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Equivariant bordism and G-bundles Grguric, Izak 2008

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Equivariant Bordism and C-bundles by Izak Grguric B.Sc., The University of British Columbia, 2000 M.Sc., The University of British Columbia, 2002 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December, 2008  ©  Izak Grguric 2008  Abstract Let G be the cyclic group of 4 elements and H the subgroup of G of order 2. We study the actions of G on manifolds modulo the equivariant bordism relation by studying the equivariant bordism relation on G-vector bundles; specifically, we focus on G-vector bundles such that G action is free away from the zero section, and the isotropy group of each point in the base is equal to H. We obtain a complete set of characteristic numbers that deter mines when such a G-vector bundle is nulibordant. Using this result, we obtain a geometric splitting of the bordism classes of these bundles into ge ometrically simpler components. Furthermore, we determine a complete set of characteristic numbers for the bordism ring of G-manifolds. Finally, we generalize our results to a larger family of finite groups G.  11  Table of Contents Abstract Table of Contents List of Figures  II 111 V  Acknowledgements Dedication 1  Introduction  2  Background 2.1 Singular G-bordism 2.1.1 Isotropy Restrictions 2.1.2 Free G-bordism 2.1.3 Bordism homology theory 2.2 G-vector bundles 2.2.1 Cyclic groups 2.2.2 Classifying spaces 2.3 G-bundle bordism ;Z bordism 2.3.1 2 r (Z r _i)-bundle  3  Characteristic Numbers 3.1 Motivation 3.2 (Z ;Z 4 )-bundle bordism 2 )-bundle bordism 2 ;Z 4 3.3 Characteristic numbers for (Z -bordism 4 3.4 Characteristic numbers for Z  4  An 4.1 4.2 4.3  Application: Splitting Motivation Smith homomorphism, extensions, and restrictions The bundle splitting results  Vu  1  10 10 13  17 19 19 20 22 24 26 26 27 33 ‘U  Table of Contents 5  6  More General Groups 5.1 Characteristic numbers for more general groups 5.2 Smith constructions, extensions, restrictions for G r 2 5.3 The bundle splitting for C = Z Proofs of Bundle Results 6.1 Characteristic numbers 6.1.1 The proof of Theorem 3.1 6.1.2 The proof of Theorem 3.3 6.2 Splitting 6.2.1 The proof of Proposition 4.15  Bibliography  =  2 Z  .  .  37 37 40 43 47 47 47 52 53 53 59  iv  List of Figures 2.1 2.2  A bordism equivalence Bordism is an equivalence relation  4.1  Surgery along U  7 7 32  V  Acknowledgements The creation of this thesis would not have happened without the support, friendship, and infinite patience of my supervisor Laura Scull. She was my guide through the world of equivariant algebraic topology; Laura has never failed to rescue me from each and every quick-sand pit I have stumbled into. I also wish to thank Kee Yuen Lam subject of global geometry.  -  for showing me the true heart of the  vi  Dedication To my parents, Tihomir and Ksenija Grguric. They have always been there for me. Hvala yam puno!  vii  Chapter 1  Introduction This thesis is concerned with G-manifolds and their boundaries, where C is a finite group. The boundary information of equivariant manifolds is encoded in the bordism ring .A/ of closed, unoriented C-manifolds. More precisely, our thesis is concerned with the module structure of this ring over the classical bordism ring of non-equivariant, closed, unoriented manifolds. This module structure depends greatly on the answer to the following question: given a C-manifold M, how does the G-bordism type of this manifold depend on its fixed-point sets MK and their normal bundles in M for various subgroups K of C? Focusing our attention to the fixed-point sets of the action and the associated normal bundles will produce a passage from JV to the non-equivariant -homology. Working in the latter setting has an advantage: there are 4 .M numerous computational tools designed for dealing with .,V-homology. For example, the set of characteristic numbers determining the non-equivariant bordism type is well known by the classical work of Thom [25], and these results have been extended to f-homo1ogy by Conner and Floyd in [8]. In the same paper one also finds extensions to f-homology of a number of standard computational homological tools, such as the Künneth formula, certain spectral sequences, and various types of exact sequences. To look at the fixed-point and normal bundle information of a given equiv ariant manifold, we will define the fixed-point homomorphism F. But first let us set some notation. Let K be a subgroup of C, then AK is the family of all subgroups of C conjugate to subgroups of K, and PK is the family of subgroups conjugate to proper subgroups of K. Let .A/(AK,PK) be the 1  Chapter 1. Introduction bordism ring of all K-manifolds with boundary, such that the isotropies of the boundary points belong to PK. Now let FK be the homomorphism of jV-modu1es FK :jV  .‘  AçK(Aj-,  PK)  defined as taking [M] E to the class represented by the tubular neighbor hood of the fixed-point set MK in M. We collect the fixed-point information extracted by FK for various subgroups K of G into a single homomorphism, called the fixed-point homomorphism: S/  F= eFK. K<G  Note that although in general the bordism rings JV(AK, PK) are equiv ariant, their .M-module structure is much simpler than that of the original J and is in fact computable since one can show that it can be expressed in terms of the .N-homo1ogy of certain classifying spaces (see [21]). Hence, given [M] € JV we have a way of computing if F([M]) = 0, but now the question becomes: to what extent does F determine the G-bordism type of M?  The answer depends on the group G and on the type of actions considered. The simplest type of actions with fixed points that we can consider are the semi-free actions: actions with every isotropy subgroup being either the entire group or the trivial subgroup. Suppose that G admits an orthogonal representation such that the action of G is fixed-point free on the sphere of that representation. Then it is known that for such a group G, the fixedpoint homomorphism F completely determines the semi-free G-bordism. For a discussion of this result and examples of finite and infinite groups which satisfy this property, we refer the reader to [23].  2  Chapter 1. Introduction On the other hand, if we make no restrictions with respect to the type of actions, the list of groups for which F completely determines the G-bordism narrows down to G = (Z 2 )‘, where r is finite (see [7]). Naturally, we might be curious as to what happens outside of this list of groups; the simplest possible case, that of G = Z , was already considered by torn Dieck and 4 Bix in [7], as well as Stong in [22], but in both papers no complete set of characteristic numbers was found. Our motivating problems will therefore be: Ml. Given [M] e such that F([M]) = 0, what extra information is needed to determine if M is a Z -boundary? 4 M2. What is the geometric significance of F([M])  =  0?  An answer to these two problems was given by Beem and Rowlett in [5]. The proofs in this paper are based on the results found in [2]; specifically, the proofs are based on a sequence of algebraic results about the ring structure of and the action of the bordism ring of free 7Z -manifolds on JV2. 