THE GEOMETRY OF THE LOCAL COHOMOLOGY FILTRATION IN EQUIVARIANT BORDISM DEV P. SINHA Abstract.We present geometric constructions which realize the local cohom* *ology filtration in the setting of equivariant bordism, with the aim of understanding free G actions on m* *anifolds. We begin by reviewing the basic construction of the local cohomology filtration, starting with * *the Conner-Floyd tom Dieck exact sequence. We define this filtration geometrically using the languag* *e of families of subgroups. We then review Atiyah-Segal-Wilson K-theory invariants, which are well-suite* *d for studying the manifolds produced by our techniques. We end by indicating potential applications o* *f these ideas. 1. Introduction Local cohomology techniques are an area of rapid development in equivariant t* *opology. Since their introduction by Greenlees [11] they have been used for example to compute the k* *u-homology of classifying spaces of abelian groups [5], they have an interesting relationship with the ch* *romatic filtration [13], and they have been generalized beyond equivariant topology as announced by Dwyer an* *d Greenlees. They give a new duality between homology and cohomology of classifying spaces, which can * *be played against the Universal Coefficient duality. See [12] for an excellent survey. One purpose of this note is to better understand the local cohomology filtrat* *ion by casting it in the language of families. We will see that the local cohomology filtration is coars* *er than the families filtration. It is more manageable, however, because the algebra which arises is that of loc* *alization. Moreover, when one "frees up" the theory, it becomes more computable and gives rise to an inte* *resting relationship between the homology and cohomology, with coefficients in any complex-oriented spectrum* * (for example ordinary cohomology or K-theory), of classifying spaces. A strong appeal of equivariant bordism as a sub-field of equivariant topology* * is in its connection with manifold theory. Our geometric constructions allow one to prove collapse result* *s by constructing particular G-manifolds which represent classes in the local cohomology spectral sequence. * *Moreover, we may also bring in Atiyah-Segal-Wilson localized K-theory invariants to answer questions * *about differentials and extensions. On the geometric side, the constructions in this paper are in part* * an answer to questions about realizing the bordism of classifying spaces geometrically. These question* *s were posed to the author by Botvinnik who has used such knowledge to solve cases of the Gromov-Lawson-Ro* *senberg conjecture concerning spin manifolds which admit positive scalar curvature metrics [3, 26]* *. Indeed, the main ideas of this paper came out of conversations with Botvinnik and Sadofsky while the a* *uthor was visiting the University of Oregon. This paper will blend well-established ideas of Conner-Floyd, tom Dieck, Atiy* *ah, Segal and Wilson with recent constructions of Greenlees and recent insight of the author to show* * that the local cohomology filtration is well-suited for addressing questions about both the algebra and g* *eometry associated to free G-actions on manifolds. We begin the paper by giving a retrospective of the Con* *ner-Floyd-tom Dieck exact sequence in Z=p-equivariant bordism. We then focus on the case of (Z=p)2, defin* *ing the local cohomology filtration. The new insights come when we interpret this filtration in the lang* *uage of families in order to give a geometric understanding of the filtration. We then show how classes we b* *uild using this filtration ___________ 1991 Mathematics Subject Classification. Primary: 55. 1 2 DEV P. SINHA are well-suited to analysis through the use of Atiyah-Segal-Wilson K-theory inv* *ariants, taking our time to define these invariants. We close with ways we see in which these techniques co* *uld be applied. Note that Igor Kriz has told us that he has proved collapse of the local coho* *mology spectral sequence for abelian groups using different methods. 2. The Conner-Floyd-tom Dieck Exact Sequence Conner and Floyd were strongly interested in studying free G-actions on manif* *olds [7]. They saw that bordism theory, newly revolutionized by Thom's work, would give information bey* *ond what homology would see. To that end, they were interested in bordism theory in which repres* *entative classes and bordisms were equipped with a free G-action. They realized that such a bordism * *theory was equivalent to the bordism homology of the corresponding classifying space. Throughout we let * *EG denote a contractible free G-space and BG denote the quotient of EG by G. Proposition 2.1.The bordism module of stably complex free G-manifolds is isomor* *phic to U*(BG). Proof.Consider the following diagram: fM --~f--!EG ?? ? y ?y M --f--!BG. Given a representative M with reference map f to BG, pull back the principal G-* *bundle EG to get fM, which is in fact a free G-manifold. Conversely, starting with a free G-manifold* * fM, there is no obstruction to constructing a map ~fto EG. Pass to quotients to obtain f :M ! BG. These maps are well-defined, as we apply the previous argument to the manifol* *ds which act as_bordisms. The composites of these maps are clearly identity maps. * * |__| Suffice it to say that the bordism module U*(BG) contains a great deal of in* *formation about free G actions. It is the main object of study in this paper. It remains mysterious fo* *r most groups, being easy to describe only for cyclic groups [20] and having been computed at all for onl* *y a few groups, including elementary abelian groups [17]. Conner and Floyd realized that this module is b* *est understood in relation to bordism in which more general actions are allowed. We carefully define this * *theory now. We begin by recalling the definition of geometric complex equivariant bordism* *. Fix UC (UR, respectively) a complex (resp. real) representation of which a countably infinite direct sum * *of any representation of G appears as a summand. Let BUG (n) (BOG(n), respectively) be the Grassmanian o* *f complex (resp. real) n-dimensional linear subspaces of UC (resp. UR), topologized as a direct * *limit of finite dimensional Grassmanians. Definition 2.2.A tangentially complex G-manifold is a pair (M, ø) where M is a * *smooth G-manifold and ø is a lift to BUG (n) of the map to BOG(2n) which classifies TM x Rk for s* *ome k. We can define bordism equivalence in the usual way to get a geometric version* * of equivariant bordism. Definition 2.3.Let U,G*denote the ring of tangentially