### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# 486 Class Note for MATH M0070 at Purdue

### View Full Document

## 22

## 0

## Popular in Course

## Popular in Department

This 17 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 22 views.

## Similar to Course at Purdue

## Reviews for 486 Class Note for MATH M0070 at Purdue

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 02/06/15

EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS JP MAY CONTENTS 1 The fundamental groupoid and categorical de nitions 3 2 Functors to the category of Gspaces over orbits 5 3 Coherent families of connected covers 7 4 Orientability of spherical G brations 8 5 Coherent families of Thom classes and Thom isomorphisms 10 6 Orientations in ordinary equivariant cohomology l3 7 Concluding remarks 16 References 16 Let G be a compact Lie group and let Ea be an ROGgraded cohomology theory on Gspaces We shall explain a sensible way to think about orientations and the Thom isomorphism theorem in the theory Ea offering an alternative to the approach given by Costenoble and Waner in Both approaches generalize the restricted theory given by Lewis and myself in 15 and both grew out of joint work of Costenoble Waner7 and myself In the study of nonequivariant bundles and their orientations an innocuous rst step is to assume that the base space is path connected The analogous equivariant assumption is that the Gset of components of the base space is a single orbit but this assumption doesn t get us very far There is an entirely satisfactory theory of equivariant Thom isomorphisms and Poincare duality under the much more stringent hypothesis that the base space X be G connected7 in the sense that each XH is nonempty and path connected This is developed in detail in 15 HI 6 and X 5 The basic problem7 then is to generalize that theory to a less restricted class of base spaces The obvious approach is to parametrize changes of ber representation on the fundamental groupoid However doing this directly leads to a fairly complicated and hard to compute7 generalization of equivariant cohomology 5 67 77 8 We seek a variant approach that allows us to work within the framework of ROGgraded cohomology theories7 so that we can apply rather than generalize the preexisting theory of 15 Our essential idea is that7 to obtain a satisfactory general theory7 it seems reason able to give up the idea that the Thom isomorphism must be a single isomorphism Rather we shall de ne it to be an appropriate family of isomorphisms More pre cisely7 we shall de ne an EEorientation of a Gvector bundle7 or more generally of a spherical G bration7 to be a suitably coherent family of cohomology classes Each class in the family will determine a Thom isomorphism7 and these isomorphisms will be nicely related If the base space is G connected7 then all of these Thom 1 2 JP MAY classes and Thom isomorphisms will be determined by one member of the family In general7 all members of the family will be determined by choosing members of the family indexed on components of xed point spaces X H which do not contain any K xed points for K larger than H We begin in Sections 173 by de ning the fundamental groupoid 71X discussing functors de ned on it and constructing the family of Hconnected covers77 of a Gspace X This construction can be expected to have other uses in equivariant al gebraic topology Many arguments in algebraic topology begin with the statement We may assume without loss of generality that X is connected Our H connected covers give a tool that often allows us to give the same start to equivariant argu ments However7 the reader is warned that there are some prices to be paid7 beyond the intrinsic complexity Probably the most signi cant is that the Hconnected cover of a nite G CW complex will in general be in nite dimensional For this reason I have not yet succeeded in obtaining a satisfactory treatment of Poincare duality that starts from the Thom isomorphism theorem given here We de ne the notion of an orientable spherical G bration in Section 4 This does not depend on our Hconnected covers7 but we use these in our de nition of an EEorientation of a spherical G bration in Section 5 Our Thom isomorphism theorem in E cohomology follows directly from the de nition and the work in 15 We specialize to ordinary ROGgraded cohomology with Burnside ring coef cients in Section 6 Nonequivariantly7 orientability as de ned topologically is equivalent to cohomological orientability with integer coef cients We prove that perhaps the most natural topological notion of equivariant orientability is equivalent to cohomological orientability with Burnside ring coef cients We brie y mention other examples in Section 7 The theory here was suggested by joint work with Steve Costenoble and Stefan Waner 5 and later Igor Kriz that began in the late 1980 s and is still largely unpublished In that work we take the more direct approach7 confronting head on the problem of parametrizing the change of ber representations of a Gvector bundle by the equivariant fundamental groupoid of the base space Rather than giving up the idea that an orientation is a single cohomology class7 we construct more complicated cohomology theories in which a single class can encode all the complexity That approach has been exploited in a series of papers on this and related topics by Costenoble and Waner 6 77 87 97 10 A fully coherent theory of orientation will require a comparison of that approach to the one given here Although Costenoble Kriz Waner7 and I sketched out such a comparison some years ago the details