2 Algebraic arguments, some of them using cohomology spectral sequences, are then utilized to obtain the desired result for Z . Although induction can 4 extend the base case to G = Z2r for r > 2, the overall algebraic method of proof is not applicable to any other family of groups. This leads us to the main question which we address in our thesis:  Question: Is there a geometric way of answering Ml and M2, preferably one that is generalizable to a wider class of groups G? We will find an affirmative answer, but first we need to slightly modify Ml; -bordant to 4 we observe that if [M] E .N and F 0, then M is Z 4 ([M]) some M’ such that (M)z4 = 0 (see Lemma 3.5). Therefore the motivating problem Ml now becomes: Ml’. What information is needed to determine if M’ is a Z -boundary? 4  3  Chapter 1. Introduction And the second motivating problem consequently becomes: M2’. ‘What is the geometric significance of 4 F ([M])  =  0?  For each component of the fixed-point set of Z 2 in M’, the normal bundle v is a Z4-equivariant vector bundle; more specifically, z-’ is a Z -bundle with the 4 property that every isotropy subgroup of the base is Z , while the action 2 of Z 4 is free on S(v), the sphere bundle of v. We call such a bundle a ;Z 4 (Z )-bundle, and define an equivariant bordism relation of these bundles 2 in Section 2.3. We collect the bordism classes into an .N-module Bk(Z4; Z ), 2 where k denotes the rank of the representatives. Let 1 k denote the disjoint union of all rank k normal bundles to the fixed-point set of Z 2 in M’. We show in Section 3.4 that if for all k, [vk] = 0 as a class in Bk(Z4;Z2), then M’ (and hence M) is a Z -boundary. 4 Therefore the “information” needed to solve the motivating problem Ml’ is contained in a complete set of characteristic numbers for B (Z ;Z 4 ). We 2 give such a set in Proposition 3.4. We combine this result with the geometric properties of the Smith construc tion of Section 4.2, to obtain an answer to M2’, the modification of the second motivating problem; the answer is stated in Theorem 4.14. Finally, we should mention that the methods we developed and used in answering the motivating problems are almost exclusively geometric: equiv ariant bundle theory, the use of the bundle-orientation class, and equivari ant surgery theory combined with the use of Smith construction, all play a prominent role, and allow us to circumvent the algebraic arguments of [2], [5], and [6]. Moreover, as we show in Chapter 5, these methods allow for a generalization to a wider class of groups than the one handled by previous theories.  4  Chapter 2  Background 2.1  Singular G-bordism  Let G be a finite group. Fix a G-space X. A singular n-dimensional G-manifold in X, is a pair (M, f) where M is a compact, unoriented n dimensional G-manifold, and f : M —* X is a C-map. A singular C-manifold is closed if t9M = 0. The map f M —* X is called a reference map for M. Two such pairs (M , fi) and (M 1 , f2) are equivalent if there is a C- diffeo 2 morphism ç: M 1 —* M 2 such that 1 M  commutes. To indicate the equivalence, we write (M , fi) 1 , f2). We 2 (M will keep an implicit assumption that all of our manifolds and maps are taken up to a G-diffeomorphism equivalence. We will mention the equivalence explicitly only where it plays an important role. Two closed singular n-dimensional C-manifolds (M, f) and (M’, f’) are said to be C-bordant, if there is a compact (n + 1)-dimensional C-manifold W, and a C-map F: W —f X, such that (öW,FIôw)  (MIIM’,fllf’).  (2.1)  5  2.1. Singular G-bordism  Now suppose that we are given two singular G manifolds (Mi, fi) and , f2) such that 1 2 (M (M fi) = (M , , f2) through some G-diffeomorphism 2 M —* M 1 . Form (Mi x I, fi) and (M 2 2 x I, h) with the G-action being trivial on I, and the maps f being defined by fj(m,t) = f(m). Next, glue M x I U M 2 x I by identifying M 1 x {1} and M 2 x {O} by . Denote the resulting manifold by V. Likewise, glue fi U f by 2 M —  and denote the resulting map by Fv. Then by construction we have that O(V,Fv) = 2 ,f (Mi,fi)ll(M ) . Therefore, if (2.1) holds, there must exists some singular G-manifold (H/, F) such that a)  OW=MIIM,’  l__\ U)  M=J,M!—J.  I;’  —  We can use the conditions a) and b) as the definition of G-bordant man ifolds and we can, if we wish, disregard the G-diffeomorphism equivalence altogether. We denote the G-bordism relation by (M, f)  (M’, f’), and the resulting  G-bordism equivalence class by [M, f ]. If we wish to explicitly mention the bounded manifold, we write (M, f) (M’, f’) In the case where (SW, FI8W) = (M, f), we write (M, f) 0 and [M, f] = 0. , f2) then [M 2 , fi] = 1 Note that by the observation above, if (Mi, fi) = (M ‘-  [M f2]; the converse however, might not hold: see Figure 2.1 (where q5 i. , 2 and denote the G-diffeomorphisms.) That the G-bordism relation is an equivalence relation is not trivial: one needs some equivariant differential topology to prove this fact most notably an equivariant version of the boundary-collar theorem. See [8] and [21] for details. Figure 2.2 depicts (on the level of G-manifolds, without mentioning the reference maps) the reflexive, symmetric, and transitive nature of the bordism relation: -  6  2.1. Singular G-bordism  Mi°  M2  0\F/ Figure 2.1: A bordism equivalence.  (R)  MMIM,  (S)  ’M 2 M , 2 then 1 M 1 ifM  (T)  if M 1  2 and M M 2  , then 3 3 M ”M where W 1 M  =  WUWfl.  Where the gluing of Wa and W is along M 2 (via an equivariant boundarycollar).  MO]M  MM 2 M 1  Ti M  3 M  1 M M 3  Figure 2.2: Bordism is an equivalence relation.  7  2.1. Singular G-bordism The collection of G-bordism classes of singular n-dimensional G-manifolds in X is denoted by .N (X). This collection carries the structure of an Abelian group: the group operation is defined by the disjoint union of representa tives: , f’j + [M 1 [M , f2] = [M 2 1 II M , fi LI f2]. 2 Notice that (MIIM,fllf)  ‘-  0, s 2[M,fj  =  0 for any [M,fj e  giving .N(X) the structure of a Z -module. 2 We write fG(X)  =  and impose a ring structure on .M.(X) by letting the Cartesian product define our multiplication: , f’] 2 1 [M [M f21 , .  =  [Mj x M , fi x f21. 2  One can check, by repeatedly applying the definition of the G-bordism re lation, that + and are well-defined. Let .Nf, stand for .Me} (*). This is the classical bordism ring of unoriented manifolds; the reference maps are superfluous since their codomain is a point space (the group acting is trivial, hence also superfluous). Notice that the Cartesian product operation makes .Af(X) into an A/-module: for any [P] e .N and [M, f] e p4(X) we define [P] .