complex G-manifolds up * *to bordism equivalence. Let U,G(X) denote the module (over U,G*) of tangentially complex G-manifolds * *equipped with a reference map to X up to bordism equivalence. Recall that one may prove that geometric bordism theory has suspension isomor* *phisms or satisfies excision using transversality arguments [21]. Because transversality holds bet* *ween G-manifolds when the range manifold has trivial action, U,G*(-) has suspension isomorphisms whe* *n the G-action on the suspension coordinate is trivial. On the other hand, it does not have isomorphi* *sms induced by suspension THE GEOMETRY OF THE LOCAL COHOMOLOGY FILTRATION IN EQUIVARIANT BORDISM* * 3 by (one-point compactified) representations in general, so we say that it is an* * equivariant homology theory "indexed on the trivial universe" (see [28] for the corresponding statement in * *terms of spectra). There is a map U*(BG) ! U,G*defined simply because a free G-manifold is a G* * manifold. Conner and Floyd realized that this map fits in a natural exact sequence. It will be best * *for us to describe this exact sequence using the language of families, as we will be using this language thro* *ughout the paper. There is an excellent discussion of families in [8]. Definition 2.4.A family of subgroups of a group G is a set of subgroups F such * *that if H 2 F and H0 is conjugate to a subgroup of H then H02 F. Definition 2.5.Let F be a family of subgroups of G. An F-space is a G-space all* * of whose isotropy groups are in F. There is a terminal object, unique up to homotopy, in the category of F-space* *s which is called EF. In simpler terms, EF is an F-space which is contractible (after forgetting the G-a* *ction). For example, if F is the family consisting of only the identity subgroup, EF is simply EG. We wil* *l use the term F-manifold to refer to a manifold which is an F-space. If F2 F1 are two families, define an (F1, F2)-manifold to be F1-manifold wi* *th (possible empty) boundary such that its boundary is an F2-manifold. Definition 2.6.Two (F1, F2)-manifolds M and N are bordant when there is an (F1)* *-manifold W such that M t-N is equivariantly diffeomorphic to a codimension zero sub-manifold of* * @W and @W -(M tN) is an F2-manifold. Let U,G*[F] (respectively U,G*[F1, F2]) denote the bordism module of tangen* *tially complex F (resp. (F1, F2))-manifolds. If F2 F1 are families, there is an inclusion map U,G*[F* *2] ! U,G*[F1] as well as a boundary map U,G*[F1, F2] ! U,G*-1[F2]. There are also inclusion maps for p* *airs of families, and in particular inclusion maps of the form U,G*[F1] ! U,G*[F1, F2], since U,G*[F1* *] ~= U,G*[F1, OE], where OE is the empty family. Proposition 2.7.Let F3 F2 F1 be three families of subgroups of G. The seque* *nce . .!. U,G*[F2, F3] ! U,G*[F1, F3] ! U,G[F1, F2] @! U,G*-1[F2, F3]* * ! . . . is exact. This exact sequence is essentially the long exact sequence of a triple. In fa* *ct, one may deduce it from the long exact sequence of a triple by showing that U,G*[F] ~= U,G*(EF) and U,G*[* *F1, F2] ~= U,G*(EF1, EF2). We now specialize to the case G = Z=p, for which there are only three familie* *s: the empty family, the family of the trivial (identity) subgroup which we call e, and the family of al* *l subgroups which we call A. We apply Proposition 2.7 using these three families and identify the terms. Not* *e that U,G*[A, OE] = U,G*. Next consider U,G*[e, OE]. By definition this is the bordism module of free G-* *manifolds, which we have identified in Proposition 2.1, and the map from it to U,G*is simply the forget* *ful map. Hence the exact sequence reads as follows. . .!. U*(BG) i! U,G*j! U,G*[A, e] @! U*-1(BG) ! . . . Conner and Floyd went further in [7] to identify U,G*[A, e]. The map j can b* *e interpreted in terms of "reduction to fixed sets". The relevant general fact is as follows. * * S Proposition 2.8.An (F1, F2)-manifold M is bordant to any smooth neighborhood N * *( H=2F2MH ) of the locus of points in M fixed by subgroups not in F2. 4 DEV P. SINHA Proof.Let W = M x [0, 1], with "straightened corners". Then @W is anSF2 manifo* *ld outside of the codimension zero submanifolds M x 0 and a tubular neighborhood of ( H=2F2MH ) * *x 1, so_W is the required bordism. * * |__| When G = Z=p, tubular neighborhoods of fixed sets are diffeomorphic to G vect* *or bundles over those fixed sets, so Conner and Floyd were able to identify U,G*[A, e] in terms of b* *ordism modules of manifolds with trivial G-action which equipped with G-vector bundles. Returing to our original question, namely as to the structure of U*(BG), we * *see that this exact sequence gives rise to two-stage filtration into the free G-manifolds which are the boun* *daries of general G-manifolds (the cokernel of @) and those which are not (the image of i). We may now state * *that a main goal of this paper is to give a similar description of the filtration on U*(BG) for more ge* *neral G which arises from Greenlees's local cohomology machinery. First, however, we wish to gain computa* *tional understanding from our current exact sequence. Unfortunately, to this day the geometric theories U,G*are mysterious, with p* *artial knowledge for abelian G and a complete computation for Z=2 which may be deduced from [25]. Hence it i* *s not computationally useful to study U*(BG) using this exact sequence. To make this exact sequence* * more algebraically manageable requires two steps. The first step was taken by tom Dieck in [9], who realized that Conner and Fl* *oyd's exact sequence was related to localization methods in equivariant K-theory being developed at * *the time by Atiyah and Segal [2]. To make this connection, tom Dieck defined a more homotopy theoretic* * version of equivariant bordism. He crafted a spectrum MUG , analogous to MU, whose corresponding infi* *nite loop space is lim-!VMaps(Sn V, T(,G|V)|)G, where V ranges over isomorphism classes of complex* * representations of G, Sn V is the one-point compactification of the Whitney sum of Cn with trivial G * *action and V , and T(,G|V)| is the Thom space of the universal complex G-bundle. See [8] for a recent trea* *tment of tom Dieck's construction. Though this spectrum has been defined for thirty years and studi* *ed actively, it is only recently that its coefficients have been understood in any cases [18, 25]. The equivariant homology and cohomology theories corresponding to this spectr* *um have suspension iso- morphisms with respect to any representation. Moreover, the cohomology theory h* *as a Thom isomorphism for complex G-vector bundles, and so has some theory of characteristic classes.