have yet to be worked out It is a pleasure to thank Costenoble Waner7 and Kriz for numerous discussions of this material This paper is a very small token of thanks to Mel Rothenberg7 my colleague and friend for the last 32 years I wish I had a better paper to offer since this one should have a sign on it saying speculative may not be useful but it is in one of the areas that Mel has pioneered eg 18 and that I have in part learned from him It has been a privilege to work with him all these years to help make Chicago a thriving center of topology EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 3 1 THE FUNDAMENTAL GROUPOID AND CATEGORICAL DEFINITIONS We here recall our preferred de nition of 77X and give some categorical lan guage that will help us de ne structures in terms of it An equivalent de nition appears in 117 107 and a de nition in terms of Moore loops is given in 177 App We assume that G is a compact Lie group7 and we only consider closed subgroups Let 0G denote the topological category of orbit spaces G H and Gmaps between them where H runs through the closed subgroups of G Let h 39g be the homotopy category of jg Of course7 0G h 39g if G is nite The following observation describes the structure of h 39g for general compact Lie groups G Recall that if a GH H GK is a G map with aeH 9K then g ng C K Lemma 11 Let j a H B be a Ghomotopy between Gmaps GH H GK Then j factors as the composite ofa and a homotopy c GH X I H GH such that CeH t ctH where 00 e and the 0 specify apath in the identity component of the centralizer GgH of H in G Proof Let jeHt gtK Since we can lift this path in GK to a path in G starting at go we may assume that the 9 specify a path in G Now g1Hgt C K for all t so we can de ne d H X I H K by dht gglhgt Since the adjoint d I H MapH K is a path through homomorphisms the MontgomeryZippen Theorem 4 381 implies that there are elements 19 E K such that ho e and dht klgalhggkt De ne C K H Homgangg K by Chh k lh k The image of C may be identi ed with K L where L is the subgroup of elements 19 such that Ce is the inclusion of gangg in K It follows that C is a bundle over its image We may regard d as a path in Homgangg K and we can lift it to a path 19 I H K with 190 5 Thus we may assume that the let specify a path in K Now de ne ct gtklggl Then jeHt CtggK co e and the C are in GgH7 as desired D De nition 12 Let X be a Gspace De ne the fundamental groupoid 77X as follows Its objects are the pairs GHx where H C G and x E XH we think of this pair as the G map x GH H X that sends eH to x The morphisms GHx H GK y are the equivalence classes aw of pairs ao1 where a GH H GK is a Gmap and 01 GH X I H X is a G homotopy from x to yoa Here two such pairs a 01 and 0 of are equivalent if there are Ghomotopies jzazo and920120 suchthat ka0txa and ka1tyojat for a E GH and t E I If G is nite then a 0 and j is constant Composition is evident De ne a functor a 8X 77X H h 39g by sending GH x to GH and sending aw to the homotopy class a of a A Gmap f X H Y induces a functor 77X H 77Y such that 8y 0 f 8X A Ghomotopy h f 2 f induces a natural isomorphism h H To be precise about orientability and orientations we need some abstract def initions and constructions7 which are taken from joint work with Costenoble and Waner The rst encodes the formal structure of the fundamental groupoid De nition 13 A groupoid over a small category 33 is a small category together with a functor a H 33 that satis es the following properties For an object I 4 JP MAY of 53 the ber W5 is the subcategory of consisting of the objects and morphisms that 8 maps to b and id5 i For each object 5 of 53 W5 is either empty or a groupoid in the sense that each of its morphisms is an isomorphism ii Source lifting For each object y E and each morphism B a H 8y in 53 there is an object at E such that 830 a and a morphism y x H y in such that 87 iii Divisibility For each pair of morphisms y x H y and y x H y in and each morphism B 830 H 8 in 33 such that 87 87 0 3 there is a morphism 5 x H x in such that 86 B and y o 5 7 Remark 14 We say that has unique divisibility if the morphism 5 asserted to exist in iii is unique This holds for fundamental groupoids when G is nite7 but not when G is a general compact Lie group When it holds7 is exactly a cate gorie br e en groupoides77 over 33 as de ned by Grothendieck 137 p 166 De nitions 15 Let be a groupoid over i is skeletal over 33 if each ber W5 is skeletal has a single object in each isomorphism class of objects ii is faithful over 33 if a is faithful injective on hom sets iii is discrete over 33 if each W5 is discrete has only identity morphisms Lemma 16 is discrete over 33 and only it is skeletal and faithful over Proof Clearly ifW is skeletal and faithful7 then it is discrete IfW is discrete7 then it is clearly skeletal and it is faithful by Remark 17ii below 1 Remarks 17 Let be a groupoid over i If is skeletal over 53 then divisibility implies that the object at asserted to exist in the source lifting property is unique If is skeletal and faithful over 53 then the morphism asserted to exist in the source lifting property is also unique If is faithful over 53 then it is uniquely divisible ii By divisibility any two morphisms x H y of over the same morphism of 33 differ by precomposition with an automorphism of at over the identity mor phism of Thus is faithful over 33 if and only if the only automorphisms in each W5 are identity maps iii If y x H y is a morphism of such that 87 is an isomorphism7 then 7 is an isomorphism7 as we see by application of divisibility to the equality a ya y 1 id If every