no p-torsion in C, then there are no nontrivial maps from M to . Proof. Let , be the generator of R(G=G). Recall that p , = ø(æ(,)), where ø and æ are the transfer and restriction maps for R. Thus, for any x 2 M(G=G), px = p,x = ø(æ(,))x = ø(æ(,x)). The submodule pM(G=G) of M(G=G) is therefore contained in the image of the transfer ø : M(G=e) ___//M(G=G). It follows that h(pM(G=G)) must be zero, and so the image of M(G=G) under h must be contained in C0. We can now prove our weak Universal Coefficient result. Proof of Proposition 11.1. We prove the analogous result for reduced ho- mology. The asserted result for unreduced homology is then obtained by replacing X by X+ . Let X be a G-space whose reduced RO(G)-graded ordinary homology eHG*(X; A)with Burnside ring coefficients is free over H*. Also, let {Hi} and {* *!i} be the collections of subgroups of G and elements of RO(G), respectively, such * *that M !i(H* )G=Hi ~=eHG*(X; A). i Recall that there is a short exact sequence 0 __//__//A __//_R __//_0 of Mackey functors. From this short exact sequence, we obtain long exact sequen* *ces for the reduced homology of both X and F = _i(G=Hi)+ ^S!i. Lemma 11.3 implies that the maps ~fH, ~fHA, and ~fHRinduce a map . ._.@F//_eHG*(F ;_)//_eHG*(F_;_A)//_eHG*(F@;FR)//_. . . (f~H)*fflffl|| ~=f(f~HA)*flffl||(f~HR)*fflffl|| . .@.X//_eHG*(X; )//_eHG*(X;_A)_//eHG*(X; R)@X//_. . . 11. A WEAK UNIVERSAL COEFFICIENT THEOREM 117 between these two long exact sequences. By Lemma 11.2, the vertical maps in this diagram are split epimorphisms. Moreover, by assumption, (f~HA)* is an isomor- phism. For any ! 2 RO(G), eHG!(F ; )is either zero or a sum of copies of . Thus, by Lemma 11.4, @F must be zero. A simple diagram chase now gives that (f~H)* is a monomorphism. Thus, (f~H)* is a isomorphism, and the five lem* *ma gives that (f~HR)* is an isomorphism. Our observation about the relation between oeYXand ~fYfor an arbitrary module spectrum Y then gives that oeXand oeRXare isomorphisms. The fact that oeLXis also an isomorphism follows immediately from the connection between the equivariant Eilenberg-Mac Lane spectra associated to L and R described in Corollaries 9.3 and 9.6. CHAPTER 12 Observations about Mackey functors This chapter supplies some facts about short exact sequences of Z=p-Mackey functors which are used in Chapters 5 and 6. Most of these observations are the sort of thing that would be left to the reader if we were working in an abelian category more familiar than the category of Mackey functors. Lemma 12.1. Let 0 ____//A_'__//D_ß_//___//0 be a short exact sequence of Mackey functors. Then ( D ~= A if the sequence splits, R otherwise. Moreover, if D ~=R , then the two components '1: A __//_R and '2: A __//_<* *Z> of the map ' are surjective. Proof. Clearly, D(G=e) = Z, and D(G=G) is either Z Z Z=p or Z Z. If D(G=G) = Z Z Z=p, then the short exact sequence must split because the restriction map æ : D(G=G) ___//D(G=e)must vanish on the Z=p summand of D(G=G). On the other hand, if D(G=G) = Z Z, then the short exact sequence obviously does not split. Thus, assume D(G=G) = Z Z. Let ~ and ~øbe the standard generators of A(G=G). Also, let z = '(~), u = æ(z), and t = '(~ø) = ø(* *u). Note that u must generate D(G=e) = Z. A simple rank argument indicates that the kernel of the restriction map æ : D(G=G) ___//D(G=e)is isomorphic to Z. Let y b* *e a generator of this kernel. Then ß(y) 6= 0. Otherwise, select some w 2 D(G=G) such that ß(w) 6= 0. There is an integer s such that æ(w - sz) = 0, so that w - sz =* * ry for some integer r. From this we get the contradiction that ß(w) = ß(ry + sz) =* * 0. Since ß(y) isn't zero, t, y, and z generate D(G=G). However, æ(t - pz) = 0, so there is an integer b such that t - pz = by, from which it follows that y an* *d z generate D(G=G). Note that ß(by) = ß(t - pz) = 0, so p divides b. Thus, there is an integer b0such that t = p(z + b0y). By replacing y by its negative if necess* *ary, we can assume that b0 0. Observe that py is in the image of ' since ß(py) = 0. In fact, because ' is injective, there is an integer c such that py = c(t - pz). Substituting in for * *t, we have that py = b0cpy. Thus b0= c = 1 since b0 0. Let z0= y + z. Then t = pz0, and u = æ(z0). It follows that z0generators a copy of R contained in D. Further* *, y generates a copy of in D, and it is easy to see that D is the direct sum of * *these two Mackey functors. Since z = z0- y0, the two maps '1 and '2 are surjective. Lemma 12.2. (a) Let 0____//L_'_//D__ß_//___//0 (12.1) 118 12. OBSERVATIONS ABOUT MACKEY FUNCTORS 119 be a short exact sequence of Mackey functors. Then ( D ~= L if the sequence splits, A[d] otherwise. Here, d is assumed to be relatively prime to p. Moreover, if D ~=A[d], then d is determined in Z=p up to sign by the fact that there is an element x 2 D(G=G) su* *ch that ß(x) generates (G=G) = Z and æ(x) = d 2 D(G=e) = Z. (b) Let 0 ____//_'_//D_ß__//R___//0 (12.2) be a short exact sequence of Mackey functors. Then ( D ~= R if the sequence splits, A[d] otherwise. Here, d is assumed to be relatively prime to p. (c) If the Mackey functor D fits into short exact sequences of both form (12.* *1) and form (12.2), then D ~=A[d] for some integer d prime to p. Proof. For part (a), note that D(G=G) must be Z Z. Let t02 D(G=G) be '(t), where t is the standard generator of L(G=G). Then æ(t0) = p 2 D(G=e) = Z. Select an element x of D(G=G) such that ß(x) generates (G=G) = Z. Clearly t0 and x generate D(G=G). Let d = æ(x) 2 D(G=e) = Z. If p divides d, then by adding some multiple of t0 to x we can obtain an element x0 of D(G=G) such that ß(x) generates (G=G) and æ(x0) = 0. In this case, the short exact sequence obviou* *sly splits. Thus, assume that d 2 D(G=e) is relatively prime to p. Any other element y of D(G=G) such that ß(y) generates (G=G) must be of the form x + at0for some integer a. Therefore, æ(y) 2 D(G=e) is also relatively prime to p, and the short exact sequence cannot split. Moreover, it is easy to check that D ~=A[d] * *by a map sending x and t0to the standard generators ~ and ø of A[d](G=G). For part (b), again note that D(G=G) must be Z Z. Let k 2 D(G=G) be the image of a generator of (G=G) = Z. Then æ(k) = 0 2 D(G=e) = Z. Select an element y of D(G=G) such that ß(y) generates R(G=G) = Z. Clearly k and y generate D(G=G). Moreover, u = æ(y) generates D(G=e). It is easy to check that ø(u) = py - ak for some integer a. If p divides a, then we can adjust y by some multiple of k to obtain an element y0 of D(G=G) such that ß(y0) generates R(G=G), u = æ(y0), and ø(u) = py0. In this case, the short exact sequence obvio* *usly splits. Thus, assume that a is relatively prime to p. Let z be any other elemen* *t of D(G=G) such that ß(z) generates R(G=G). Then z = y + bk for some integer b and æ(z) = u. Further, ø(u) = pz - a0k for some integer a0which is congruent to a modulo p. But then a0 is relatively prime to p, from which it follows that the exact sequence cannot split. Select an integer d such that ad is congruent * *to 1 modulo p. It is easy to check that D ~=A[d] by a map sending y and k to the standard oe and ~ generators of A[d](G=G). For part (c), it suffices to show that the Mackey functors R, L , * *and A[d] are pairwise nonisomorphic. Clearly, L is not isomorphic to either of* * the other two because the the restriction map is surjective in R and A[d], but* * not in L . Further, R is not isomorphic to A[d] because every element in ( R)(G=G) which is in the image of the transfer is p-divisible, whereas the* *re are elements in the image of the tranfer in A[d](G=G) which are not p-divisible. 