The RO(G)Graded Equivariant Ordinary
Homology of GCell Complexes with
EvenDimensional Cells for G = Z=p
Kevin K. Ferland
L. Gaunce Lewis, Jr.
Author address:
Department of Mathematics, Bloomsburg University, Bloomsburg,
PA 17815
Email address: kferland@bloomu.edu
Department of Mathematics, Syracuse University, Syracuse NY
132441150
Email address: lglewis@syr.edu
Contents
Introduction 1
Part 1. The Homology of Z=pCell Complexes with Even
Dimensional Cells 7
Chapter 1. Preliminaries 8
1.1. Mackey functors for Z=p 8
1.2. RO(G)graded Mackey functorvalued homology 12
1.3. The homology H* of a point 14
1.4. Modules over H* 18
1.5. Generalized Gcell 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 singlecell freeness results 42
4.1. A proof overview for the dimensionshifting theorem (Theorem 3.3) 43
4.2. Simplifying the cellattaching long exact sequence 44
4.3. Characterizing dimensionshifting long exact sequences 48
4.4. Constructing the comparison dimensionshifting 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. Dimensionshifting long exact sequences 65
6.1. Preliminary observations about dimensionshifting sequences 65
6.2. The reduction to complexity one dimensionshifting 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 !iP !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 CWcomplex with cells only in even
dimensions is free. The equivariant analog of this result for generalized Gcel*
*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 Gcell complexes construct*
*ed
using the unit disks of finite dimensional Grepresentations as cells. Our main*
* result
is that, if X is a Gcomplex containing only evendimensional 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=pactions. Our new result applies more generall*
*y to
equivariant complex Grassmannians with linear Z=pactions.
Two aspects of our result are particularly striking. The first is that, even
though the generators of HG*(X; A)are in onetoone 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. BredonIllman 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 Gcell complex having only evendimensional 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 Gcell complexes. This choice yield*
*s a
theorem which is trivial to prove, but turns out to be inapplicable to any inte*
*resting
Gspaces. An alternative type of Gcell (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 BredonIllman
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 BredonIllman homology. Let V be a nontrivial
complex representation of G. Its onepoint 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 BredonIllman homology HGn(SV ; M)of
SV with respect to a coefficient system M contains ptorsion 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
BredonIllman 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 Zgraded 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 evendimensional, then the BredonIllman homology of SV would be
torsionfree. Since this homology is not torsionfree, we know that it is not p*
*ossible
to build even as nice a space as SV out of only evendimensional cells of the f*
*orm
G=H x Dn. Indeed, it seems likely that very few spaces can be constructed using
only such evendimensional cells.
1
2 INTRODUCTION
There is a simple explanation for these difficulties with Zgraded Bredon
Illman homology. For any reasonably wellbehaved 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 Grepresentations. In fact, in those dimensions, HM* is full of*
* torsion
at the primes dividing the order of G. The reduced BredonIllman homology group
eHGn(SV ; M)is a part of HM* and so reflects this ptorsion.
This explanation for the lack of a good freeness theorem for Zgraded Bredon
Illman homology leads us to both the right homology theory and the right notion
of freeness. If the coefficient system M is ringvalued, then its EilenbergMac*
* Lane
spectrum is a ring Gspectrum, 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 Gspace 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 ringvalued, 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 Grepresentations V as our cells rath*
*er
than cells of the form G=H x Dn. For such a cell DV , the appropriate meaning
of "evendimensional" is that, for each subgroup H of G, the fixed point space *
*V H
is an evendimensional 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 wellunderstood 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 Grepresentation V , or even cells of the form G xH DW , for
a subgroup H of G and an Hrepresentation 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 functorvalued. 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 Zgraded BredonIllman 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 Gspace
INTRODUCTION 3
whose RO(G)graded equivariant ordinary homology is free over H* with even
dimensional generators. Let Y be a Gspace obtained from B by adjoining a single
evendimensional cell DV . If we could show that the homology of Y must also
be free, then an inductive argument indicates that any finite generalized Gcell
complex with only evendimensional 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 Gcell complex having evendimensional 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 Gactions 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
3planes in a typical 6dimensional complex Grepresentation 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 cellattaching 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 Gspace. 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 Gcell structures yield a nonzero boundary map in the long exact
sequence associated to attaching the 4cell to the 2skeleton. 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 cellattaching long exact sequences were Zgraded, 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 cellattaching 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 evendimensional representation cells, then the homolo*
*gy
HG*(X; A)of X with Burnside ring coefficients is free over H*. There is a onet*
*oone
correspondence between the cells of X and the generators of HG*(X; A). However,
the dimensionshifting 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 dimensionshifting 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 finitetype 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 cellattaching 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 nonfree 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 finitetype, 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 cellattaching 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=pactions (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 cellattaching 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 Gcell 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 Gspace 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=paction.