endomorphism of every object of 33 is an isomor phism7 as holds in jg then every endomorphism of every object of is an isomorphism Construction 18 Let be a groupoid over We construct the discrete groupoid over 33 associated to i Construct a faithful groupoid with the same objects as by setting Ime H dx The quotient functor H is the universal map from into a faithful groupoid over ii Construct a skeletal subgroupoid W of by choosing one object in each isomorphism class of objects of W5 for each object 5 of 33 and letting W be EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 5 the resulting full subcategory of The inclusion b H is an adjoint equivalence of categories over 53 its left inverse is a retraction p obtained from any choice of isomorphisms from each object of each W5 to an object of We call 5 a skeleton of v By Remark 17iii passage from to creates no new isomorphisms so that we can make the same choices of objects for and for when forming skeleta Then W78 We call this category the discrete groupoid over 33 associated to Lemma 19 Joyal A discrete groupoid b39 over determines and is determined by an associated contravai iant functor F 33 H Sets Proof Given 539 de ne F as follows For an object 5 of B Fb is the set of objects of 53 For a morphism B a H b of 33 and an object y of 53 is the unique object of Wu that is the source of a map covering Given F de ne as follows Its objects are pairs 5 y where b is an object of 33 and y E Fb A morphism a x H b y is a morphism B a H b of 33 such that x Composition and the functor a H 53 are evident D Now return to the fundamental groupoid Notations 110 Let 770 X denote the discrete groupoid over h 39g associated to the fundamental groupoid 71X The quotient functor 77X H 77Xa identi es aw and ozw whenever a 0 so that functors de ned on 77X factor through 77Xa if their values on morphisms are independent of paths The cate gory 770X is obtained from 77Xa by choosing one point in each component of each xed point space The notation 770X is justi ed since the associated con travariant functor h 39g H Sets can be identi ed with the evident functor that sends an orbit GH to the set of components 770XH 2 FUNCTORS TO THE CATEGORY OF G SPACES OVER ORBITS In the next section we show how to construct a system of interrelated H connected covers77 associated to a given Gspace X The interrelationships will be encoded in terms of functors de ned on 71X We describe the target category and the abstract nature of the functors we will be concerned with in this section Let lt2 be the category of compactly generated weak Hausdorff spaces and let G W be the category of G spaces De nition 21 De ne Glt2jlg to be the category of Gspaces over Gm bits The objects of this category are G maps X X H GH and the morphisms are commutative diagrams of Gmaps X H Y XL Li G H H G K There is an evident notion of a homotopy between such morphisms and a resulting homotopy category hG W quotG Let a hGlt2jlg H h 39g be the evident augmen tation functor it forgets the G spaces and remembers the Gorbits 6 JP MAY We think of a Gmap X X H GH as having the total space X and base space GH although we do not require X to be a bration Let V X 1eH C X Then V is an Hspace and the action of G on X induces a Gmap 5 G X V H X that is easily veri ed to be a bijection To avoid pointset pathology we agree to restrict attention to completely regular eg normal Gspaces For such X 5 is a homeomorphism of G spaces The following remark gives a more concrete but less canonical description of the category hGlt2jlg Remark 22 For a commutative diagram of Gmaps GXHVHfgtGgtltKW Li GH asaK with aeH 9K de ne f V H W by g 1fv Then f is an Hmap where H acts on W by hw g lhgw and 23 MW 091 for j E G We call f the Gmap associated to the pair g Suppose that maps f0 and f1 over homotopic Gmaps a0 and al are associated to pairs f0gg and f1gl Then f0a0 and f1a1 are homotopic if and only if there is a path ft connecting f0 to f1 in the space of Hmaps V H W The point is that by Lemma 11 the homotopy a0 2 a1 can be written in the form ateH CtggK where C is a path in GgH starting at e and the conjugate Haction on W is then the same throughout the homotopy De nition 24 Let be a groupoid over h 39g A ltzquot space is a functor Y H hGlt2jlg over h 39g Given a map A H b of groupoids over h 39g a map c5 Y H Y from a ltitfspace Y to a ltquotrmspace Y is a natural transformation zYHY voverhquotG There is a less conceptual but perhaps more easily understood version of this def inition in terms of our concrete description of G Wj39g We shall work throughout in terms of this alternative version Lemma 25 A ltzquot space Y determines and is determined by the following data i An Hspace Zx for each object as in the ber KgH of ii A homotopy class of H maps Z yg Zx H Zy for each morphism y x H y of and element 9 ofG such that a yeH 9K where 87 GH H GK here H acts on Zy by ha g lhga for a E These data must satisfy the following properties iii In Z ygk 2 k 1Z yg for k E K iv Zid e 2 id and Z y g o Zyg 2 Zy o ygg when M o y is de ned Given A H K a map c5 Y H Y from a ltzquot space Y to a ltitmspace Y determines and is determined by H maps C Cm Zx H Z Ax for objects 90 E KgH such that Z Ayg o 0 21 0 Z yg for pairs 79 as in Proof Given the speci ed data set Yx G X Zx and let Y y be the homo topy class of the Gmap associated to the pair Z ygg property iii ensures that Y y is independent of the choice of 9 Conversely given Y let Zx C Yx EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 7 be the Hspace over the orbit 5H and let Z y g Zx H Zy be the composite of YW and