* * An interesting class of G-vector bundles are complex representations V , considered as bundles over a p* *oint. The associated Euler classes are denoted eV 2 MUmG(pt.) = MUG-m. There is a Pontryagin-Thom map from* * U,G*to MUG*, but it is not an isomorphism, since transversality arguments fail in an equivariant* * setting [9] (in particular the Euler classes eV are non-zero classes in negative homological degrees where geo* *metric bordism is zero by definition - see [8]). Reflecting on the Conner and Floyd exact sequence for G = Z=p, tom Dieck cons* *idered the cofiber sequence EG+ ! S0 ! gEG, whose long exact sequence in U,Gtheory is the Conner-Floyd exact sequence. We * *say that the resulting long exact sequence in MUG theory is the tom Dieck exact sequence. tom Dieck realized that the map from MUG*to MUG*(gEG) had an interpretation i* *n terms of localization (and hence a connection with the work of Atiyah and Segal [2]). The key observa* *tion is that one model for EG for cyclic G is as the unit sphere in the representation 1 V which we d* *enote S( 1 V ), where V is the standard representation of G as the roots of unity in the complex number* *s. Hence a model for gEG is the one-point compactification of that representation, which we denote S1V . For any commutative ring R and element e 2 R let R[e-1] denote the localizati* *on of R obtained by inverting e. Lemma 2.9. For any G, ^MUG*(S 1 V) ~=MUG*[e-1V] as rings. THE GEOMETRY OF THE LOCAL COHOMOLOGY FILTRATION IN EQUIVARIANT BORDISM* * 5 Proof.The left-hand side ^MUG*(S 1 V) is a ring because S 1 Vis an H-space thro* *ugh the equivalence S 1 V^ S 1 V~=S 1 V. To compute the left-hand side, apply ^MUG*to the identification S 1 V = lim-!S * *nV. After applying the suspension isomorphisms ^MUG*(S kV) ~=M^UG*+|V(|S k+1V), the maps in the result* *ing directed system are multiplication by the eV. * * |___| tom Dieck also uses the fact that transversality arguments go through in the * *presence of free G actions to show that MUG*(EG) ~=MU*(BG). Here one could also use Adams's transfer argum* *ent [1] to show (MUG ^ EG+)G ~=MU ^ BG+. Hence the tom Dieck exact sequence reads . .!.MU*(BG) ! MUG*! MUG*[eæ-1] ! . ... Moreover, just as Conner and Floyd computed the third term in the geometric set* *ting, one can compute the localization MUG*[eæ-1]. The fact that one can compute such a localization * *is the starting point for the computation of MUG*for abelian G [25], which reveal that the structure of these* * rings is quite complicated. In fact, their structure is complicated to an extent which renders the tom Diec* *k exact sequence not usable for computation of MU*(BG). The next step one must take to make this exact sequence more computable is to* * "free up" the spectrum MUG . Consider the mapping spectrum Maps(EG+, MUG ). Its homotopy groups are th* *e homotopy groups of the G-maps from EG+ to MUG . Because EG+ is free, we have MapsG(EG+, MUG ) = Maps(BG+, i*(MUG )) = Maps(BG+, MU) where i* takes a G-spectrum and passes to the underlying spectrum. Note as wel* *l that because EG+ is simply S0 non-equivariantly we have that i*(Maps(EG+, MUG )) = MU as well. H* *ence by Adams's transfer argument [1] we have that EG+ ^G Maps(EG+, MUG ) = BG+ ^ i*(Maps(EG+, MUG )) = MU ^ BG+. Finally, let tGMUG denote the spectrum Maps(EG+, MUG ) ^ gEG. For cyclic G the * *coefficients of tG are obtained by inverting the Euler class of the standard representation, as in Lem* *ma 2.9, in the coefficients of Maps(EG+, MUG ) which again are just MU*(BG). Thus if we smash the spectrum Map* *s(EG+, MUG ) with tom Dieck's cofiber sequence and take homotopy groups we get . .!.MU*(BG) ! MU*(BG) ! MU*(BG)[eæ-1] ! . . . The terms in this exact sequence are all computable. Landweber used the Gysin* * sequence for the fiber bundle S1 ! BZ=p ! CP1 to show in [19] that MU*(BZ=p) ~=MU*[[x]]=[p]F(x) where* * [p]F(x) is the p-series in the formal group law over MU (see [23]). More topologically, pF(x) * *is the Euler class of the pth tensor power of the tautological bundle over CP1 , whose associated sphere * *bundle is the fiber bundle above. Hence we may finally be explicit about the exact sequence, which reads (1) . .!.MU*(BG) ! MU*[[x]]=[p]F(x) ! MU*((x))=[p]F(x) ! . .,. where in general we use R((x)) to denote R[[x]][x-1]. We may now compute MU*(BG* *), as was first done by Landweber [20]. Note that MU*(BG) is a ring with unit, since it is isomorphi* *c to MUG*(EG) and EG is an H-space. Theorem 2.10. MU*(BZ=p) is generated as a module by the unit class and by class* *es yiwith relations pyi+ a1yi-1+ . .+.ai-1y1 = 0, where aiis the coefficient of xiin the series pF(* *x). 6 DEV P. SINHA Proof.We compute the cokernel and kernel of the map MU*[[x]]=[p]F(x) ! MU*((x))* *=[p]F(x) in the sequence of Diagram 1 above. The cokernel is generated by the negative powers o* *f x. Let yidenote the image of x-iin MU*(BG). By multiplying the p-series by negative powers of x, we* * have that px-i+ a1x1-i+ . .+.ai-1x + non-negative powers=of0x, so we deduce the stated relation. Clearly all relations arise from multiplying * *the p-series by a Laurent polynomial in x, so these relations suffice to generate all relations. The kernel is generated by the series pF(x)=x. It is a free MU* module on one* * generator. Because the cokernel consists entirely of MU*-torsion (p-torsion, in fact) an* *d the kernel is free, we see that MU*(BG) splits as this sum of these modules. Moreover, since the ideal gen* *erated by the unit class is non-zero and free, it must map isomorphically to its image under i, namely t* *he_ideal_generated by pF(x)=x. * * |__| It is interesting to note that the Atiyah-Hirzebruch spectral sequence to com* *pute this module has as many non-trivial extensions as possible. To summarize the exact sequences we have introduced, for G = Z=p we have a co* *mmutative diagram as follows. U*(BG) --i--! U,G* --j--! U,G*[A, e] ? ? ? ~=?y PT?y PT?y MU*(BG) ----! MUG* ----! MUG*[eæ-1] ?? ? ? y ?y ?y MU*(BG) ----! MU*(BG) ----! MU*(BG)[eæ-1] flfl fl fl fl flfl flfl MU*(BG) ----! MU*[[x]]=[p]F(x)----!MU*((x))=[p]F(x) The middle two rows of this diagram are known as the Tate diagram [15]. We may use this commutative diagram to give geometric representatives of the * *classes in U*(BG). The key observation is the following. Proposition 2.11 (Lemma 3.1 of [8]).The class e-1æ2 MUG*[eæ-1] is the image und* *er the Pontryagin- Thom map of D(æ), the unit disk in the representation æ, which is a G-manifold * *with A-boundary. Sketch of Proof.The appropriate setting for the proof is that of "stable manifo* *lds". Given a manifold with boundary M, @M and a map of pairs from this manifold to D(V ), S(V ) for s* *ome representation V there is a Pontryagin-Thom map which produces an element of ßdimM-dimV(MUG ). N* *on-equivariantly, such a map does not produce any classes beyond those produced by the standard P* *ontryagin-Thom map (it cannot, since the usual Pontryagin-Thom map is an isomorphism). Equivariant* *ly, this procedure does produce new classes, in fact giving all of MUG*[4]. For example, eV is the image under this map of the stable manifold in which M* * is a point which maps to 0 2 D(V ). It is then conceivable that the inverse of this class in the loca* *lization is given_by the image under the Pontryagin-Thom map of D(V ) mapping to a point (zero-disk). * * |__| Corollary 2.12.The generator yi2 U2i-1is represented by S( iæ). It is quite straightforward to geometrically understand the unit class in MU** *(BG) as well by running through the isomorphism of Proposition 2.1. Proposition 2.13.The unit class in MU*(BG) is represented by G itself, as a zer* *o-dimensional manifold. THE GEOMETRY OF THE LOCAL COHOMOLOGY FILTRATION IN EQUIVARIANT BORDISM* * 7 Hence, we not only have an algebraic computation of U*(BG) but we have expli* *cit geometric under- standing of most of the generators as (quotients of) free G-manifolds which are* * boundaries of arbitrary G-manifolds. Note that having a free G-manifold as the boundary of an arbitrary* * G-manifold is precisely the setting for studying the free G-manifold using the K-theoretic invariants o* *f Atiyah and Segal [2] as further developed by Wilson [29]. We will develop these ideas below. Our goal for more general G is to develop similar understanding of U*(BG), w* *ith as much control as possible over both the algebra and the geometry. We will outline a program for * *doing so in this paper, describing a computational tool which generalizes the discussion of this sectio* *n, namely the local coho- mology filtration. There are in fact two ways in which one may generalize the e* *xact sequences developed in this section, the other known as the families filtration, which is described* * very well in [8]. Briefly, one filters EG by a sequence of maps EG ! EF1 ! . .E.Fn, where Fi+1is obtained from* * Fiby adding one (conjugacy class of) subgroup. For many theories such as bordism, the equivaria* *nt homology of the cofiber of EFi! EFi+1is equivalent to (non-equivariant) homology of some sort of classi* *fying space (much as in Proposition 2.1). The local cohomology filtration is coarser than this filtr* *ation but surprisingly much more computable once one "frees up" the theories involved. 3.The Local Cohomology Filtration Just as the survey [6] successfully illustrated equivariant theory by concent* *rating on Z=2, we are going to focus on (Z=p)2, giving an indication of how things would generalize as we g* *o along. In the last section it was useful for compuations to notice that the third te* *rm Conner-Floyd-tom Dieck sequence, namely the one associated to S 1 æ, is a localization. To continue to* * have localizations involved in our computations, we are going to assemble EG from pieces of the form S 1 Vf* *or various V . For G = (Z=p)2, EG is no longer given by an action on a sphere but on a produ* *ct of spheres. Let G be generated by elements x and y (so that px = py = x + y - x - y = 0). Let Vx * *(respectively Vy) be the standard representation of Z=p pulled back to G through its projection to G=* * (respectively G=). We have that EG = S( 1 Vx) x S( 1 Vy). Hence we smash the cofiber sequence S( 1 Vx)+ ! S0 ! S 1 Vxwith the corresponde* *nce sequence with Vx replaced by Vy to get a commutative diagram as follows: EG+ ----! S( 1 Vx)+----! S( 1 Vx)+ ^ S 1 Vy ?? ? ? y ?y ?y (2) S( 1 Vy)+ ----! S0 ----! S 1 Vy ?? ? ? y ?y ?y S( 1 Vy)+ ^ S 1 Vx----! S 1 Vx ----! S 1 (Vx Vy). In this situation, 2EG+ is known as the total cofiber of the square diagram * *consisting of the last two spaces of the last two rows in the above diagram. All of the spaces in question* *, except for S0, are of the form S 1 Vas desired. We pause for a moment to recall basic definitions of cubical diagrams and the* *ir total cofibers (from [10]) and give a canonical filtration on those total cofibers. Let n_= {1, . .,.n} and let P(n_) denote the category of subsets of n_with mo* *rphisms given by inclusion. A cubical diagram of spaces (of dimension n) is a functor from P(n_) to based s* *paces. We often let Xo denote such a functor, XS denote the value of this functor on S 2 ob(P(n_)) and* * fS,S0denote the unique 8 DEV P. SINHA morphism from XS to XS0when S S0. A morphism of cubical diagrams is defined a* *s usual for a diagram (or functor) category. Cubical diagrams are a convenient diagram category in part because a morphism* * of cubical diagrams may be viewed as a cubical diagram of dimension greater by one. Let i be an ele* *ment of n_+_1_and by let 'idenote the order-preserving inclusion from n_to n_+_1_for which i is not in t* *he image. We may define an n + 1-dimensional cubical diagram Zo from a morphism Xo ! Yo by letting Z'i(S)=* * XS, Z'i(S)[i= YS and the map from Z'i(S)to Z'i(S)[ibe the map from XS to YS. Conversely, given a* * cubical diagram of dimension n one may view it as a morphism of cubical diagrams of dimenstion n -* * 1 in n ways. We now define the total cofiber of a cubical diagram, which is a functor from* * cubical diagrams to spaces we will use extensively. For a cube In whose vertices are naturally labelled by* * subsets of n_let @SIn for a subset S of n_denote the face whose vertices are subsets S. Definition`3.1.The total cofiber of an n-dimensional cubical diagram Xo is the * *quotient of the union XS x @n_-SIn through the identifications o For x 2 XS and S S0, x x @n_-S0In x x @n_-SIn is identified with fS,S0* *(x) x @n_-S0In. o Points of the form * x @AIn, where * is the basepoint of XS and A is a fac* *e which does not meet the initial vertex (labelled by the empty set), are all identified to a basepo* *int. There are Puppe sequences for cubical diagrams. Given a map of cubes Xo ! Yo * *one may take the Puppe sequence for each map XS ! YS. These sequences fit together to define a * *sequence of cubical diagrams which we also call a Puppe sequence. Moreover, we have the following. Proposition 3.2.Let Ck be the total cofiber of the kth cube in the Puppe sequen* *ce of a map Xo ! Yo. The sequence C0 ! C1 ! . .i.s a sequence of spaces weakly equivalent to the Pup* *pe sequence associated to the induced map from the total cofiber of Xo to the total cofiber of Yo. There is a canonical filtration on the total cofiber of a cubical diagram. Proposition 3.3.Let h be a homology theory and let Xo be a cubical diagram of d* *imension n. There is an n-column spectral sequence whose E1 term is given by M E1p,q= hq-#S(XS) S n_ #S=n-p and 0 0 d1|h*(XS)= S0 S(-1)#S .