[M, f] as [P x M, f’J e .iV(X) where f’(p, m) = f(m) for (p, n) e P x M. Usually the .Af-module structure of .N(X) is easier to determine than the ring structure, and allows for a more unified approach of computation by families: see [21] for a description of the method of computation by families, and [12] for a survey of some introductory results of this sort on the bordism of Abelian groups. The structure of the ring f’(X) is very much dependent on the choice of C and X; for example, for G = {e} and X connected, the ring has an identity  8  2.1. Singular G-bordism element, given by [*, i] where i : * —* X is an inclusion. On the other hand, if C and X are non-trivial, the ring might not have an identity. The structure of these bordism rings can be very difficult to compute: even such basic question as, for example, the existence of zero-divisors, are generally hard to answer. For results in this direction, see [3], [4], [11], [20], [27], and [18].  2.1.1  Isotropy Restrictions  Sometimes it is desirable to restrict the type of manifolds occurring in the definition of the singular G-bordism to those with their isotropy groups belonging to a family of subgroups of G. A family F in G, is defined to be a collection of subgroups of G such that: a) b)  ifHEFandKcH,thenKeF,and eF. 1 ifHeFandgeG,thengHg  Given a family F in G, a compact manifold M is F-free (or an F-manifold) if for every z e M, the isotropy group G belongs to F. (Note that the part b) of the definition of F is chosen in a minimal way that is consistent with the orbit structure of a G-manifold.) Fix a family F in G. Repeat the definition of singular G-bordism as before, but now let all manifolds in sight be F-free. We thus obtain the singular F-free bordism, denoted JJ (X) (F). Let K  C. We will mostly focus on four families of subgroups in G: (i) FG = {{e}}, (ii) PG = {H (iii) AG = {H (iv) AK = {H  the free family, G and H G}, the proper family, G}, the family of all subgroups of C, and K}.  We denote f(X)(FG) by iQ(X), and evidently N(X)(AG) J°(x). The inclusion FG c AG induces a forgetful ring homomorphism JVG(X). If X is a free C-space, k is an isomorphism. k :f?(X) 9  2.2. G-vector bundles  2.1.2  Free G-bordism  Let G be a finite group. The correspondence between the principal G bundles and free G-manifolds induces the isomorphism of Nt-modules  JIG f(BG) where EG —* BG is the universal principal G-bundle. Given a class [M] in JG M is a free G-manifold and the quotient M —+ M/C is a principal G-bundle. We classify this bundle by f : M/G —* BG, then [M/G, f] is the class in f(BG) corresponding to [M} under the isomorphism. The inverse isomorphism is obtained by the construction of the induced bundle.  2.1.3  Bordism homology theory  We only mention that AIK ( ) is a generalized homology theory (satisfying all of the Eilenberg-Steenrod axioms but the dimension axiom). We will use the fact that for nice spaces X and Y, the Künneth isomorphism holds x Y) N(X)®(Y) (see [8] for a proof). Note that although for nontrivial C, .N( ) too is a gen eralized (equivariant) homology theory, the Künneth isomorphism might not hold (however, according to [9] there will be a Kftnneth spectral sequence).  2.2  C-vector bundles  We mention some standard facts about C-vector bundles. Let C be a finite group and p: E — M a smooth, locally trivial, real vector bundle such that M and E are C-manifolds, and p is a smooth C-map. If for each g e C, the left multiplication map L 9 : p (x) —* p (gx) is a linear isomorphism, then p: E —* M is called a smooth C-vector bundle. We will sometimes denote a vector bundle by the 3-tuple (E, p, M). All our  10  2.2. G-vector bundles  G-bundles will be real, smooth, locally trivial, and of a finite rank. We will be using two types of morphisms of G-vector bundles: G-bundle maps, and G-isomorphisms. We define these here for the sake of clarity. Given two C-vector bundles = (E,p, M) and ‘ = (E’,p’, M’), a bundle map (f, f) consisting of C-maps, is called (i) a C-bundle map, if x e M,  fIE  :E  —*  E() is a linear isomorphism for each  (ii) a C-isomorphism, if it is a C-bundle map, and if M’  =  M,  f=  idM.  Sometimes we refer to f itself as the bundle map covering f. We signify isomorphic C-bundles by or simply by ‘ if the equivariance is clear from the context.  e  We will also be using the usual constructions such as Whitney sum, tensor product, or exterior power. These constructions are functorial, therefore when applied to C-vector bundles, they again yield C-vector bundles. The functoriality also yields the usual isomorphisms such as and and if we let A denote the determinant bundle of (meaning, A = A k where k equals the rank of ) and A a C-line bundle, then for odd-rank bundles and A  (®)  A®Ai,  and  A  (®A)  A®AA  A®A.  Let the base of be M, and define the equivariant trivial bundle of rank m by = M >< R”, where 9 E C acts by g(m,v) = (gm,v) on (m,v) eM Note that a rank m bundle over M could be geometrically trivial (i.e. non equivariantly isomorphic to the trivial bundle), yet not C-isomorphic to a” . 2 11  2.2. C-vector bundles  If A is a C-line bundle, then A®A  C  Note that A and ® commute with pulibacks: suppose we have C-bundles = (Ej,p, M) where i = 1,2, and and a C-map f :M’ —* M. Then  f*(®)  f*(e)Øf*()  and  f*(A)  It is worth mentioning here that the C-action on the induced bundle is given by the inclusion f*(j) C M’ x E.  f*()  We will also occasionally need to make use of the C-bundle gluing (or clutch ing) construction, which we outline here for convenience. Let UX 1 X=X 2  and  flX 1 A=X , 2  where all the spaces are compact C-spaces. Suppose that we are given Cbundles A) for i = 1,2, and a G-isomorphism :E 1 —* E . We define 2 a new C-bundle P2 1 U P2 2 by gluing E 1 to E 2 via .  We will need the following proposition, the proof of which can be found in [1]. Proposition 2.1. If M is a free C-manifold, C-vector bundles over M correspond bijectively to vector bundles over M/G by the quotient P2 —* P2/C. If M is a free C-space, then necessarily P2 is a free C-space since the projection map p: B —* M is equivariant. If r : M — M/G denotes REMARK:  the quotient map, then Proposition 2.1 gives that any C-vector bundle = (E,p, M) corresponds to some 7r*(j) where = (F, q, M/C) is a non equivariant vector bundle. Conversely, every vector bundle i = (F, q, M/G) is the quotient /C of some C-vector bundle = (B, p, M). Also note that by functoriality of the quotient (and the pullback),  ‘  if and only if  ‘/C. For details and more facts about C-bundles, the reader is 12  2.2. G-vector bundles referred to [1] and [17].  2.2.1  Cyclic groups  Note that for any g e G, the left multiplication Lg : E —÷ E is a G-bundle map covering L IM p o Lg 0 S (where o denotes the zero section.) In 9 the case when G = Zm, a cyclic group, we might wish to be explicit about the bundle map defining the action: we denote the G-vector bundle by = (E, &,p, M, a) or (E, , M, a) (omitting the projection map). Here (a, a) is the bundle map E UE  M  >M,  which defines the action of G on : in other words, Lg generator of C = 7Zm). Note that in this case (sm, ) tm a  & (where g is the =  (idE, idM).  We will mostly be interested in the case of cyclic groups of order 2.  Definition 2.2. A 2 r (Z r _i)-bundle ;Z is a 4-tuple (E,&,M,a) such that p : E —* M is a YZ2r-vector bundle, and the generator of Z2r acts on p: E — M  by the bundle map (&, a). Let o : M —* E denote the zero section. We require that for every x e M the isotropy G = Z r, while for every 2 Z x e E — so(M) the isotropy is G {e}. REMARK: Note that since the action is given by bundle maps, so is equiv ariant, and so(M) is Z r-invariant in E. The equivariance of o and p gives 2 . 