multiplication by 9 1 Similarly given C let on Yx H Y Ax be the Gmap associated to the pair Came and conversely given c5 let Cm be the restriction Zx H Z Ax of 1 3 COHERENT FAMILIES OF CONNECTED COVERS A standard tool in equivariant algebraic topology is to study G spaces by means of their diagrams of xed point spaces On the diagram level it is quite trivial to give a notion of an H connected cover The following de nition encodes that trivial starting point of our work De nition 31 An jgspace is a continuous contravariant functor 0G H 24 and a map of jgspaces is a natural transformation Let jgW denote the cat egory of jgspaces For a Gspace X de ne the xed point jgspace ltIgtX by DX XH For a xed point x E XH let XHx denote the component of x in XH De ne the H connected cover of ltIgtX at x to be the sub j39Hspace ltIgtXx of the j39Hspace X such that ltIgtX Xx for J C H We can lift this essentially nonequivariant structure to the equivariant level by means of a construction due to Elmendorf 12 see also 16 V 3 and VI 6 We shall gradually make sense of and prove the following result in this section Theorem 32 Let X be a Gspace There is a 77Xspace X such that for x GH H X Xx is the Hconnected cover of X at do There is a natural map of 71Xspaces X H X where X is regarded as a constant 77Xspace For Jf C H 5 Xx H X is the composite of a canonical weak equivalence Xx H Xx and the inclusion Xx H X Here we are thinking of 77Xspaces in terms of the data speci ed in Lemma 25 We recall the main properties of Elmendorf s construction Theorem 33 There is afunctorIJ jgW H G W and a natural transformation a ltIgtIJ H Id such that for an jgspace T each GH IJTH H TGH is a homotopy equivalence IfX is a GCW complex then XIJTG E DXTG For an jgspace T evaluation of a at G 5 gives a Gmap 8Ge IJTG H TGe When T PX so that TGe X 8Ge 8GJ IJltIgtX H X Thus 8Ge IJltIgtX H X is a weak Gequivalence for any G space X and 8Ge is a G homotopy equivalence if X is a G CW complex De nition 34 The Hconnected cover of X at x E XH is the Hspace Xx IJltIgtX Thus we have homotopy equivalences 8HJ Xx H Xx for J C H Applying IJ to the inclusion of j39Hspaces ltIgtXx C ltIgtX and composing with 8Ge IJltIgtX H X we obtain an Hmap Em Xx H X such that 5 is the composite of the homotopy equivalence 8HJ and the inclusion X x H X 8 JP MAY It remains to discuss the functoriality and naturality of this construction which is the crux of the matter We recall that the functor IJ jgW H Glt2 is given by a categorical twosided bar construction qu BTi39g339g Here 3b 0G H lt2 is the covariant functor that sends the object GH of 0G to the space GH The construction is suitably functorial in all three of its variables An alternative description of IJT may make the functoriality clearer De ne a small topological Gcategory EXT G as follows The object Gspace of EXT G is the disjoint union of the G spaces TGH gtlt GH where G acts on the orbit factors A morphism a 750 H if6 is a Gmap a GH H GH such that ac c and at t where the subscript and superscript s indicate the evaluation of covariant and contravariant functors There is an evident topology and Gaction on the set of morphisms such that the source7 target identity and composition functions are continuous Gmaps Up to canonical homeomorphism of Gspaces szT 13mm G For a homomorphism u G H G a Gfunctor EXT G H ltTCG induces a Gmap IJT H IJT where G acts on the targets by pullback along 4 similarly a Gnatural transformation induces a Ghomotopy Let aw x H y be a morphism in 71X Let a GH H GK be given by aeH 9K and let Cg H H K be the conjugacy injection that sends h to g lhg By Lemma 11 if we change a in its homotopy class then we replace 9 by cg for some 0 in the identity component of GgH Therefore7 although Cg depends on the choice ofg in its coset it does not depend on the choice of a in its homotopy class The homomorphism Cg determines a functor quotH H quotK that sends HJ to Kg IJg and we also have the Hmap HJ H Kg ljg that sends hJ to g lhg g ng Using the functoriality of the twosided bar construction there results an H map Xla wl791XHXy The properties speci ed in iii and iv of Lemma 25 are sati ed Here iii is not obvious since a homotopy is required7 but it is easy to check that the two maps speci ed there are obtained by passage to classifying spaces of categories from naturally equivalent functors and are therefore homotopic Intuitively7 this transport along paths ensures that our Hconnected covers are related by the evident commutative diagrams to inclusions of components of xed point spaces The naturality of the construction with respect to Gmaps X H Y is checked similarly 4 ORIENTABILITY OF SPHERICAL G FIBRATIONS Nonequivariantly there is only one sensible de nition of an orientation of a vector bundle7 but this is a calculational fact that does not extend to the equivariant setting The point is that Z2 g 770000 g 7T0PL71 g 7T0T0Pn g 7700301 for all n 2 1 including 71 00 Nothing like this holds equivariantly There are at least eight different reasonable orientation theories for Gvector bundles cor responding to the linear piecewise linear topological and homotopical categories EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 9 and their stable variants Similarly7 there are six orientation theories for PL G bundles7 four for topological Gbundles7 and two for spherical G brations We shall focus on the stable spherical G bration case7 but the modi cations for the other cases are easily imagined A general framework is given in We begin in this section with the simpler notion of o entability Even this depends on the type