~(S,S()fS,S0)*, where ~(S, S0) is the element of S0not in S, which converges to the homology of* * the total cofiber of Xo. Partial proof.We supply a proof, which is immediately generalizable, in the cas* *e of square diagrams. Consider the sequence of square diagrams * ----! * * ----! X1 XOE ----! X1 ?? ? f ? ? f ? ? y ?y !1 ?y ?y !2 ?y ?y * ----! X1,2 X2 ----! X1,2 X2 ----! X1,2. These maps give rise to a sequence of maps between total cofibers of those diag* *rams, which we denote T0 f1!T1 f2!T2. As there is for any such sequence of maps, there is a three-co* *lumn spectral sequence whose E1 term consists of the homology of T0, the cofiber of f1 which we denote* * cof(f1), and cof(f2). Let gi denote the map from Ti to cof(fi) (with g0 the identity map by conventio* *n) and @i the map in the Puppe sequence from cof(fi) to Ti-1. The d1 of this spectral sequence is g* *iven by gi-1O @i. This spectral sequence converges to the homology of T2. THE GEOMETRY OF THE LOCAL COHOMOLOGY FILTRATION IN EQUIVARIANT BORDISM* * 9 We have that T0 is simply X1,2. Next we see that the cofiber of the map from * *T0 to T1 is the total cofiber of the cube defined by f1 as a map of squares. This total cofiber is in* * turn the total cofiber of the square cofiber(* f1!*)----! cofiber(* f1!X1) * ----! X1 ?? ? ?? ?? y ?y = y y cofiber(* f1!X2)----!cofiber(X1,2f1!X1,2),X2----!* which is simply X1^ X2. Similarly, the cofiber of the map from T1 to T2 is 2* *XOE. Finally, we identify the d1 differential. As we stated above, it suffices to * *understand the boundary maps in the Puppe sequence of total cofibers. By Proposition 3.2, we may instead loo* *k at the Puppe sequence of cubes. Unraveling the definitions, we see that the boundary maps used to define* * d1 in a spectral sequence of a filtration are a wedge of suspensions of structure maps from the original * *cube, with signs introduced by the interchange of suspension coordinates. * * __ * *|__| We apply this proposition to the square diagram S0 ----! S 1 Vy ?? ? y ?y S 1 Vx ----! S 1 (Vx Vy). where the homology theory in question is either MUG or respectively Maps(EG+, M* *UG ), and make identifications using Lemma 2.9 in order to deduce the following theorem origin* *ally due to Greenlees. Once again we restrict our attention to the case of (Z=p)2. Theorem 3.4 (Greenlees).There are three-column spectral sequences converging to* * MU*(B(Z=p2)) whose E1 terms are given by MUG*[e-1Vx, e-1Vy] +,--MUG*[e-1Vx] MUG*[e-1Vy] - MUG*, and (3) MU*[[x, y]][(xy)-1]=(pF(x), pF(y)) +,--MU -1 -1 *[[x, y]][x ]=(pF(x), pF(y)) MU*[[x, y]][y ]=(pF(x), pF* *(y)) - MU*(BG) = MU*[[x, y]]=(pF(x)), pF* *(y)), where the grading defined so that these columns are in degrees -2, -1 and 0 and* * where the maps above are the canonical maps arising from the fact that each of the columns, which ar* *e graded rings, is obtained from the next by localization. This spectral sequence is called the local cohomology spectral sequence, beca* *use in fact the E2 term is precisely what is known as the local cohomology of the ring MUG*, respectively * *MU*(BG), at the ideal generated by the Euler classes eVxand eVy[16]. The algebra of local cohomology * *and its relevance to this situation is well-documented in the work of Greenlees and his collaborators [11* *, 12, 14]. In short, local cohomology groups of a ring R at an ideal I are the derived functors of the I-t* *orsion functor. Because of its connection with well-developed homological algebra, the local cohomology fi* *ltration has lead to many advances in equivariant topology, and the topology of classifying spaces in par* *ticular. 10 DEV P. SINHA 4.A Geometric Version of the Local Cohomology Filtration Here we will take a geometric approach by recasting this spectral sequence in* * the language of families. First we replace our diagram of nine spaces from Diagram 2 - the cofiber sequen* *ce of cofiber sequences - with a diagram of bordism modules defined by families. Then, we give geometric defin* *itions of the differentials in this spectral sequence. Such analysis leads to conditions for collapse of th* *e spectral sequence, which we verify for G = (Z=p)2. Let e and A continue to denote the families (of subgroups of (Z=p)2 now) cons* *isting of only the trivial group and of all subgroups, respectively. By abuse, let denote the family c* *onsisting of the subgroup generated by x along with the trivial subgroup, and similarly for . Let * * [ denote the family consisting of the subgroups generated by x and by y along with the trivial subg* *roup. Consider the following diagram of bordism modules. U,G*[e, OE]h11----! U,G*[,hOE]12----! U,G*[, e] ? ? ? v11?y v12?y v13?y (4) U,G*[,-OE]h21---! U,G*[A,hOE]22----! U,G*[A, ] ? ? ? v21?y v22?y v23?y U,G*[,-e]h31---! U,G*[A,h]32----! U,G*[A, [ ]. There are boundary maps which continue the sequences defined by the first two* * rows and columns. For example, h13: U,G*[, e] ! U,G*-1[e, OE] by defined by taking the boundary o* *f a representative class. These are simply the boundary maps of Proposition 2.7. There are also boundary maps f* *or the last column and row. Given an (A, [ )-manifold M representing a class in bordism, the bo* *undary map of the last column sends [M] to [ ((@M)x)] the class represented by a tubular neighborhood * *(in @M) of the x-fixed set of @M. Proposition 4.1.The rows and columns of Diagram 4, continued with the boundary * *maps defined above, form long exact sequences. For the first and second rows and columns this is an application Proposition * *2.7. For the last row and column, the verification is similar to that of the long exact sequence of a pai* *r in bordism. As mentioned above, we are constructing an analog of Diagram 2. Proposition 4.2.The sequences of Diagram 4 map to those of Diagram 2 through th* *e Pontryagin-Thom map. Our cubical diagram formalism allows us to use Diagram 4 to construct an anal* *og of the local cohomology spectral sequence. Theorem 4.3. There is a spectral sequence whose E1 term is given by U,G*[A, [ ] v23-h32- U,G*[A, ] U,G*[A, ] h22-v22 U,G* **[A, OE] = U,G*, which converges to MU*(BG). This spectral sequence maps to those of Theorem 3.4. Because the maps in this spectral sequence are geometrically defined we may f* *urther our understanding of how