2 G = () 30 = Z G Consider a (Z ;Z 4 )-bundle (E, a, M, a). By Definition 2.2, a is an involution 2 2 = idM), and denoting the fiber above x by E, we get a linear map (a â : 2 E  >E  13  2.2. G-vector bundles  which (after selecting some Z -invariant Riemmanian metric on the bundle) 4 takes (x,v) to (x,—v) for all (x,v) E E. We will indicate this by saying that locally &2  Definition 2.3. A (Z r_i ; Z 2 r-i )-bundle is a 4-tuple (E, 9, M, 9) such that 2 p:E —* M is a 7Z r-i-vector bundle, and the action on the base M, given by 2 9, is Z r—i-free. 2 By Proposition 2.1, there is a one-to-one correspondence between ._i; Z 2 (Z _i)-bund1es over M and non-equivariant bundles over M/Z 2 r_i = 2 M/9. REMARK:  2.2.2  Classifying spaces  Let Vect (M) denote the equivalence classes of G-vector bundles over M under the G-isomorphism relation. For a finite group G, we can always construct the universal real k-plane G-bundle, denoted EOk(G) J,’fk(G)  BOk(G). This bundle is “universal” in the usual sense: any rank k C-vector bundle over M is C-bundle isomorphic to f*(7k(G)) for some C-map f : M —f BOk (C). In fact, a stronger conclusion holds Vect(M)  [M,BOk(C)]G  where [ , ]c stands for C-homotopy classes of maps. The space BOk(G) is called the classifying space for C-vector bundles of rank k. For more facts on the universal G-vector bundle, as well as for the proof of universality, the reader is referred to [28].  14  2.3. G-bundle bordism Let T = e’2r_1 be the generator of Z r acts on C by complex 2 . Then Z 2 multiplication. Definition 2.4. Let BOk(C°°) be the Grassmannian of real k-planes in C°°. Let r : BOk(C°°) —* BO(C°°) denote the map giving the Z r-i action on 2  this space induced from the complex multiplication by T on C°°. 1 = —1; the multiplication The map r is a Z r_i action since T2’ 2 by T takes a k-plane in C°° to itself. Over BOk(C°°) we have the universal k-plane bundle EOk(C’°) which consist of pairs (p, v), where p is a k-plane in C, and v is a vector in p. The group Z r acts on EOk(C°°) by 2 : (p,v) —* (r(p),T v). We denote the bundle EOk(C°°) — BOk(C°°) by REMARK:  .  7k(C°°). Proposition 2.5. BOk(C°°) is the classifying space for (Z r; Z 2 r_i)-bundles 2 of rank k. PROOF.  See [22j.  We have the following Lemma 2.6. BOk(C°°) is a free Z -space if k is odd. 2 PROOF.  2.3  Again, see [22].  G-bundle bordism  Given a smooth G-bundle of finite rank, over the base W which is a smooth compact manifold, we will denote  O()  Iaw.  Definition 2.7. Two G-bundles and ‘ of rank k are called G-bordant if there exists a G-bundle also of rank k, such that  e  15  2.3. G-bundle bordism REMARK:  We indicate the G-bordant bundles by  Note Lemma 2.8. If PROOF.  e  ‘  then  Let  M  be the G-isomorphism, and let I denote the unit interval with the trivial G-action. Glue x I and ‘ x I (both of which are now bundles over the baseMxI)byidentifyingMx{1}cxIwithMx{O}c’xI. Now use b to clutch IMx{1} to We obtain a C-bundle such that  We can also use C-bundle clutching to show that the relation is transitive. (Reflexivity and symmetry follows from the use of  REMARK: ‘  disjoint union II in Definition 2.7.) We have the following Lemma 2.9. Suppose : M’  —  M is a G-diffeomorphism, and let  be a  G-bundle over M. Then  The proof is similar to the proof of Lemma 2.8, except now we glue x I and *() x I by identifying M x {1} with M’ x {O} by q. But then by the definition of induced bundle, the clutching isomorphism over the glued ends is an identity. I PROOF.  Definition 2.10. Let B denote the collection of G-bordism equivalence  classes of C-bundles of rank k. 16  2.3. G-bundle bordism  L3 is an Abelian group under LI. Given [] € L3 where explicitly = (E,M), and given some [F] E .Af define [P] [] = [E x P,M x P]. The action on [E x F, M x P] is defined as follows: if (e, p) e E x F, then for any g E G we have g(e,p) = (ge,p). This product gives an f-module structure REMARK:  to B. There is a fairly obvious connection between L3, and the singular G-bordism of manifolds with reference maps into BOk(G). Let [M, f] e .N(BOk(G)) and consider f*(7k(G)), which is G-bundle of rank k over the base M. , f2]. Therefore, there is some [W, F] such that 2 [M Suppose [M , fi] 1 8W = M 1 HM 2 and FIM = f. But this implies that F*Qyk(G))IM = f(7k(G)) or  = f(’yk(G)) HfQyk(G)).  (2.2)  Comparing (2.2) and Definition 2.7, we get  Lemma 2.11. 13° G(Bok(G))  as  -modules.  As in the case of singular bordism, we could restrict our attention to bundles with base or fiber isotropies belonging to a specific family of subgroups of  C.  2.3.1  r; Z 2 (Z r_i)-bundle bordism 2  Definition 2.12. Suppose we have two (Z r; Z 2 r_i)-bundles and ‘. We 2 r; 2 2 rZ _i)-bordant if there is a (Z will say that they are (Z r; Z 2 r_i)-bundle 2 such that 8= H ‘, and denote this by r ;Z 2 (Z r_i) 2  We have the following  Definition 2.13. Let Bk(Z2r; Z r_i) denote the J-module of (Z 2 r; Z 2 r_i) 2 r; 1 2 ,._ )-bundles. 2 Z bordism classes of (Z 17  2.3. G-bundle bordism Notice that the bundle of Definition 2.12 must be a (Z r; Z 2 r_i)2 bundle, and not only a Z r-bundle. The symbol ““ on k(Z2r;Z 2 r_1) is 2 there to remind us of this fact. This condition on amounts to an isotropy restriction on the bundles. REMARK:  Combining Proposition 2.5 and Lemma 2.11 we get Lemma 2.14.  r1) 2 Bk(Z2r; Z  fZ2r(BO(coo))  as .N, -modules. Given a (Z ; {e})-bundle i of rank k, by Definition 2.2 the Z 2 2 action on the base of is trivial. Taking some Z -equivariant Riemannian 2 metric, we see that the action on the fibers must be by multiplication by (—1), hence canonical. Therefore the classifying space for (Z ; {e})-bundles 2 of rank k can be taken to be the ordinary (non-equivariant) classifying space REMARK:  BOk.  Therefore we obtain Lemma 2.15. 13k(Z ; 2 {e})  N(BOk)  as M -modules.  18  Chapter 3  Characteristic Numbers 3.1  Motivation  We refer to a set of characteristic numbers as “complete” for a particu lar bordism theory B, if they are able to distinguish between non-bordant representatives of bordism classes in B. For example, the Stiefel-Whitney characteristic numbers are a complete set of characteristic numbers for the unoriented bordism iV. The objective of this chapter is to find a complete set of equivariant char acteristic numbers for the representatives of the Z -bordism classes in 4 This problem has been studied before from the perspective of equivariant homotopy bordism by Bix and torn Dieck [7] (also see [26]), and the geo metric point of view by Stong [22]. In each case however, the characteristic numbers obtained were not sufficiently fine to distinguish all Z -bordism 4 types. A complete set of such numbers was obtained by Beem and Rowlett in a more general setting of the group 7Z r (see [5]). Unfortunately, their arguments 2 are not generalizable to other (for example, non-abelian) groups. It was the search for a generalization that has led us to the results presented in this chapter. We obtain a complete set of characteristic numbers for 7Z -bordism; the 4 and geometric equivariant significance of these numbers differs from that of the numbers found in [5]. The procedure involved in the calculation of the new numbers shows (what we feel to be) a very pretty interaction between 19  3.2. (Z ;Z 4 )-bundle bordism 2 the local geometry of the manifold around the fixed point set, and the group action. These numbers complement those of Beem and Rowlett, potentially allowing for an easier computation of the Z -bordism type. 4 We find these new numbers to be of a better theoretical use in describing the geometric structure of elements of And perhaps more significantly, we find our approach to apply to a more general family of finite groups G.  3.2  ;Z 4 (Z )-bundle bordism 2  Suppose we are given a (Z ;Z 4 )-bundle = (E, &, M, o) and a (Z 2 ;Z 4 )-line 2 bundle A = (L,âL,Mu). Observe that ®A is a 2 ; (Z ) -bundle: Z taking some Z4-invariant Riemannian metric, we see that locally 2 (&oaL)  =  (—1)®(—1)  =  101,  showing that the bundle map J®UL covering u, is an involution. But now using the properties of tensor products of G-bundles, as summarized in Section 2.2, we get that (®A)®A ; 4 (Z ) -bundles. Z as 2 Hence we obtain a construction which allows us to move from the setting of ;Z 4 (Z )-bundles to that of (Z 2 ;Z 2 )-bundles and back, without loosing any 2 equivariant or geometric information (as long as we keep track of the line bundle A). This construction is well-defined on (Z ;Z 4 2 )-isomorphism classes of bundles. Once we are in the setting of 2 (Z Z ; )-bundles, we can use the conclusion of 2 Proposition 2.1: we quotient by the action to obtain (®A)/Z , which due 2 to the fact that the Z -action is free, is again a real vector bundle. We call 2 this quotient bundle r. Notice that pulling back along r : M —* M/a gives the original (Z ;Z 4 )-bundle back: K*() = ®A by definition of the quotient. 2 Now, provided we have kept track of A, we can form 7r*()®A to retrieve the original (Z ;Z 4 )-bundle 2 .  20  3.2. (Z ; 7Z 4 )-buiidle bordism 2  The question arises whether this process is well-defined on the 4 (Z Z ; )2 bordism classes. The answer is not immediately obvious since, for example, ;Z 4 (Z )-bundle isomorphism relation is a finer equivalence relation than 2 that of (Z ;Z 4 )-bordism: 2 ‘ certainly does not imply that as “  2 ; 4 (Z ) Z  (Z Z ; 4 )-bundles. Furthermore, there is the question of the line bundle A: 2 given a (Z )-bundle = (E, , M, u), is there a canonical way to select 2 ;Z 4 a (Z ;Z 4 )-line bundle A over (M, o), such that this process of selection is 2 well-defined on the (Z ;Z 4 )-bordism classes? 2 In the case where rank() = k is odd, the orientation bundle A provides such a canonical (Z ;Z 4 )-line bundle, and the process we have outlined 2 induces the isomorphism 4 of the following theorem: Theorem 3.1. :Bk(Z4;Z2)  A’(BSOk  X  ) 4 BZ  is an isomorphism of J.f,-modules for all odd k. REMARK: We postpone the proof until Section 6.1.  ;Z 4 )-bundle 2 Given a (Z = (E, , M, o), we will denote its orientation (determinant) bundle by A = (AE, A, M, o), omitting the rank of the bundle from the notation of the exterior product (i.e. A stands for A, where k = rank()). Note that locally (A) ( 2 xi A  ...  A Xk)  =  (x1) A 2 â  ...  A  (xk) 2 â  =  (—xi) A  =  (_l)kxi A  ...  A (—xk)  ...  A Xk,  giving that A is a (Z ;Z 4 )-bundle if k is odd; however A is a (Z 2 ;Z 2 )2 bundle if k is even. Therefore in the case of a (Z ;Z 4 )-bundle of an even 2 rank, we can not use the orientation bundle to send to a (Z )-bundle, 2 ;Z 2 21  3.3. Characteristic numbers for (Z ; 7Z 4 )-bundle bordism 2 since ®A would again be a (Z ;Z 4 )-bundle (hence Proposition 2.1 does not 2 apply the quotient of ®A by the action would not be a vector bundle). -  Given a 4 (Z 7Z ; )-bundle = (E, &, M, a), we will say that the base of 2 supports a (Z ; 7Z2)-line bundle if there exists some (Z 4 ;Z 4 )-line bundle over 2 (M,u). Let 2 ;Z be a collection of 2 4 £(Z ) ; 4 (Z ) -isomorphism Z classes of bundles given by  [a’]  Definition 3.2. ;Z 4 ) if there exists ‘ such that [] = 2 e £(Z as ;Z 4 (Z )-bordism classes, and the base of ‘ supports a (Z 2 )-line bundle. 2 ;Z 4  [a’]  Therefore if e £(Z ;Z 4 ), we can pick some ‘ such that [] = 2 and the base of ‘ supports some (Z )-line bundle A. We form ‘®A and 2 ;Z 4 proceed as in the case of odd k (but now utilizing A instead of At). Then we can state REMARK:  assign a class  xe  [] e  5,(Z; Z ) and e £(Z 2 ;Z 4 ), then we can 2 ) to [j, such that if x = 0 then [j = 0. 4 d(BO2k x BZ  Theorem 3.3. Suppose  The proof will be given in Section 6.1. The choice of the line bundle A is not canonical, hence the class x depends on this choice, and unlike in the case of odd k, we do not get a well-defined homomorphism ) —* .Af(BO 2 ). However, x (together with 4 of Theo 4 B2k(Z4; Z k x BZ 2 rem 3.1) still allows us to obtain a complete set of characteristic numbers for 2 ;Z 4 (Z ) . REMARK:  Characteristic numbers for (Z ;Z 4 )-bundle 2 bordism  3.3  Let  =  (E, &, M, a), then we have the following  ;Z 4 £(Z ) . A set of Bk(Z4;Z2) be such that e 2 characteristic numbers which completely determines the bordism type of [] is given by the products of the following classes in H*(M/a; Z ): 2  Proposition 3.4. Let  (1) w(T(M/a))  -  []  E  the Stiefel- Whitney classes of the tangent bundle of M/u, 22  3.3. Characteristic numbers for (Z ; 7Z 4 )-bundle bordism 2 (2) w 3 (j) - the Stiefel- Whitney classes of the (orientable) bundle i, and (3)  and /3 - the characteristic classes of the principal 7Z -bundle 4 S(A) — M/a,  where i = (®A)/Z , and A is as described in the remark preceding Theo 2 rem 3.3. The products are formed as is usual for characteristic numbers (see for example [8]), and then evaluated on the fundamental Z -homology 2 class of M/a. REMARK:  By Theorem 17.3 of [8], the characteristic numbers of the singular bordism class [M, Ii] e V(X) are given by products of w(TM) and h*(x), for all x é H*(X; Z ). In our situation, the Kiinneth isomorphism gives 2 PROOF.  ) 4 .N(BSOk x BZ  ) 4 AI*(BSOk) ® .N(BZ  The isomorphism is given by ir U ir, where ‘ri :BSOk x BZ 4 —* BSOk and ir :BSOk x BZ 4 —f BZ 4 are the projections. Taking [M/o,f x g] = we get  (f x 9)*(* U ir)  = =  f*R* U g*1r f* U g*  (the same conclusion holds if we were to replace BSOk by BOk, except instead of using ( we would use the class x given by Theorem 3.3).  [c])  But f classifies i in BSOk if k is odd (or in BOk if k is even), while g classifies the principal Z -bundle S(A) —+ M/o if k is odd (or the principal 4 -bundle S(A) —* M/u if k is even)’. Now 4 Z  2 ; 4 H*(BZ ) Z  [,/3]/(a Z 2  =  1),  23  3.4. Characteristic numbers for 7Z -bordism 4 = g*(), where Ic = 1 and II = 2 (see for example [11]). So and = g*()• f*() ) 2 Likewise H*(BSOk;Z ,. ,w], and 2 Z[il = w(ij) by the definition of f and the universal Stiefel-Whitney classes (if k is even, ) Z 2 the universal classes are coming from H*(BOk;Z [wi,... ,wk].) • 2 .  3.4  .  Characteristic numbers for Z -bordism 4  Given [M] E  we have the fixed point homomorphism 4 :jV F  —*  .M._k(BO(Z4))  of [21], which sends [MI to the bordism class determined by the normal bundles of Mz4 in M. The characteristic numbers giving the bordism type in the image of F 4 are determined by the equivariant splitting of the normal bundles. For details on the computation of these characteristic numbers see for example [14]. Lemma 3.5. If[M]  e .AJ  is such that F -bordant 4 ([M]) = 0, then M is Z 4 to some M’, such that for every x E M’, G Z 2 <Z . 