of Gbundle or G bration we consider By a G bration7 we understand a map that satis es the Gcovering homotopy property GCHP De nitions 41 Let j39gGlt2quotG be the category of Gspaces 5 X H GH with sections 039 GH H X and sectionpreserving maps of Gspaces over orbits For an Hrepresentation V let SV be the onepoint compacti cation of V We have a G bration G X SV H GH with section given by the points at in nity De ne the category 9 of n sphere G brations to be the full subcategory of j39gGlt2quotG whose objects are the G brations with section that are ber G homotopy equivalent to some G X H SV ii A homotopy between maps in 9 is a sectionpreserving homotopy compare Remark 22 The homotopy category h is a groupoid over the category h 39g iii De ne the stable homotopy category sh of n sphere G brations over orbits to have the same objects as h and stable homotopy classes of maps Then sh is also a groupoid over h 39g and we have a canonical map i h H sh of groupoids over h 39g To control the colimits implicit in iii7 let U be the direct sum of countably many copies of each irreducible orthogonal representation of G since any representation of H C G extends to a representation of G on a possibly larger vector space U is also the sum of countably many copies of each irreducible representation of H For a G representation W G X SW GH gtlt SW over GH7 and we have the berwise smash products X A W of spherical G brations of dimension n and such trivial G brations The set of stable maps X H Y is the colimit over W C U of the set of maps of spherical G brations X A W H Y A W Restricting objects and morphisms appropriately7 we obtain analogous de ni tions for vector bundles or better7 their berwise onepoint compacti cations7 piecewise linear bundles7 and topological bundles We need a lemma before we can explain what it means for a spherical G bration p E H B to be orientable Lemma 42 An nsphere G bmtion p E H B determines a functm pquot 773 H h over h 39g A map D E qL LP ATE f H of nsphe7 e G bmtions determines a natuml isomomhism j 1 H pquot o f of functm s 71A H h over h 39g If DthHgtE AXITgtB 10 JP MAY is a homotopy between maps of spherical G brations f0d0 and fhdl then ff NHL 0 f3 I Proof This is an exercise in pulling back spherical G brations along Gmaps x G H H B and using the Gcovering homotopy property D De nition 43 A spherical G bration p E H B is orientable the stable sense if the composite of pquot 77B H h and i h H sh has the property that ipao1 ipoo for every pair of morphisms aw and oz40 such that a 0 Intuitively over a given a the stable homotopy class of the map of bers over orbits induced by a path between orbits is independent of the choice of path In the language of Construction 18 orientability requires ip to factor through the universal faithful groupoid associated to 77B We must distinguish between orientability in the stable sense and orientability in the unstable sense since it is possible to have ipao1 ipoo but paw pa o The following lemma is easily veri ed no matter how we de ne orientability Lemma 44 Each G gtltH SV is an orientable spherical G bration Nonequivariantly when de ning orientations of bundles we implicitly compare bers to R with its standard two orientations This amounts to choosing a skeleton of the category of n dimensional vector spaces Equivariantly we must orient the G X SV and use their orientations as references and we must start by xing a skeleton of sh We have already discussed how to do this in Construction 18 De nition 45 Let Sph n denote the discrete groupoid over h 39g associated to sh and let p sh H Sph n be the canonical equivalence of categories Explicitly for each H C G choose one homomorphism f H H 0n in each conjugacy class and let Vf R with H acting through Choose one SV in each stable homotopy class of such Hlinear nspheres The objects of Sph n are the resulting nsphere G brations G gtltH SV For each nsphere G bration X over GH we have an isomorphism A X H G X SV in Sh n and these chosen isomorphisms determine p De nition 46 Let p E H B be an nsphere G bration De ne pit to be the composite of 19 71B H sh and p sh H Sph n We continue to write pit for its restriction to a skeleton sk7139B of 77B Now recall Notations 110 The following immediate observation gives a con ceptual characterization of orientability in terms of the relationship between the fundamental groupoid and the component groupoid of B Lemma 47 The nsphere G bration p E H B is orientable and only pit shrrB H Spit factors through the associated discrete groupoid 770B 5 COHERENT FAMILIES OF THOM CLASSES AND THOM ISOMORPHISMS Let E be a commutative ring Gspectrum in the classical homotopical sense we have a unit map S H E and a product E A E H E satisfying the usual unity associativity and commutativity diagrams in the stable homotopy category of Gspectra of 15 see also 16 X111 5 We are interested in the ROGgraded cohomology theory E G represented by E Evaluated on Gspaces we understand EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 11 the unreduced theory7 writing for the reduced theory on based Gspaces We shall make use of the precise treatment of ROGgrading given in 167 XIII 12 We may regard E as an Hring spectrum for any H C G7 and we write E for the theory on H spaces represented by E For an H space Y and Grepresentation p EpcG X Y E ElfJAY We begin with a generalization of the notion of a cohomology class of a Gspace De nition 51 Let be a groupoid over h 39g let q H Sph n be a map of groupoids over h 39g and let Y H Glt2hjlg be a ltiffspace For an object as E KgH write qx G X SW and describe Y