the various sub-quotients in the local cohomology filtration give rise t* *o classes in MU*(BG). For example, we see that the module U,G*[A, [ ] maps to U,G*[e, OE] * *= MU*(BG) by sending a class [M] to [@( (@M)x)], which is the boundary of a tubular neighborhood (in @* *M) the the x-fixed set of @M. We may verify this by starting at (A, [) and composing a verticle arr* *ow with a horizontal arrow in Diagram 4. If M1 = @W1 and M2 = @W2 are free Z=p-manifolds which bound arbit* *rary Z=2-manifolds, THE GEOMETRY OF THE LOCAL COHOMOLOGY FILTRATION IN EQUIVARIANT BORDISM* * 11 then W1x W2 is naturally a (Z=p)2-manifold with boundary which represents a cla* *ss in U,G*[A, [ ]. Under the map to MU*(BG), this class maps to [M1x M2]. We may understand the d2 differential as the map defined by h12applied to a l* *ift of a class [M] 2 U,G*[A, OE] which maps to zero under v22and h22. Geometrically, this differen* *tial determines whether a manifold [M] which is cobordant to both a manifold with no -fixed points a* *nd to another with no -fixed points is actually cobordant to a free manifold. Of course, in order to prove that a differential is zero it suffices to find * *non-zero classes in MU*(BG) to which the classes in the spectral sequence coincide. Theorem 4.4. The spectral sequence of Diagram 3 in Theorem 3.4 collapses at E2. Proof.The kernel of the d1 map MU*[[x, y]]=(pF(x)) ! MU*[[x, y]][x-1]=(pF(x), pF(y)) MU*[[x, y]][y-1]=* *(pF(x), pF(y)) is the ideal generated by the series pF(x)=x . pF(y)=y. We claim that this seri* *es is in the image of the map from MU*(BG) to this quotient in the filtration, and hence cannot support a* * d2 differential. For dimensional reasons, no other differentials are possible. The class which maps to the series pF(x)=x . pF(y)=y is the unit class in MU** *(BG), represented by G itself, a zero-dimensional manifold. This follows from the fact that the unit c* *lass in MU*(BZ=p)_mapped_ to pF(x)=x in MU*(BZ=p) in the Tate sequence, as we established in Theorem 2.10* *. |__| Perhaps the most fun and interesting aspect of this analysis occurs for in th* *e middle column of the spectral sequence, even though it cannot support a differential simply for dime* *nsional reasons. If two classes [M] 2 U,G*[A, ] and [N] 2 U,G*[A, ] have the same image in U,G* **[A, [ ] then if we take [@M] 2 U,G*[, OE] it must go to zero under v21and hence lift to U,G*[* *e, OE] = MU*(BG). Geometrically, becuase [M] and [N] have the sameSimage in U,G*[A, [ ]* * it means that there is a cobordism W betweenSa tubular neighborhood of H6=orMH , which we denot* *e M , and a tubular neighborhood of H6=orNH , which we call N . The free G-manifold to whic* *h @M is cobordant may be constructed by glueing together @M - ((@M)y), @N - ((@N)x) and S( (@W)x). It is easy to be more explicit if we suppose now that the tubular neighborhoo* *ds M and N are dif- feomorphic and that we can extend that diffeomorphism to include a neighborhood* * of (@M)y in M and (@N)x in N. Then one may plumb M and N together by identifying these neghborhoo* *ds (and ör unding the corners" in a canonical way) to define a manifold with boundary whose bound* *ary has a free G-action. This boundary represents the coset in MU*(BG) associated to [M] and [N] in this* * filtration as above. For example, let G = Z=22. Given a complex representation W let P(W) denote t* *he space of complex one-dimensional subspaces of W with inherited G-action. We consider P(C Vx),* * where by abuse C denotes the one-dimensional trivial representation, which is just the Riemann s* *phere where x acts trivially and y acts by multiplication by -1. Then P(C Vx) x D(Vy) represents a class * *in U,G4[A, ] and, similarly, P(C Vy) x D(Vx) represents a class in U,G4[A, ]. These classes* * map to the same class in U,G4[A, [ ] since neighborhoods of their (x + y)-fixed sets are both di* *ffeomorphic to two copies of D(Vx Vy). These diffeomorphisms which do extend to the boundaries as needed a* *bove. If we plumb P(C Vx) x D(Vy) and P(C Vy) x D(Vx) along these neighborhoods we get a four* *-manifold P whose boundary is free. It is useful to picture the real analog in which two bands - * *S1 x D1 - are glued along S0 x I2 to get a surface diffeomorphic to S2 with four open disks removed. We will show in the next section the the class [@P] 2 MU3(B(Z=2)2) is non-zer* *o. In fact, it must then repsent a öT r class", by which we mean a class which is not in the image * *of the tensor product of MU*(BZ=2) with itself under the Künneth map. As far as we know, ours is the fir* *st construction of a representative for a class in MU*(BA) for any abelian group A which is not a un* *ion of products of spheres. Our plumbing constructions greatly increase the tools with which one can create* * free A-actions. 12 DEV P. SINHA Our techniques clearly generalize beyond Z=p2. One may chase through the anal* *og of Diagram 4 to define classes generated by processes composed the basic ones we have used in this sec* *tion: taking boundaries, taking tubular neighborhoods of fixed sets, and using cobordisms between equiva* *lent classes. 5.Atiyah-Segal-Wilson Invariants In this section we describe some powerful invariants which may be used to stu* *dy free G-manifolds which are boundaries of arbitrary G-manifolds, as are most manifolds constructe* *d from the geometric local cohomology filtration. These invariants are essentially characteristic nu* *mbers in localized equivariant K-theory. We will take a geometric approach, though it will be clear how to sta* *bilize as suggested to us by May. Recall that in ordinary non-equivariant homology, characteristic numbers enco* *de the homomorphism Ø: MU*(X) ! H*(BU x X), which sends the fundamental class in homology of M to its image under (ø x f)* * *in H*(BU x X). When X is an H-space (in particular, when X is a point), this homomorphism is a ring* * homomorphism. There is a corresponding construction at the level of spectra (see for example [27]). Proposition 5.1.The homomorphism Ø corresponds with the Boardman homomorphism h: ß*(MU ^ X+) ! ß*(MU ^ H ^ X+). Proof.Let be the normal bundle of M embedded in some sphere Sk+N, and conside* *r the following commutative diagram Hk+N(Sk+N) --c*--!Hk+NT( )-T(øxf)*----!Hk+N(T(,N ) ^ X+) x x (5) Thom~=?? Thom~=?? Hk(M) (øxf)*----!Hk(BU(N) x X), where c is the collapse map onto as in the Pontrijagin-Thom construction. The* *n the top composite is the homology Boardman homomorphism, and the bottom map is Ø. Commutativity of the r* *ight square and the fact that c* is a homology isomorphism in dimension k + N gives the