4  PROOF. Standard by an equivariant surgery on the normal bundles of the fixed point sets. See [21]. • -  But now note that the normal bundles of the Z 2 fixed-point sets are (Z ;Z 4 )2 bundles. We show that it is these bundles that now determine the Z 4 bordism type of M’. Without loss of generality, suppose (M/)Z2 = F where dimF = dimM’ — k >0. Let v be the normal 2 ; 4 (Z ) -bundle Z ofF in M’. Now suppose i’ = äi for some Z -bundle 4 We can then glue M’ x I and D(1) by identifying D(v) x {1} and D(01). This produces a 7Z -bordism 4 between M’ and M”, where M” is a free Z -manifold. But then [M”] = 0 4 in .iV?. (To see this, form M” , and let A be the associated 2 M”/7Z )-line bundle. Then S(A) 2 ;Z 4 (Z M”, but ÔD(A) = S(A).) Therefore the 7Z -bordism type of M’ is completely determined by the image of [ii] in 4 fZ2 (BOk (C°°)) .  —  24  3.4. Characteristic numbers for Z -bordism 4  jV2(Bok(C00)) Let k : k(Z4;Z2) homomorphism. We have  Lemma 3.6. If  []  E 13k(Z4;Z2)  and  —*  J..f*Z2(Bok(coO)) be the forgetful  ;Z then 4 £(Z ) 2  []  E  kerk.  In [3] it is shown that ekJVk(BOk(C°°)) splits as a non-direct sum of rings K and L, where K is generated by e and r,te exten sions from ekJV_k(BOk) ekBk(Z2;{e}). But as we show in the proof of Theorem 4.14, these extensions must support (Z ;Z 4 )-line bundles. By 2 Lemma 2.1 of [4], we know that Z,, is in the kernel of k. • PROOF.  Therefore it is the combination of the characteristic numbers of )) and the characteristic numbers given by Proposition 3.4 that 4 .M(BO(Z gives a complete set of characteristic numbers for REMARK:  The 4 -Z characteristic numbers found by Beem and Rowlett in [5] are given by the following Proposition 3.7. F([M])  0 and F([8 1 x M]) 2 z  =  0 if [M]  =  0 in  Here F denotes the fixed-point homomorphism defined in the Introduction. REMARK:  PROOF.  See [5].  •  The characteristic numbers which can be obtained through Propo sition 3.7 are different from those that we have obtained. For example, the geometric data coming from the fixed-point sets and normal bundles of the -twisting construction on a given manifold M, is different from the data 1 S obtained by the bundle constructions of the Proposition 3.4 on the normal bundles of the fixed-point sets of MZ2. REMARK:  25  Chapter 4  An Application: Splitting 4.1  Motivation  In [5] we can find the following Proposition 4.1. Every [M] E .A/4 (PZ ) is equal to 4 4 x M [Z ] + [S 1 1 x M ] + [K], 2 2 z  for some [Mi] and [M ] in .N2 and the class [K] is in the kernel of the 2 forgetful homomorphism (PZ ) 4 —  PROOF.  This is Proposition 3.3 of [5]. I  This result oddly resembles the conclusion of the remark follow ing Proposition 4 of [15], where a similar splitting is obtained on free Z manifolds. However, the proof of this latter result is purely geometric, while the proof of the former result relies heavily on algebraic structure of the bordism algebra of involutions (as described in [24] and [10]), and the action of this algebra on N2 (as described in [2]). Even more discomforting is the fact that certain key steps of the argument come from a preprint by Stong that was actually never published. The argument also relies on [6]. REMARK:  It was our desire to find a simpler, purely geometric proof of Proposition 4.1 that has led us to the results described in this chapter. We demonstrate that the splitting is given by a surgery along the submanifold given by the Smith construction.  26  4.2. Smith homomorphism, extensions, and restrictions  4.2  Smith homomorphism, extensions, and restrictions  The following construction can be found in [20]. Let X be a 7Z-space, and let x e Ji? (X) be represented by a singular -manifold (M, o, f), where ci denotes the action on M and f : M —* X 2 Z is an equivariant reference map. The quotient M —* M/ci is classified by some class in H 1 (M/a; Z ), which under Poincaré duality corresponds to 2 a submanifold N C M/ci of codimension 1. Let N — N be the induced double cover. Since by the definition of an induced cover we can take N to -invariant submanifold of N x M, we get an induced involution ci’ 2 be a 7Z on N, given by 1 x ci. Let i : N —* M be the equivariant inclusion obtained by projecting N C N x M onto M. Definition 4.2. The above construction defines a homomorphism of f modules >NZ2(X)  LS2(X)  of degree —1, which sends [M, ci, f] to [N, a’, f o i]. That L is a well-defined homomorphism is shown in [20]. This construction is usually called the Smith construction, and the homomor the Smith homomorphism. We will sometimes abuse the nota phism REMARK:  tion slightly and denote the Z -invariant submanifold i(N) C M by (M). 2 Therefore in this notation [N, ci’, f oi] = [(M), I(M) fI(M)1. There is another, equivalent way of defining (M), given in [8]. Let R -representation (R, (—1)) and let SN = S(R’) for N >> 2 denote the Z dim M. Then since both M and SN are fixed-point free, we can find some equivariant map f : M —* SN which is transverse regular to the equa N1 tor 5 c SN: for example, we descend to the quotients, take a non equivariant map  J: M/ci  pN_l  and then lift to obtain  c  R.PN  —+  2 SN/Z  =  RpN which is transverse regular to  f. 27  4.2. Smith homomorphism, extensions, and restrictions Then by the usual results on transverse regularity, f_i (SN—i) is a submani fold of codimension 1 in M, and by the equivariance of f it is a Z -invariant 2 submanifold. As observed in Section 24 of [8], the Poincaré dual of the ] E H_i(M/a;Z 2 homology class [f_1(sN_i)/z ), where n = dim(M), 2 classifies the double cover M M/u, and therefore by Definition 4.2, we f_i (SN_i). can take (M) to be This definition is easier to use in certain —  situations. Again, let (M, a) be a smooth, fixed-point free involution. We use the second definition of L to prove the following  Lemma 4.3. The normal bundle of(M) in M is equivariantly isomorphic to /(M) x R_. Furthermore, there exists a compact, regular submanifold 0 c M, such that M M 0 U a(Mo) = M and M 0 fl a(Mo) = (3M 0 = Ni the normal bundle of 5 Then equivariantly But by definition, f is transverse regular so the nor mal bundle of f_i(SN_) in M is equivariantly isomorphic to f*(vs) f*(SN—1 x IR_) = f_i (SN_i x R_ = (M) x IR ) PROOF. Let vs be N_1 x R_. 5  Now, let U be some open, Z -invariant tubular neighborhood of i(M) in 2 M. Then by definition of an equivariant tubular neighborhood, U is Z 2 SN_SN_i diffeomorphic to (M) xD(R_). But ImfIM_u c which consists N of two components such that the action on 5 takes one into the other. By the equivariance of f, we conclude that M U = Ma II M where = M and the two manifolds are disjoint in M. But using U to form an equivariant isotopy, we see that Ma is equivariantly isotopic to some such that OM = (M). Therefore, we can take M 0= The conclusion now follows. • —  Lemma 4.4. The  of Definition 4.2 induces a well-defined homomorphism  of .N. -modules  :L3k(Z4;Z2)  >  2 ; 4 B(Z ) Z 28  4.2. Smith homomorphism, extensions, and restrictions which sends  []  =  [E,&,M,] to [I(M)l  PROOF. Let [M, o, f] correspond to  []  =  [EI(M),&I(M),(M),J[(M)].  