as in Lemma 25 in terms of a system of Hspaces An E cohomology class 1 indexed on q of the ltzquot smzce Y consists of an element 1x E EEEZ30 for each object as E KgH The 1x are required to be compatible under Testn39ction in the sense that Zgvy We where y x Hgt y is a morphism of b and g is an element of G such that a yeH 9K compare Lemma 25ii Here mgr E wzw H E ltzgtltzltxgtgt is the composite of restriction Ef www H EEltygtltZltygtgt along Cg H H K and the map E2ltygtltzltygtgt H EEltzgtltZltxgtgt induced by theNHmap Z yg Zx H Zy and the inverse of the stable Hequivalence f SW H SW9 such that g determines the stable G equivalence q y G X SW H G gtltK SW9 compare Remark 22 The simultaneous functoriality in the grading and the space that we have used is explained in 167 XIII 12 Essentially this is just an exercise in the use of the suspension isomorphism in ROGgraded cohomology We shall apply this de nition with 773 taking q to be the functor p 773 H Sph n associated to an nsphere G bration p The relevant WBspace is the Thom WBspace Tp given by the following de nition De nition 52 Let p E H B be an nsphere G bration De ne the based Thom Gspace Tp to be the quotient space EO39B For example7 if we start with a Gvector bundle 5 then its Thom space is obtained from the berwise onepoint compacti cation of 5 by identifying all of the points at in nity We have the map of WBspaces 5 B H B of Theorem 32 De ne the Thom WBspace Tp by letting Tp Tf9ac7 x 6 BH be the Thom Hspace of the pullback 1330 of p along 5 Bx H B The point of the de nition is that the Hspace 330 is Hconnected and7 as we now recall orientation theory for n sphere G brations over Gconnected base spaces is well understood We rst de ne orientations of spherical G brations over orbits then de ne orientations of spherical G brations over Gconnected base spaces7 and nally give our new de nition of orientations of general spherical G brations 12 JP MAY De nition 53 The Thom Gspace of 5 G X SV H GH is G AH SV and E G AH 8V 2 EEW 2 EM An E orientation or Thom class 4 of 5 is an element 4 E EgG AH 8V that maps under this isomorphism to a unit of the ring Eg pt De nition 54 Let p E H B be an nsphere G bration over a Gconnected base space B For any x 6 BG p 1x is a based Gspace of the homotopy type of SV for some n dimensional representation V of G and V is independent of the choice of 30 Moreover for all x GH H B the pullback of p along x is ber Ghomotopy equivalent to G X SV An E on39entation or Thom class 4 ofp is an element 4 E Tp that pulls back to an orientation along each orbit inclusion ac De nition 55 Let p E H B be an n sphere G bration An EEorientation or Thom class ofp is an E cohomology class 4 indexed on p 713 H Sph n of the Thom WBspace Tp such that7 for each x 6 BH ux E EgzTx is an orientation of the pullback 1330 ofp along a Bx H B We say that p is E orientable if it has an EEorientation Here for x 6 BH Vx is the ber Hrepresentation at at so that SW is stably Ghomotopy equivalent to p 1x the equivalence being xed by the speci cation of p Observe that the equivalence xes a stable H map 56 SV 2 197130 H The following observation should help clarify the force of the compatibility condition required of our orientations on H connected covers Lemma 57 Let p be an nsphe7 e G bmtion The following diagmm commutes for a momhism aw x H y in 773 with aeH 9K EEM ltpgtltygtgt gtEXltygtltSVltygtgt e EEHM T TltPgtltlagtwlgtgi LPWWMLQV LCM EElt gtltTltpgtltxgtgt H EEltmgtltSVltzgtgt E529 my HZ Proof The map pa 01 g is de ned exactly as was Tpa 01 g in De nition 51 and the left square is a naturality diagram The right square commutes by a direct unravelling of de nitions D Remark 58 If the horizontal arrows are isomophisms7 then the left vertical arrow is determined by the right vertical arrow and the compatibility reduces to a question of compatible units in the rings comprising the Mackey functor E0 with E0 Eg pt As we shall see in the next section this is exactly what happens when p is orientable and we specialize to ordinary cohomology with Burnside ring coef cients Compatible Thom isomorphisms follow immediately from 15 X 5 where a generalization of the following theorem is proven EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 13 Theorem 59 Let p ENH B be an nsphere G bration over a Gconnected base space B and let a E EETp be a Thom class Then cupping with a de nes a Thom isomorphism e 91 E B a EgVTp for all p E ROG Again we refer to 167 XIII 12 for precision about the grading Theorem 510 Let p E H B be an nsphere G bration and let be a Thom class ofp Then the Mac as 6 BH give rise to Thom isomorphisms mm 2 Emma H ELVlt gtltT ltpgtltxgtgt where the H space SW is stably equivalent to p 1x Moreover the following diagrams are commutative for aw x H y where aeH 9K andp E ROK B 3agtwlgtgL L p lawlu B Em ltxgtgt E2Vltmgtltfltpgtltxgtgt Here the vertical arrows are as speci ed in De nition 51 6 ORIENTATIONS IN ORDINARY EQUIVARIANT COHOMOLOGY We have formalized the intuitive geometrical notion of orientability in De nition 43 and have expressed this notion categorically in Lemma 47 It is natural to hope that this notion coincides with the notion of orientability with respect to a suitable cohomology theory Nonequivariantly7 the relevant theory is integral cohomology The real reason this works is that orientability is a stable notion and Z coincides with the zeroth stable homotopy group of spheres Equivariantly the analogue of Z is the Burnside ring 14G7 which is the zeroth equivariant stable homotopy group of spheres As was rst explained by Bredon 3 ordinary