120 12. OBSERVATIONS ABOUT MACKEY FUNCTORS Lemma 12.3. For any nonzero map ß : R __//_, Kerß ~=L. Proof. Let K = Kerß. Clearly, K(G=G) ~=Z ~=K(G=e). The restriction and transfer maps for K are easily computed from the embedding of K into R. From this, it follows immediately that K = L. Bibliography 1.G. E. Bredon, Equivariant cohomology theories, Lectures Notes in Math., vol.* * 34, Springer- Verlag, Berlin, 1967. 2.A. W. M. Dress, Contributions to the theory of induced representations, Alge* *braic K-theory, II: "Classicalä lgebraic K-theory and connections with arithmetic (Proc. Co* *nf., Battelle Memorial Inst., Seattle, Wash., 1972), Lectures Notes in Math., vol. 342, Sp* *ringer, Berlin, 1973, pp. 183-240. 3.A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and * *algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Math* *ematical Society, Providence, RI, 1997, (with an appendix by M. Cole). 4.H. Fausk, L. G. Lewis, Jr., and J. P. May, The Picard group of equivariant s* *table homotopy theory, Adv. Math. 163 (2001), no. 1, 17-33. 5.K. K. Ferland, On the RO(G)-graded equivariant ordinary cohomology of genera* *lized G-cell complexes for G = Z=p, Ph.D. thesis, Syracuse University, 1999. 6.J. A. Green, Axiomatic representation theory for finite groups, J. Pure Appl* *. Algebra 1 (1971), no. 1, 41-77. 7.P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley & S* *ons, 1978. 8.H. Hiller, Geometry of Coxeter groups, Research Notes in Mathematics, vol. 5* *4, Pitman (Advanced Publishing Program), Boston, Mass., 1982. 9.S. Illman, Equivariant singular homology and cohomology. I, Mem. Amer. Math.* * Soc. 1, issue 2 (1975), no. 156. 10.L. G. Lewis, Jr., The theory of Green functors, Mimeographed notes, 1980. 11._____, The RO(G)-graded equivariant ordinary cohomology of complex projectiv* *e spaces with linear Z=p actions, Algebraic topology and transformation groups, Proceeding* *s, Göttingen 1987, Lecture Notes in Math., vol. 1361, Springer, Berlin, 1988, pp. 53-122. 12._____, The equivariant Hurewicz map, Trans. Amer. Math. Soc. 329 (1992), no.* * 2, 433-472. 13._____, Change of universe functors in equivariant stable homotopy theory, Fu* *nd. Math. 148 (1995), no. 2, 117-158. 14._____, The category of Mackey functors for a compact Lie group, Group repres* *entations: cohomology, group actions and topology (Seattle, WA, 1996), Proc. Sympos. Pu* *re Math., vol. 63, Amer. Math. Soc., Providence, RI, 1998, pp. 301-354. 15.L. G. Lewis, Jr. and M. A. Mandell, Equivariant universal coefficient and Kü* *nneth spectral sequences, In preparation. 16.L. G. Lewis, Jr., J. P. May, and J. E. McClure, Ordinary RO(G)-graded cohomo* *logy, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 208-212. 17.L. G. Lewis, Jr., J. P. May, and M. Steinberger (with contributions by J. E.* * McClure), Equi- variant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Spr* *inger, Berlin, 1986. 18.J. P. May, Equivariant homotopy and cohomology theory, CBMS Regional Confere* *nce Series in Mathematics, vol. 91, Published for the Conference Board of the Mathemati* *cal Sciences, Washington, DC, 1996. 19.J. W. Milnor and J. D. Stasheff, Characteristic classes, Annals of Mathemati* *cs Studies, vol. 76, Princeton University Press, 1974. 20.J. Th'evenaz, A visit to the kingdom of the Mackey functors, Bayreuth. Math.* * Schr. 33 (1990), 215-241. 21.A. G. Wasserman, Equivariant differential topology, Topology 8 (1969), 127-1* *50. 121