under the identification  Bk(Z4;Z2) =A2(BOk(C0O)).  (4.1)  By definition of this identification, f*(7k(Coo)) = equivariantly. Apply the homomorphism of Definition 4.2 to [M, r, f] to get  But under (4.1) this latter class identifies to (fI(M))*(yk)  =  f*Qy)()  =  We will sometimes denote the choice of the bundle I(M) by Therefore, the bordism class ([j) is represented by a (Z ;Z 4 )-bundle 2  REMARK:  Given a 2 ; 4 (Z ) -bundle Z = (E,&,M,u), let 2 ; {e})-bundle 2 rz ( ) be the (Z obtained by letting Z 2 act on through the inclusion Z 2 7Lj. Therefore 2 = (—1) on 2 acts trivially on the base of , and via a multiplication by the fibers. We denote this explicitly by rz (e) = (E,(—1),M). Note that 2 2 commutes with the operation of taking of the boundary, and therefore rz takes (Z ;Z 4 )-bordant bundles to (Z 2 ; {e})-bordant bundles. 2 Definition 4.5. Let :Bk(Z4;Z2) 2 rz  >  13(7Z;{e})  be the restriction homomorphism of .Af-modu1es which sends ,Mj. 2 to [rz ()] = {E,u 2  []  =  [E, , M, u]  Lemma 4.6.  Im  C  . 2 kerrz  Let [] e Bk(Z4; Z ) and let = (E, &, M, o’). By Lemma 4.3, fOr 2 any choice of .(M) there is a submanifold M 0 C M such that PRooF.  29  4.2. Smith homomorphism, extensions, and restrictions 0 U u(Mo) M of to M . 0  M and M 0 fl a(Mo) = (M). Let o denote the restriction &2(o) Then Also = o since 2 = idM and & covers a. non-equivariantly, Oeo = oIaMo = I(M) = Therefore a(rz (o)) = 2 r()).• =  Let Z 4 be represented by the powers of the complex root of unity i; i.e. Given a (Z = {1,i,—1,—i}. ;{e})-bundle i = (F,(—1),P) of rank k, 2 define the free Z -bundle x Z by (F x Z 4 , (—1) x i, P x Z 4 , 1 x i). The 4 quotient ( x Z )/((—1) x (—1)) is again a vector bundle of rank k since 4 xZ 4 is a free Z -bundle. Let Z 4 4 act on the quotient by 1 x i. An argument on the representatives of the orbits shows that the Z 4 action by 1 x i is free away from the base, and has Z -isotropy on the base. Therefore 2 the quotient is a (Z )-bundle and we denote it by e(i). The base of 2 ;Z 4 is the free Z -manifold (P x Z 2 , 1 x (—1)). Therefore 2 e(ij)  =  )/((—1) x (—1)),1 x i,P x Z 4 ((F x Z ,1 2  x  (—1)).  Since taking of the quotient by a free action preserves boundaries, e is well defined on the bordism classes of (Z ; {e})-bundles. 2 Therefore, we obtain the following Definition 4.7. The above construction defines the extension homomor phism of ..M-modules e:Bk(Z2;{e})  which sends  []  =  k(Z4;Z2)  [(F, (—1),P] to [e(?7)].  Lemma 4.8. Ime C ker rz . 2 By construction of e(), we have rz (e(?7)) Hi (notice that 2 the action of Z 2 Z 4 on the base is given by (1 x (_1))2 = 1 x 1). Therefore oe([7]]) = rz2([e(11)]) = [rz2(e())] = {iHi] = 2 [j = 0. • PROOF.  30  4.2. Smith homomorphism, extensions, and restrictions  We have the following Lemma 4.9.  ker Z  =  Ime.  PROOF. We first show that Ime C ker Let [ii] e L3k(Z2; {e}) and consider e (n). By definition of e (j), the basis of this bundle is (M x Z , 1 x (—1)) for some non-equivariant manifold M. Let x e 2 = N 8 S(IR’) and define an equivariant map f :Mx7Z 2 by f(m, 1) = x and f(m, —1) = —x. The points x and —z being antipodal implies that there N such exists some SN c 5 that (Imf)flSN_l = 0. Therefore (MxZ ) = 2 0 and the definition of the homomorphism gives (e([i])) = 0. .  —  Now we show that ker C Ime. Let = (E, , M, o) be a (Z )-bundle 2 ;Z 4 and suppose that z([]) = 0. Therefore, there is some (Z )-bundle 2 ;Z 4 such that 9 = Let W be the base of , so OW = (M). Let U (M) x D(R_) be the equivariant tubular neighborhood of (M) in M. We glue x I (as a bundle over M x I) to x D(R) (as a bundle over W x D(]L)) by identifying )u x {1.} and O x D(W.). See Figure 4.1 for a depiction of how this surgery is performed on the bases of x D (R) and x I. The restriction of the resulting bundle to the boundary of the base is equal to  II ‘ and we will show that  e’  =  e() for some (Z ; {e})-bundle 2  Ti.  First note that Lemma 4.3 implies that M U = M 0 H o-(Mo) for some non-equivariant manifold 0 M (except now M 0 fl u(Mo) = 0). Let o = elM 0 then ‘o = CoH&(Co). But ‘ is equal to o U(Cx D(R))lwz 2 by the gluing. Let ‘ be the bundle map giving the Z4-action on C’. Then if we —  denote  Co  U  aWx {i}  (x D(R))Iw{l}, we get that  C’  =  H u’(). But  (Z {e})-bundle which we denote by Ti By definition of e we ; is a 2 have that C’ = e(Ti). •  (Cs’ &12)  31  4.2. Smith homomorphism, extensions, and restrictions MxI W x D(k) M  M  M’  M  n  M’  J Figure 4.1: Surgery along U.  Similar to the bundle case, we can define the extension and the restriction homomorphism for the bordism of free manifolds. These are also denoted by e and rz , and whether we are talking about the bundle variant or the 2 manifold variant of each homomorphism will be clear from the context. Given a free 2 -7Z manifold (M, u), note that the construction x (—1)), 1 x i) gives a free Z ((M x -manifold. We denote this con 4 struction by e (M, o), or simply by e (M) when it is clear from the context -manifold and we wish to avoid mentioning the action. 2 that M is a free Z Definition 4.10. Let  ) 2 e :J(BZ  ) 4 N(BZ  be the extension homomorphism of JV-modules which sends [M, o] to [e(M, o)}. The construction preserves boundaries, therefore this homomor phism is well defined on bordism classes. REMARK:  32  4.3. The bundle splitting results  Similarly, we have the following Definition 4.11. Let  ) 4 2 :.N(BZ rz  ) 2 .M(BZ  be the restriction homomorphism of jV-modules which sends [N, 9] to [N, 92]. Then we can obtain the following Lemma 4.12. If [M,cr] eAI(BZ ) then rz 2 2 oe([M,o])  =  0.  PROOF. By definition, e(M,u) = ((Mx Z )/(o x (—1)), 1 x i). Therefore 4 ) / (a x (—1)), 1 x (—1)) 4 2 (e (M, u)) = ((M x Z rz (M, a) II (M, o). Hence 2 oe([M,u]) = 2 [M,a] = 0. • rz  4.3  The bundle splitting results  Given S 21 = S(C’), let T denote the 4 Z action by multiplication by i. Let a denote the antipodal action on S . Then we have the following 2 Proposition 4.13. (Katsube) ) is a free A/ 4 .NK (BZ , -module with basis 1  {[S ’ 2 ,T],e([S 21 ’,a])  In  0}.  PROOF. The proof of the proposition can be found in [11]. We spend the rest of this section showing that this .,V-module structure, combined with the geometry of the Smith construction z, induces a geomet ric splitting of the representatives of (Z )-bordism classes. Theorems 3.1 2 ;Z 4 and 3.3 are instrumental in our proofs. Furthermore, we show a connection between z and 2 ;Z 4 £(Z ) .  33  4.3. The bundle splitting results Theorem 4.14. If ;Z 4 £(Z ) 2 , then  [] e  ) is such that z([]) 2 ;Z 4 (Z  (Z;Z)  0 and  E  e (ji) + re’ (772)  where i, ‘72 are (Z ; {e})-bundles. Furthermore, this bordism can be obtained 2 by an ezplicit geometric surgery on  We will need the following Proposition 4.15. If ;Z 4 E(Z ) 2 , then  {]  ) is such that 2 ;Z 4 B(Z =  for some (Z ; {e})-bundle 2 REMARK:  /.([])  0 and  [eQ)j  .  The proof is given in Section 6.2.  