equivariant cohomology theories are indexed on coef cient systems7 namely contravariant functors M h 39g H Ab where Ab denotes the category of abelian groups We have the Burnside ring coef cient system A such that AGH As was proven in 14 the ordinary cohomology theory indexed on M extends to an ROGgraded theory if and only if the coef cient system M extends to a Mackey functor See 157 V 9 or 167 IX 4 for a discussion of Mackey functors in the context of compact Lie groups The Burnside ring coef cient system does so extend7 hence we have the ordinary ROGgraded cohomology theory H HA It is represented by an EilenbergMac Lane Gspectrum HA 167 XIII 4 and HA is a commutative ring Gspectrum We abbreviate HAEorientability to Aorientability and HEXA to We proceed to relate orientability to Aorientability7 beginning with the case of G brations over Gconnected base spaces Theorem 61 Letp E H B be an nsphere G bration where B is Gconnected Let x 6 BG let V be the ber Grepresentation at x and consider the map i SV 2 p 1x C Tp The following statements are equivalent 14 JP MAY i p is orientable ii p is Aorientable iii iquot H gw v E AG is an isomorphism Proof By GCW approximation we may assume without loss of generality that B is a G CW complex with a single G xed base vertex 30 Let 3 1 be the q skeleton of B let Eq p 1Bq and let pq be the restriction of p to Eq Let Cg TpqTpq 1 Observe that SV 2 Tp0 If c GHC gtlt Dq H 3 1 is the characteristic map of a qcell of B then the pullback of p along 0 is trivial and is thus equivalent to G HC gtlt Dq gtlt SV Moreover the equivalence is determined by a choice of path connecting x to Ce 0 These equivalences determine an equivalence between the wedge over all q cells c of the G spaces GHCLr A Sq A SV and the quotient Gspace Cg Consider cohomology in degrees V i7 where i is an integer We have g iGH A 8 1 A 8V HEW This is zero unless i 2 q and it is AH when i q by the dimension axiom We conclude by long exact sequences and lim1 exact sequences that HV WTW EV iTP 0 for i 2 1 and there is an exact sequence 0 a HgTpigtHgSVHg101 Here 5 may be viewed as a map AG H HAHC of 14Gmodules7 where the product runs over the lcells c A lcell c is speci ed by a loop at x in BHC The component of 5 in AHC can be interpreted geometrically as the difference between the identity map of SV and the stable Hequivalence of SV obtained by the action of this loop on SV The three statements of the theorem are each equivalent to the assertion that 5 0 D Observe the relevance of our de nition of orientability in the stable sense The conclusion would fail if we de ned orientability in the unstable sense Before generalizing this result7 we recall a standard fact about conjugation homomorphisms between Burnside rings Let a GH H GK be given by aeH 9K and consider Cg H H K Since clc AK H AK is the identity for k E K by inspection of the the standard inclusion of AK into a product of copies of Z eg 15 V 2 we see that cg AK H AH is independent of the choice of g in its coset 9K It is also independent of the choice of a in its homotopy class7 by Lemma 11 We write cg ca Theorem 62 Let p be an nsphere G bration The following statements are equivalent i p is orientable ii Each 1330 is orientable iii Each 1330 is Aorientable iv p is Aorientable Moreover an HAorientation a ofp is speci ed by a collection ofunits 1x E AH for points x 6 BH of the discrete groupoid 7103 that satisfy the compatibility condition cauy 1x for a map 7 x H y of 7103 with 87 a Equivalently a is speci ed by an automorphism of the fanctor p shrrB H Sph n over h 39g EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 15 Proof Since the notion of orientability of p depends only on the behavior of the pullbacks of p along paths and paths lie in connected components the equivalence of i and ii is immediate from the properties of H connected covers given in Theorem 32 The equivalence of ii and iii is part of the previous theorem and it is trivial that iv implies iii by consideration of pullbacks Thus assume iii and consider the diagram of Lemma 57 with E HA Its horizontal arrows are isomorphisms by the previous theorem and Remark 58 applies to give the speci ed description of an Aorientation in terms of units of Burnside rings In particular we may take 1x to be the identity element for all x and this shows that p is Aorientable Finally the group of automorphisms of an object G X SV of Sph n is canonically isomorphic to the group of stable Hequivalences of SV and thus to a copy of the group of units of the Burnside ring For our functor pif sk7139B H Sph n the compatibility condition on units required of an Aorientation can be interpreted as the naturality condition required of an automorphism of functors 1 Let AOrp denote the set of Aorientations of an orientable nsphere G bration p Corollary 63 By multiplication of units or equivalently by composition of au tomorphisms of the functor pif skrrB H 53910an over h 39g AOrp acquires a stiuctui e of commutative group Nonequivariantly there are both topological and cohomological notions of an orientation and these notions coincide Equivariantly we have explained a coho mological notion of an orientation There is also a topological notion de ned in However these two notions do not coincide To explain this we sketch the de nition given in Working in the category of groupoids over h 39g consider maps into Sph n In 5 we construct and characterize a particular map p it H Sph n such that it is faithful over h 39g and any map from a faithful groupoid over h 39g into Sph n factors up to isomorphism through at least one map into 