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 EilenbergMac 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 EilenbergMac 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=pMackey functors.
Part 1
The Homology of Z=pCell
Complexes with EvenDimensional
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 Gequivariant with respect to the trivial
action on M(G=G). Moreover, the composite æ O øPis required to be the trace of
the Gaction 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 Gaction. Whenever M(G=e) is a pfold 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 Gaction 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 Gmap, 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) = gk1xk and eM(y) = (y, g1y, . .,.g1py).
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 functorvalued homology
Throughout this paper, we work with RO(G)graded, Mackey functorvalued
equivariant ordinary homology theories (see [1618]). 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 Gspace X, virtual Grepresentation !, and Mackey functor M, the Mackey
functorvalued 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 BredonIllman 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 Grepresentation 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 EilenbergMac Lane spectrum HM (see [13, 14,
1.2. RO(G)GRADED MACKEY FUNCTORVALUED HOMOLOGY 13
16]). If ! is represented by the formal difference V  W of Grepresentations 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 Gstable category. The restriction and transf*
*er
maps for HG!(X; M) come from the stabilization of spacelevel maps between or
bits and the transfers associated to those spacelevel 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 Gspace 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 spectrumlevel definition. In particular, equivariant ordi*
*nary
homology satisfies the following axioms:
(i)Additivity: Disjoint unions of Gspaces are carried to direct sums.
(ii)Exactness: Cofibre sequences of Gspaces 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 Grepresentation V ,
element ! of RO(G), and based Gspace X.
(v)Dimension: For n 2 Z, regarded as the trivial Grepresentation 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 EilenbergMac 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 Gspace
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 Gspace 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 ! = !0and !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 welldefined. 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 Grepresentations
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 welldefined 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
Grepresentations 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 Gmap 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 Grepresentations. Let
Y
d(!) = di.
i
Also, let d(0) = 1. Observe that this gives a welldefined 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 Grepresentation, and let jk be
its kfold complex tensor power for some integer k relatively prime to p. Then
j  jk 2 RO0(G) and djjk 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 !86= 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, andÆ = 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 = dÆ0Æ0ffl!0,Æ0if ! + Æ = !0+ Æ0
(m) ffl1Æ~! = ffl1Æ0~!0 if !  Æ = !0 Æ0
(n) ~!0(ffl1Æ~!) = ffl1Æ~!+!08
>~!+Æ0Æ ifÆ  Æ0= 0
:pffl!+Æ0ÆifÆ  Æ0> 0
(p) (ffl1Æ~!)(ffl1Æ0~!0) = p(ffl1Æ+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) (ffl1Æ~!) 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 dimensionshifted copy of M. Since H* is a graded ring, a set
of projective generators for H*Mod obviously ought to include dimensionshift*
*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 SMod 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
dimensionshifted 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 Ghomeomorphism
between the Gspaces 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 dimensionshifted 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 welldefined 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 fflffl0
( !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= ÆG2or 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 evendimensional
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 evendimensional if its dimension is an evendimensional element *
*of
RO(G). A projective generator is said to be evendimensional if its dimension m*
* is
an even integer.