desired* * correspondence_between homomorphisms. * * |__| Remark.When using the natural basis for ß*(MU^H^X+) these numbers correspond to* * the characteristic numbers of the stable normal bundle, not tangent bundle, of M. A more homotopy theoretic point of view of characteristic numbers facilitates* * their definition for gen- eralized, possibly equivariant, cohomology theories which have a Thom isomorphi* *sm for complex vector bundles. Given M a stably complex manifold, E a cohomology theory for which the* * Thom isomorphism theorem holds, and x 2 E*(M) we can define "evaluation of x on the fundamental * *class of M", as motivated by Equation 5. Definition 5.2.Let x 2 E*(M). With notation as in Proposition 5.1, define the n* *umber x[M] to be the image of x under the composite eE*(M+) ~=!eE*+|(|T( )) c*!eE*+|(|SV ) ~=!E*-n(pt.), where n is the dimension of M. This composite map goes by many names in the literature, including "Gysin map* *ä nd "wrong-way map". The class x[M] is a bordism invariant of M when x is a characteristic cla* *ss of the tangent bundle of M. As in the case of ordinary cohomology, these numbers encode a ring homomo* *rphism. To avoid discussion of the fundamental class and facilitate computation, we may define t* *his ring homomorphism THE GEOMETRY OF THE LOCAL COHOMOLOGY FILTRATION IN EQUIVARIANT BORDISM* * 13 by giving explicit formulae for the characteristic classes dual to multiplicati* *ve generators of E*(BU). We define this homomorphism now in the case of equivariant K-theory, which for fin* *ite complexes may be defined in an entirely analogous way to ordinary K-theory by using G-vector bun* *dles (see [24] for a full discussion). Remark.Equivariant K-theory characteristic numbers play an important role in ge* *ometry, by work of Atiyah, Segal and Singer [2]. For example if fl is a holomorphic vector bundle * *over a complex G-manifold M, let Hi(M, O(fl)) denote the complex G-vector space which is the cohomology o* *f the sheaf of holomorphic sections of fl. Then the equivariant Riemann-Roch theorem of [2] states that X (-1)iHi(M, O(fl)) = fl[M]. i This sheaf cohomology will often vanish for i > 0 so we will have that fl[M] is* * isomorphic to the G-vector space of holomorphic sections of fl. Definition 5.3.Let fl be an n-dimensional complex G-vector bundle over X, and l* *et fl also denote the corresponding class in K0G(X). The characteristic class fii(fl) is the coeffici* *ent of tiin the power series 0 1 X log@ ~i(fl - n) .2tiAeK0G(X)[[t]]. i 0 Recall the coefficient ring for equivariant K-theory, which satisfies a stron* *g form of Bott periodicity. K0Gis isomorphic to the the representation ring R(G), while K1Gis zero. Definition 5.4.Let M represent a class in U,Gn. Define the ring homomorphism ~* *: U,G*! R(G)[[bi]] by setting X ~([M]) = fiI(TM)[M] . bI I=(n1,n2,...nk) Q ni n n n where fiI(E) = 1 i k(fii(E))and bI = b11b22. .b.kk. The homomorphism ~ extends to the Boardman homomorphism ß*(MUG ) ! ß*(MUG ^ K* *G ). We think of the coefficients of ~ as invariants of U,G*with values in the re* *presentation ring. A standard approach to the representation ring is through character theory. We can ask wha* *t parts of the structure of a G-manifold M we need to retain in order to understand the value of E[M] at* * a single conjugacy class of G. For example, the value of E[M] at the identity element, namely its dimens* *ion, will only depend on the underlying manifold M and not the G-action. Atiyah and Segal show in [2]* * that the characters associated to these invariants at a single conjugacy class depend only on the f* *ixed-set structure of M at that conjugacy class, which is given by the image of [M] in U,H*[A, P], where * *H is the cyclic subgroup of G generated by a representative of the conjugacy class and P is the family of p* *roper subgroups of H. Atiyah and Segal noticed that the geometry of reduction to fixed sets coincid* *ed in algebra to localization, which we have already seen in Lemma 2.9. Definition 5.5.Given a conjugacy class C of the representation ring R(G) let p(* *C) be the prime ideal of elements of R(G) whose characters do not vanish at C. Let R(G)p(C)denote the lo* *calization of R(G) at p(C), and let ~: R(G) ! R(G)p(C)denote the canonical map of R(G) into this loca* *lization. Definition 5.6.Let evC:R(G)p(C)! C denote the map which gives the value of a lo* *calized character at C. This map is well-defined because characters of denominators in R(G)p(C)do no* *t vanish on C. Theorem 5.7 (Atiyah-Segal).Let C be a conjugacy class of G and let h 2 C. Then * *the composite U,G*~!R(G)[[bi]] ~!R(G)p(C)[[bi]] evC!C[[bi]] 14 DEV P. SINHA factors though the homomorphism U,G*! U,H*[A, P]. In other words, there is a * *homomorphism ~C: U,H*[A, P] ! C[[bi]] such that ~C O 'h = ev O ~ O ~, where 'h is the canonical map from U,G*to U,H* **[A, P]. We will give a formula for ~C below. Now note that the maps from U,G*to U,H*[A, P] all factor through U,G*[A, e* *]. This observation in conjunction with Theorem 5.7 leads to powerful invariants for MU*(BG). Consider* * the following diagram U,G*[A,-OE]ff---! U,G*[A,fe]i----! U,G*-1[e, OE] = MU*-1(BG) ? ? (6) ~?y ~0?y L R(G)[[bi]]-Ø---!C C[[bi]], where by abuse we are now using Ø to denote the character map in representation* * theory. By exactness of the first row and commutativity of the square we may deduce the following. Theorem 5.8. If class [M, @M] 2 U,G*[A, e] has image under ~0 which is not in * *the image of Ø then [@M] is non-zero MU*-1(BG). In [29], G. Wilson shows that the converse of this theorem is true for groups* * with periodic cohomology. We now present the cohomological definition of ~C, as given in [2]. First not* *e that by Proposition 2.8 an element [M] of U,H*[A, P] is represented by a tubular neighborhood of MH , * *which by the equivariant tubular neighborhood theorem is diffeomorphic to the total space of an H vector* * bundle over MH . We will be viewing classes as represented by such bundles for the purposes of defi* *ning our fomulae. We first define an equivariant version of the Chern character for G-bundles over trivial* * G-spaces. Definition 5.9.Let E be a complex G-vector bundle over a trivial G-space X, whi* *ch decomposes as æi2Irr(G)Eæi(all such G-bundles do so - see [24]). For each g 2 G define X ch(E)(g) = Øæi(g) . ch(Eæi), æi2Irr(G) where Øæi(g) is the trace of g acting on æi. We next introduce the Todd class which is needed to translate from K-theory t* *o cohomology. Let ci denote the ith Chern class of a complex vector bundle fl in H2i(X). Form the ri* *ng