6 this is a weak universal property of p which intuitively is a kind of universal orientation Fix an orientable n sphere G brationp E H B for the rest of the section The functor p skvrB H Sph n factors through the discrete groupoid 7703 and we now agree to write pif for the resulting functor de ned on 710 The topological notion of an orientation is a pair C 7 consisting of a functor C 7103 H it over h 39g together with a natural isomorphism n p H p o Let 07 denote the set of such orientations of p Precomposing with automorphisms of pif for xed C we obtain a free right action of AOrp on Orp Call the orbit set OrpA it can be identi ed with the set of those functors C 7103 H it that can be part of an orientation Cm Let Fp be the set of all functors C 710 B H it over h 39g such that pif and p oC agree on objects In general not all such functors are components of orientations and we have an inclusion a OrpA H Let AX h 39g H Ab be the contravariant functor that sends GH to the group of units of AH and continue to write AX for its composite with a H h 39g for any groupoid over h 39g By analyzing the obstruction to the construction of n such that C n is an orientation one arrives at the following proposition We omit the proof as it is not very illuminating The essential ingredients are the cited weak universal property of p and the fact that Sph n is a uniquely divisible groupoid over h 39g 16 JP MAY Proposition 64 There is an exact sequence of pointed sets gt OrpAigtFpigtH17mBAX gt Thus H1770B AX measures the difference between topological and cohomo logical orientations if B is a bijection7 the notions are equivalent 7 CONCLUDING REMARKS Whenever one has cohomological orientations of a class of Gvector bundles that are suf ciently natural in G one will have cohomological orientations in the sense that we have de ned Since this paper was written around the deadline for submissions to this volume7 I have not had time to check details of the following two examples7 but they are surely correct Here it makes sense to use the variant of the theory appropriate to unstable Gvector bundles rather than to stable G brations Clearly orientations in the former sense give rise to orientations in the latter sense The methods of 2 should give the following result compare Example 7 1 Complex G vector bundles admit canonical KUgorientations Real Gvector bundles with Spin structures and dimension divisible by eight admit canonical KO Gorientations Tautological orientations should give the following result Example 72 Complex G vector bundles admit canonical M U orientations Real Gvector bundles admit canonical M OEorientations At the most structured extreme7 as in the nonequivariant case7 we have the following observation Example 73 A spherical G bration is Eorientable if and only if its pullbacks to Hconnected covers are stably ber homotopy trivial with suitably compatible trivializations To obtain a Poincare duality theorem along the present lines7 one would have to prove an Atiyah duality theorem for the H connected covers of smooth compact Gmanifolds M That is if M embeds in V with normal bundle 1 one might hope that the Hspaces TV and are Vdual for x 6 MH Although is in nite dimensional one has complete homotopical control on its xed point spaces7 which are homotopy equivalent to smooth manifolds I have not explored this question REFERENCES 1 MF Atiyah Bott periodicity and the index of elliptic operators Quart J Math 191968 113140 2 MF Atiah R Bott and A Shapiro Clifford modules Topology 31964 Supplement 1 338 3 G Bredon Equivariant cohomology theories Lecture Notes in Mathematics Vol 34 Springer Verlag 1967 4 RE Conner and EE Floyd Differentiable periodic maps Ergebnisse der Math N S 33 Academic Press 1964 5 SR Costenoble JP May and S Waner Equivariant orientation theory Undistributed preprint 1989 6 SR Costenoble and S Waner Equivariant orientations and Gbordism theory Paci c J Math 1401989 6384 7 8 9 14 15 16 17 18 EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 17 SR Costenoble and S Waner The equivariant Thom isomorphism Paci c J Math 1521992 21 39 SR Costenoble and S Waner Equivariant Poincare duality Michigan Math J 391992 325351 SR Costenoble and S Waner The equivariant Spivallt normal bundle and equivariant surgery Michigan Math J 391992 415424 SR Costenoble and S Waner The equivariant Spivallt normal bundle and equivariant surgery for compact Lie groups Preprint 1996 T tom Dieck Transformation groups Studies in Mathematics Vol 8 Walter de Gruyter 1987 A Elmendorf Systems of xed point sets Trans Amer Math Soc 2771983 275284 A Grothendieck Revetements etale et groupe fondemental Lecture Notes in Mathematics Vol 224 SpringerVerlag 1971 LG Lewis JP May and JE McClure Ordinary ROGgraded cohomology Bull Amer Math Soc 4 1981 1287130 LG Lewis JP May and M Steinberger with contributions by J E McClure Equivariant stable homotopy theory Lecture Notes in Mathematics Vol 1213 SpringerVerlag 1986 J P May et al Equivariant homotopy and cohomology theory CBMS Regional Conference Series in Mathematics Number 91 American Mathematical Society 1996 JP May G spaces and fundamental groupoids Appendix to An equivariant Novikov con jecture by J Rosenberg and S Weinberger Journal of Ktheory 4 1990 503 M Rothenberg and J Sondow Nonlinear smooth representations of compact Lie groups Paci c J Math 841979 427 444

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "I used the money I made selling my notes & study guides to pay for spring break in Olympia, Washington...which was Sweet!"

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.