(b) An element ! of RO(G) is said to be spacelike if !  !G  0. A
purely free generator of a free H*module is said to be spacelike if its dimen*
*sion
is spacelike. A projective generator is said to be spacelike 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 ! = !0and !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 dimensionshifting 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
dimensionshifting 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 evendimensional 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 evendimensional 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 evendimensional 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 !  !0is an odd0positive integer and !G  (!0)Gis 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 dimensionshifting 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 !  !0is
an odd positive integer0and !G  (!0)Gis 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
fflÆÆ02 HÆÆ0 gives a monomorphism from ( ! H*)Æ0to ( ! H*)Æ by Lemma 8.7.
It follows trivially that fÆ is nonzero.
1.5. GENERALIZED GCELL COMPLEXES 23
1.5. Generalized Gcell complexes
Our initial definitions in this section apply to any compact Lie group G. A
generalized Gcell complex X is a Gspace X together with an increasing sequence
of Gsubspaces Xn of X such that X0 is a disjoint union of orbits, Xn+1 is form*
*ed
from Xn by attaching Gcells, X = [nXn, and X has the colimit (or weak) topology
derived from the subspaces Xn. The Gcells 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 Hrepresentation V . Such a cell is attached to Xn in t*
*he
process of forming Xn+1 via an attaching Gmap 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 evendimensional
if the fixed point subset V K is evendimensional 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 Gcell complex X are those of the form DV ,*
* for
some Grepresentation 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 Gcell structure on the Grassmannian
manifold G(V, k) of kplanes in some Grepresentation 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 onepo*
*int
compactification of V . The point at infinity in SV is given trivial Gaction*
* 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 cellattaching 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 Gsubspace 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 !nand V G< !G1. In this case, it is possible for the boun*
*dary
map
@ : eHG*(SV ; A)//_eHG*1(B; A)
in the cellattaching 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= !Gi1, 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


 !n1


 .
 ..





 !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 dimensionshifting 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 evendimensional cell DV is added to a space B whose homology
is free over H* with evendimensional generators and the boundary map
@ : eHG*(SV ; A)//_HG*1(B; A)
in the cellattaching 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 Gcell 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
Grepresentation. We want to form a generalized Gcell complex X containing one
2cell 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 cellattaching 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 cellattaching 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 dimensionshifting 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 Gcell complexes formed from evendimension*
*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 Gcell complex formed from evendimensional 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 2sphere 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 4sphere, we can map it to a wedge S4 _ S4 of two 4spheres by taking one
copy of the suspension of f going into each of the two 4spheres. Now consider *
*an
infinite wedge _m2Z S4 of 4spheres indexed on the integers. Form a new space X
by adjoining a Zindexed 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 Zindexed collect*
*ion
of generators plotting at the point (2, 4), and a Zindexed 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 Gcell complex formed from only evendimensional 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 functorvalued equivariant or
dinary homology of X with Burnside ring coefficients is free over H* . Moreover,
there is a onetoone 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 kdimensional subspaces of a complex Grepresentation V
is a generalized Gcell 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=pcell complex X w*
*ould
be that, for each nonnegative 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=pcell 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
Gcell complex formed from evendimensional 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 evendimensional Grepresentation
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 Gspace B is free over H* with
evendimensional generators and the Gspace Y is formed from B by adding a
single evendimensional cell of the form DV or G x Dm , then HG*(Y ; A)is also
free over H* with evendimensional 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
spacelike. 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,fidec*
*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 ffindexed ge*
*nerator
of Cfiunder the map ~fi,fl: Cfi__//_Cflcould be a multiple of the ffindexed 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 ordinalindexed 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+1and !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 ffin*
*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 ffindexed 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 ffindexed
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 Gcell 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 Gcell 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 evendimensional spaceli*
*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 cellattaching 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 evendimensional spaceli*
*ke
generators. Our inductive assumptions imply that the finite subsets Jff(ffi + 2*
*) of
J (ffi + 2) are wellpositioned. 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
cellattaching 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 ffindexed 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 ffindexed 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 ffiindexed 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> !nand !Gffi,ffi+1=*
* !Gnare
satisfied. But !n is !ffn,ffi, which satisfies the conditions !ffn,ffi= !ffn*
*,ffn+1and
!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 spacelike 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 !iand (!00j)G !Gi,
(ii)!00j= !iand (!00j)G> !Gi, or
(iii)!00j< !iand (!00j)G= !Gi.