H*(X)[a1, . .,.an]=(si(a1, . .,.an) = ci), where sidenotes the ith elementary symmetric polynomial in the variables indica* *ted. Thus, any symmetric polynomial in the aiwill be a polynomial in the ci. Definition 5.10.Let M be a unitary manifold. The Todd class of M, denoted td(M* *), is defined by taking fl to be a complex vector bundle which is a lift of the stable tangent b* *undle of M. Then set nY a td(M) = ____i__-ai. i=11 - e * * 2ßi_) Now let H be a cyclic group of order n, and fix a generator h of H. Let i = e* *( n . And let æibe the representation of H in which the generator h acts on C by multiplication by ii. Definition 5.11.Let Eæibe a complex H-vector bundle over a base with trivial H-* *action and fiber isomorphic to kæifor some k. Define Y ` ii- 1 ' Ui(Eæi) = _______i-aj. 1 j k i - e THE GEOMETRY OF THE LOCAL COHOMOLOGY FILTRATION IN EQUIVARIANT BORDISM* * 15 If E = 1 j, X is two copie* *s of X1,æ1, where æ1 is the sign representation of Z=2. By the above example ~g2(X) is integral with co* *nstant term one. Thus, from the constant terms of the ~C(X) we construct a localized character whose v* *alues are (0, 1, 0) at g, g2 and g3, respectively. Even though each entry of this localized character is an * *integer, the character itself is not integral in that it is not in the span of (1, 1, 1), (i, -1, -i), (-1, 1* *, -1) and (-i, -1, i), the values of characters of irreducible representations of G at g, g2 and g3. Thus X is not t* *he fixed set of any complex Z=4 surface. Finally, we apply these invariants to the Z=22manifold P defined at the end o* *f the previous section. P has two fixed points, whose normal bundles are Vx Vy in the notation of the* * previous section. If we look at the conjugacy class of x + y, there are only these two fixed points and* * the computations proceed as in the first example above to give a value of 1=4 + 1=4 = 1=2 multiplied by * *some integral power series in bi, which shows that the boundary of P represents a non-trivial class in MU3(BZ* *=22). At the conjugacy class x (respectively y) the fixed set is a P1, whose Todd genus is 1 + c1(P1)=* *2 = 1 + a, where a is the generator of H2. As in the first example, there is a two in the denominator com* *ing from the action of x on the normal bundle, and so we once again get that the value of the localized * *character is 1=2 multiplied by an integral series. 16 DEV P. SINHA 6.Directions for Further Work We leave with a couple suggestions for projects which may use these technique* *s. o Do computations in the local cohomology spectral sequence for MU*(BA). Com* *pare this filtration to the Künneth filtration (they seem to be very closely related), which is* * one of the main tools for computations so far [17]. Atiyah-Segal-Wilson invariants may be useful in * *solving extension problems. o Translate these constructions to the Spin setting in order to approach the* * Gromov-Lawson-Rosenberg conjecture for finite groups [3, 26]. Part of the approach of [3] is to fi* *nd arbitrary G-manifolds whose boundaries generate free G-bordism. References [1]J.F. Adams. Prerequisites (in Equivariant Stable Homotopy Theory) for Carls* *son's Lecture. In Algebraic Topology, Aarhus 1982, Volume 1051 of Lecture Notes in Mathematics, Springer, Berlin, * *New-York, 1984. [2]M.F. Atiyah, G.B. Segal and I. Singer. The Index of Elliptic Operators II, * *III. Annals of Mathematics 87 (1968), 531-578 [3]B. Botvinnik, P. Gilkey, S. Stolz. The Gromov-Lawson-Rosenberg conjecture f* *or groups with periodic cohomology. Journal of Differential Geometry 46 (1997), no. 3, 374-405. [4]T. Brcker and E. Hook. Stable equivariant bordism. Math. Z. 129 (1972), 269* *-277 [5]B. Bruner and J.P.C. Greenlees. Connective K-theory of finite groups. Prepr* *int, 2001. [6]G. Carlsson. A Survey of Equivariant Stable Homotopy Theory. Topology 31 (1* *992), 1-27. [7]P.E. Conner and E.E. Floyd. Differentiable Periodic Maps. Springer, Berlin-* *Heidelberg-New York, 1964. [8]S. Costenoble. Chapter 15 of [22]. [9]T. tom Dieck. Bordism of G-Manifolds and Integrality Theorems. Topology 9 (* *1970), 345-358. [10]T. Goodwillie. Calculus II: Analytic Functors. K-Theory 5 (1992) 295-332. [11]J.P.C. Greenlees. K-homology of universal spaces and local cohomology of th* *e representation ring. Topology 32 (1993), 295-308. [12]J.P.C. Greenlees. Local cohomology in equivariant topology. Preprint, 2000. [13]J.P.C. Greenlees and H. Sadofsky. Tate cohomology of theories with one-dime* *nsional coefficient ring. Topology 37 (1998) 145-174. [14]J.P.C. Greenlees and J.P. May. Localization and completion theorems for MU-* *Module Spectra. Annals of Mathematics 146 (1997) 509-544. [15]J.P.C. Greenlees and J.P. May. Generalized Tate cohomology. Memoir of the A* *merican Mathematical Society #543, 1995. [16]R. Harthshorne. Local Cohomology. Lecture Notes in Mathematics, No. 41, (19* *67). [17]D.C. Johnson, W.S. Wilson, and D.Y. Yan. Brown-Peterson homology of element* *ary p-groups, II. Topology and Its Applications, 59 (1994) 117-136. [18]I. Kriz. On Complex Equivariant Bordism Rings for G = Z=p. To appear in Pro* *ceedings of Conference in Honor of Michael Boardman [19]P. Landweber. Cobordism and classifying spaces. Proc. Symp. Pure Math. 22 (* *1971), 125-129. [20]P. Landweber. Unitary bordism of cyclic group actions. Proceedings of the A* *MS 31 (1972) 564-570. [21]I. Madsen and R.J. Milgram. Classifying Spaces for Surgery and Cobordism of* * Manifolds. Volume 92 of Annals of Mathematics Studies. Princeton University Press, Princeton, 1979. [22]J.P. May et. al. Equivariant Homotopy and Cohomology Theory. Volume 91 of t* *he CBMS Regional Conference Series in Mathematics. AMS Publications, Providence, 1996. [23]D. Ravenel. Nilpotence and Periodicity in Stable Homotopy Theory. Volume 12* *6 of Annals of Mathematicas Studies. Princeton University Press, Princeton, 1992. [24]G. Segal. Equivariant K-Theory. Publications of the IHES 34 (1968), 129-151. [25]D.P. Sinha. Computations of Complex Equivariant Bordism Rings. To appear in* * the American Journal of Mathematics, 2001. [26]S. Stolz. Simply connected manifolds of positive scalar curvature. Annals o* *f Mathematics (2) 136 (1992) 511-540. [27]R.E. Stong. Notes on Cobordism Theory. Princeton University Press, Princeto* *n, 1968. [28]S. Waner. Equivariant RO(G)-graded bordism theories. Topology and its Appli* *cations 17 (1984), 1-26. [29]G. Wilson. K-Theory Invariants for Unitary G-Bordism. Quarterly Journal of * *Mathematics Oxford (2) 24 (1973), 499- 526 Department of Mathematics, Brown University, Providence, RI 02906 E-mail address: dps@math.brown.edu