Note that, since ffi is assumed to be in the wellpositioned set Jff(ffi + 1), *
*!i must
satisfy the conditions
!i !ff,ff+1and!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 fijindexed 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 fijindexed generat*
*or
to eliminate the need for that fijindexed 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+1indexed 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 evendimensional spacelike *
*genera
tors. Given this, Proposition 3.7 indicates that HG*(Xffi; A)~=colimff !ff0,fiandVfGi< !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+1and!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)andVfGi< 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> Vffand 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 evendimensional spacelike generators. From*
* this,
we conclude that HG*(Xffi; A)is a free H*module with evendimensional spaceli*
*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 evendimensional spacelike 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 singlecell freeness results
Throughout this chapter, B is assumed to be a Gspace whose homology is
free over H* with evendimensional spacelike generators. We also assume that t*
*he
Gspace Y is formed from B by adding a single evendimensional cell of the form
DV or G x Dm . Associated to this attachment we have a homology cellattaching
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
evendimensional spacelike generators. If the cell attached to B is of the fo*
*rm
G x Dm , then the boundary map @ in the cellattaching long exact sequence is z*
*ero.
Lemma 4.2. If the boundary map @ in the cellattaching 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 DIMENSIONSHIFTING THEOREM 43
4.1.A proof overview for the dimensionshifting theorem
(Theorem 3.3)
For the remainder of this chapter, we assume that the cell attached to the G
space B to form the Gspace Y is of the form DV and that the boundary map @ in
the associated cellattaching 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 cellattaching long exact sequence for HG*(Y ; A). This*
* new
long exact sequence is essentially identical to the cellattaching 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 SINGLECELL 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____//. . .
= ` = =
fflfflfflffl 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 wrapu*
*p of
the proof of Theorem 3.3.
4.2. Simplifying the cellattaching 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 cellattaching 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 CELLATTACHING 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, bx1ya) 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= !2and !G1= !G2,
<~ + cffl if! < ! and !G = !G,
(~1) = > 1 1 2 1G 2G
>>:~1 + c, if!1= !2and !1 > !2,
~1 + cffl,if!1< !2and !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 Gspace whose homology HG*(B; A)is free over
H* with evendimensional spacelike generators. Assume that the Gspace Y is
46 4. PROVING THE SINGLECELL FREENESS RESULTS
formed from B by adding a single evendimensional cell of the form DV , and that
the boundary map
@ : eHG*(SV ; A)//_HG*1(B; A)
in the associated cellattaching 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 spacelike,
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< !G1since @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 Gspace whose homology HG*(B; A)is free over
H* with evendimensional spacelike generators. Assume that the Gspace Y is
formed from B by adding a single evendimensional 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 CELLATTACHING 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 cellattaching
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 SINGLECELL 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 dimensionshifting long exact sequences
In this section, we assume that !1, !2, . . . , !n, !01, !02, . . . , !0n+1,*
* and V are
evendimensional spacelike 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= !Gi1, 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 cellattaching 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 DIMENSIONSHIFTING 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 !iP !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 SINGLECELL 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 jthsummand
has the form
A __//_ __//_.
Here, the two copies of in the middle come from the generators of J in
dimensions !j1 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 onetoone
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 !iP !0j) 1 mod p.
4.4. Constructing the comparison dimensionshifting 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 DIMENSIONSHIFTING SEQUENCE 51
and
(!0i)G= !Gi1, fori 2.
Our first task is to verify that there are dimensions !0isatisfying these co*
*nditions
in which Q is sufficiently wellbehaved 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 spacelike evendimensional 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 lefthand 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 SINGLECELL 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
n177
ppp
~ØVp1pppp `V1
ppp fflffl
. ._._____//A__@0___//n_0___//_n1_______//0.
V ØV 1
Thus, `V 1is an isomorphism.
4.4. CONSTRUCTING THE COMPARISON DIMENSIONSHIFTING 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
fflfflØ0 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 Gspace whose homology HG*(B; A)is free over
H* with evendimensional spacelike generators, and the Gspace Y is formed from
B by adding a single evendimensional cell of the form DV . We assume that the
boundary map in the associated cellattaching 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 cellattaching 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 cellattaching long exact se
quences, we obtain the commuting diagram
.. . . .
. .. .. ..
fflfflØL fflffl_L fflffl L fflffl
. ._.//_HG*(B; L)_//_HG*(Y ;_L)//_eHG*(SV ;@L)//_HG*1(B;_L)//_. . .
fflfflØ fflffl_ fflffl fflffl
. ._.//_HG*(B; A)_//_HG*(Y ;_A)//_eHG*(SV ;@A)//_HG*1(B;_A)//_.(.5..1)
fflfflØ 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 cellattaching long
exact sequence and the maps @0 and .
For any Mackey functor S, denote the RO(G)graded homology of a point with
Scoefficients 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 cellattaching 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
.. . . .
. .. .. ..
fflfflØ0fflffl_0 fflffl@0 fflffl
. ._.___//_JL_____L//_QL___L//_NL____L//_ JL____//_. . .
fflfflØ0fflffl_0 fflffl@0 fflffl
. ._.____//_J_____//_Q______//_N______//_ J_____//_. . .(5.4)
fflfflØ0fflfflfflffl_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 Gspace 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)!=!iand !G = !Gi1for 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 Grepresentation ,. 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  ,)Gcan 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_____//
fflfflØ0!fflffl_0!~=fflfflfflffl@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 ! = !iand
( G
!G = !i1 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!_____//____//_
fi2flfflØ0 fflffl0 ~=fflffl0 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
fflfflØ0!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
fflfflØ0!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= !i1 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
fflfflØ0!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 ptorsion 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
fflfflØ0!fflffl1_0!fflffl1
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
fflfflØ0!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
fflfflØ0!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
Dimensionshifting long exact sequences
In this chapter, n is a positive integer and !1, !2, . . . , !n, !01, !02, .*
* . . , !0n+1,
and V are evendimensional spacelike 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= !Gi1, 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 dimensionshifting 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 Gand 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 dimensionshifting 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 dimensionshifting 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. DIMENSIONSHIFTING 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 !i1 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 DIMENSIONSHIFTING 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. DIMENSIONSHIFTING 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!01V (~Ø2)!1(~02) = e02,!02V.
Here ,!1!01, ffl!1!02, ffl!01V, and ,!02Vare 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 DIMENSIONSHIFTING SEQUENCES 69
However,
(_~!1O ~Ø!1)(~)= e1e01ffl!01V,!1!01+ e2e02ffl!1!02,!02V
= (e1e01+ e2e02d(V +!1!01!02))ffl!01V,!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 evendimensional spacelike 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 dimensionshifting 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 n1, 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. DIMENSIONSHIFTING 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= n1i=1Jiand ~J0= n1j=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 DIMENSIONSHIFTING 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
~ØJ~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 n1,
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 righthand 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. DIMENSIONSHIFTING 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
~ØJ~: ~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 = (~ØJ~)(~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
(~ØJ~)(~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 cellattaching 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 Grepresentations 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
dj1j 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~+j1(!02!01) 1 mod p.
Applying Lemma 1.14 again gives us the isomorphism
0!0 2~+j1
!2 1H* ~= H*
of H*modules which allows us to replace !02 !01by 2  ~ + j1. 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  ~ + j1. Via suspension by ~, this special case is
equivalent to the special case of the quadruple j + ~, 2, ~, and 2 + j1.
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. DIMENSIONSHIFTING LONG EXACT SEQUENCES
Lemma 6.8. Let j be a nontrivial irreducible complex Grepresentation. Then,
there are maps
1
bØ: 2H* ___// jH* 2+j H*,
_b: jH* 2+j1H* ___// 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 Grepresentation W , denote the associated complex
projective space with a linear Gaction by P (W ). Also, denote the trivial com*
*plex
Grepresentation with complex dimension n by nC. By Proposition 3.1 of [11], the
reduced homology of P (2C+j1) is a free H*module with generators in dimensions
j and 2 + j1. This description of eHG*(P (2C + j1);iA)s obtained by viewing t*
*his
space as being obtained from P (1C + j1) by attaching the obvious 4cell. If,
instead, we view this space as being obtained from P (2C) by attaching a differ*
*ent
4cell, then the associated cellattaching long exact sequence has the form
. ._._// 2H* ___//eHG*(P (2C + j1);_A)// 2jH* _@_//_ 3H* ___//. ...
Replacing eHG*(P (2C + j1);bA)y the isomorphic H* module jH* 2+j1H*
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+j1H* are obviously not isomorphic.
If j 6= ~, then we must use a stunted projective space with a linear Gaction
to produce the desired long exact sequence. Observe that, since j and ~ are both
nontrivial irreducible complex Grepresentations, there is an integer k such th*
*at
~ = jk and 1 < k < p. Let W be the complex Grepresentation 1C + j + . .+.jk1.
Note that neither ~ nor j1 = jp1 is contained in W . Applying the results of *
*[11]
to the stunted projective space P (W + j1 + 1C)=P (W ) provides the desired lo*
*ng
exact sequence.
Lemma 6.9. Let j be a nontrivial irreducible complex Grepresentation, and k
be an integer such that 1 < k < p. Then, there are maps
k 2+j1
bØ: 2H* ___// j H* H*,
b_: jkH* 2+j1H* __//_ j+jkH*,
and
b@: j+jkH* __//_ 3H*
such that the sequence
k 2+j1 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 + j1 + 1C)=P (W ) and ! = j + j2 + . .+.jk1.
By Proposition 3.1 of [11], the reduced homology of X is a free H*module with
generators in dimensions ! + jk and 2 + ! + j1. This description of eHG*(X; A)
is obtained by viewing this space as being obtained from P (W + j1)=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 cellattaching 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+!+j1H* 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+!+j1H* 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 + j1. 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 Gby 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. DIMENSIONSHIFTING LONG EXACT SEQUENCES
first stage to eliminate the constraint on V  !1by 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 evendimensional spacelike 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!01V +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 dimensionshifting 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 OOOO
     
~_1_01 Ø02_001Ø002~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. DIMENSIONSHIFTING 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@0flfflpØ00pppp
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. DIMENSIONSHIFTING 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@00flfflpØ0pppp
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 !iP !0jj 81
6.5. The congruence condition on d(V +P !iP !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 !iP !0jthat is the sole obstruction to the existence of a dimensions*
*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
j1X
fij = V + !i !0i,
i=1
for 1 < j n + 1. Observe that fij= V and fiGj= (!0j)Gfor 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+1V 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!01V,!1!01+ e01e002ffl!1!02ffl!02fi2,fi2*
*V
= (e1e001+ e01e002)ffl!01V,!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!0ifii,!i!0i,fiiV+ e0ie00i+1ffl!i!0i+1ffl!0i+1*
*fii+1,fii+1V
= (eie00i+ e0ie00i+1)ffl!0ifii,fii+1V.
82 6. DIMENSIONSHIFTING 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!0nfin,!n!0n,finV+ e0ne00n+1ffl!n!0n+1,!0n+1*
*V
= (ene00n+ dfin+1!0n+1e0ne00n+1)ffl!nfin+1,fin+1V.
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 Grepresentation and k is a positive integer, then the Gra*
*ss
mannian manifold G(V, k) of complex kdimensional subspaces of V carries an ob
vious Gaction derived from the action of G on V . Nonequivariantly, G(V, k) is*
* a
CWcomplex 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 Gspace with the str*
*uc
ture of a generalized Gcell 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 Gcell structure on G(V, k), we assume initiallyL
that V is a finite dimensional complex Grepresentation 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 Gaction 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+i1M
W = OE1ai+iOEj.
i=1 j=1
j62{a1+1,...,ai1+i1}
Here, OE1ai+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 0cell <0,*
* . .,.0>
as the 0filtration 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  Xn1 = { ai= n}.
i
However, there are other useful filtrations. If V V 0, then clearly G(V, k)*
* is a
subGcell complex of G(V 0, k). Hence, G(V, k) carries an obvious Gcell 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