We now present the proof of Theorem 4.14. PROOF. Let [] e 2 ;Z be such that ([]) 4 (Z ) Denote (E, &, M, o). By Proposition 4.15, =  [e(ii)}  0 and  ;Z 4 (Z ) . e2  (4.2)  where rn is some (Z ; {e})-bundle. Let U be a closed equivariant tubular 2 neighborhood of (M) in M. By the definition of z, U ((M) x D(R_),uI(M) X (—1)). We begin by constructing a (Z ;Z 4 )-bundle , such that O= H ‘, where 2 ‘ is a bundle with the property that (‘) = e(’71). By (4.2), there is a )-bundle ç such that 2 ;Z 4 (Z 0c  Denote ç  =  =  () He(i).  (4.3)  (F,O,W,O).  34  4.3. The bundle splitting results We glue xl (as a bundle with base Mx I) and ( x D(R_),6 x (—1)) (as a bundle over base(C) x D(R_)) by identifying elu x {1} and (I(M) x D(R), OL(M) x (—1)). Denote the resulting (Z ;Z 4 )-bundle 2 by , then II ‘, and by construction = =  (4.4)  e(1).  Next we need to find Denote ‘ = (E’, a’, M’, a’), and let E = re( ) = 1 1 x )/((—1) x (—1)),i x 1) where we take 1 = 5(C). Let N be the ((S base of E. By construction of E, =  e(i).  (4.5)  Now, let UE be the equivariant tubular neighborhood of (N) in N, and UM’ be a similarly defined neighborhood in M’. Denote N = N U and M’ = M’ each consist of two UM’. The bases of E and equivariantly diffeomorphic components, and in each case, the bundles over these components are interchanged by the action of Z . 4 —  —  Therefore, gluing one end of ‘ x I to an end of E x I along EIu = (which are equal by (4.4) and (4.5)) produces a bordism between ‘ and E II 1’ where T = T 0HT 1 is a (Z )-bundle such that the action of Z 2 ; 7Z 4 4 interchanges the bundles T 0 and T . 1 Let be the bundle map inducing the (Z )-action on T. Then (To, IT 2 ;Z 4 ) 0 is a (Z ; {e})-bundle (omitting the base from the notation), which we denote 2 ‘72 But T = e(2) as desired. • We obtain an immediate corollary which demonstrates the relationship be )bundles in £(Z 2 tween (Z ;Z 4 ;Z 4 ) and the Smith construction 2 .  Corollary 4.16. Suppose [] e 2 ;Z and 4 (Z ) ;Z 4 (Z ) . 0 if and only if E 2  ([])  0. Then  2([])  =  35  4.3. The bundle splitting results Let [] Suppose  PROOF.  (=)  ). 2 e B(Z ;Z 4  ([e]) 2  =  0, and assume 0. Since by Lemma 4.9 ker = Ime, we get that ([j) e([(i]) = [e()] for some (Z ; {e}) 2 o Let S = bundle e (ci). Then ([S]) = [s(S)] = [e ()] = e by the definition of L. .  ([c’])  Therefore  ([J + [5]) = ([]) + i([S]) = e([1]) + e([cij) =0. But this shows that [] + [5] ; {e})-bundle 2 for some (Z  = e([C2j),  or  [] = e([c2]) +  Let N 1 be the base of , and N 2 be the base of  o  e([ci))  Take  A=e(Ni xR_,1 x (—1))llroe(N 2 xR_,1 x (—1)), ;Z 4 )-line bundle, showing that 2 which is a (Z  ;Z 4 £(Z ). 2  ;Z 4 £(Z ) . Then by Proposition 4.15, ([]) = [e((j] Suppose e 2 for some (Z ; {e})-bundle c’. But 2([]) = L([e(C’)]) = [(eQj)j = 2 [O]=0. I  ()  36  Chapter 5  More General Groups 5.1  Characteristic numbers for more general groups  Consider a finite group G such that (Gi) (G2)  there is a K G such that K Z , and 2 foreachgeG,ifg=2thengeK.  r for r 2 For example, Z 1 satisfies the above conditions. Also 1*, the binary icosahedral group, satisfies the conditions. (Note that 1* is a non-abelian group.) REMARK:  Now, suppose  =  (E, M) is a C-bundle such that  (a) (b)  if x  e M then G  =  K, and  ifvEEandvO,thenG={e}.  Call such a bundle a (G; G/Z )-bundle. By the property (G2) above, K 2 is the subgroup of order two in G, so we denote it Z 2 without a chance of confusion. Lemma 5.1. If g e C is of odd order, then (g), the cyclic group generated by g, must act freely on  Suppose g E C and II = 2n + 1 where n> 1, is such that g acts trivially. But this contradicts (a) since I (g) I > 2. Therefore elements of odd order can not act trivially. But every element of (g) is of odd order, therefore (g) acts freely. • PROOF.  37  5.1. Characteristic numbers for more general groups  Let = (E’, M) and 2 = 2 (E M) be two (C; G/Z , ).-bundles over the same 2 —* base. Let g E G and let : be the bundle map induced by the left multiplication by g. If = 2n, then locally on the fibers we have that (L’)®L ) 2 )  =  (L))Th®(L ) 2 )  =  (—1)®(-1)  =  1®1  since gfl = 2 and for x e M, gx = x. (Note: as always we can take some G-equivariant Riemannian metric, so that locally (L) can be taken to be an element of 0(k), where k stands for the rank of the bundle .) Therefore is a (G/7Z ; G/Z 2 )-bundle (i.e. a free G/Z 2 -bundle, or 2 ® equivalently, a G-bundle in which the isotropy at every point is K = Z ). 2 Let Bk(G; G/Z ) denote the bordism ring of (G/Z 2 ; G/Z 2 )-bundles, then we 2 have the following Theorem 5.2. There is an isomorphism of Jf-modules  ) 2 :Bk(G;G/Z  f*(BSOk  ><  BG)  for all odd k. PROOF. The proof is conceptually similar to that of Theorem 3.1. We only  verify certain key points. )-bundle of odd rank k. Then A is a (C; G/Z 2 be a (G; G/Z )-bundle, 2 and as in the proof of Theorem 3.1, the action on can be verified to be orientation preserving. Classify (®A)/(G/Z ) into BSOk, and classify 2 S(A)/G into BG. We obtain a class in .iV(BSOk x BG) and a S(Ae) homomorphism  Let  —  4:13k(G;G/Z2)  >f(BSOk xBG)  38  5.1. Characteristic numbers for more general groups which can be checked to be well-defined (by an argument similar to the one found in the proof of Theorem 3.1). Conversely, if ij is a (G/Z ; G/Z 2 )-bundle and A is a (G; G/Z 2 )-line bun 2 dle, ®A is a (C; G/Z )-bundle, and by the general properties of C-vector 2 bundles, we obtain (in the case when = ®A) (®A)®A  C  C  Given a classifying map g : M — BG, let M = g*(EC). M is a free G manifold, and we can associate a (C; C/Z )-line bundle A to the principal 2 -bundle 7t : M —* M/Z 2 Z . It is not hard to see that if M = S(Ae)/C and 2 g classifies S(A), we get that S(A) S(Ae) (which is an isomorphism of principal C-bundles). Given f : M —* BSOk, form = f*(k), and let 7 = 1r*(q) which is a ; C/Z 2 (C/Z )-bundle. Form [‘®A], a class in Bk(C; G/7Z 2 ). Call the homo 2 morphism induced by this construction :JV(BSOk x BG)  13k(C;C/Z ) 2 .  As in the proof of Theorem 3.1, we can check that IJ is well defined, and .• 1 that= Let £(C; G/Z ) be defined as follows: let be a (G; G/Z 2 )-bundle, then is 2 ) if there exists a (C; C/7Z 2 in £(C; C/Z )-bundle ‘ such that ‘ is(C; C/Z 2 )2 bundle bordant to and ‘ supports a (G; C/Z )-line bundle. 2  e  Then we obtain Theorem 5.3. suppose [] E B2k(C; C/Z ), and 2 ), then we 2 £(C; C/Z can assign a class x E .N(BO2k x BG) to such that if x = 0 then [] = 0.  [a],  PROOF. The proof is conceptually identical to that of Theorem 3.3, and the  generalizations involved are completely analogous to those of the proof of 39  5.2. Smith constructions, extensions, restrictions for G Theorem 5.2.  =  r 2 Z  •  The characteristic numbers for [] E Bk(G; G/Z ) are similar to 2 those of Proposition 3.4, except that the characteristic classes involved in the classification of the quotient of the base of the bundle now come from ). 2 the generators of H*(G;Z REMARK:  5.2  Smith constructions, extensions, restrictions r 2 for G=Z  As always, let G be a finite group acting smoothly on a compact manifold M. Let V be an orthogonal, real representation of G of dimension k. We say that a submanifold N C M is dual to V if (1) (2)  N is C-invariant in M, and the normal bundle v of N in M is C-isomorphic to N x V.  For arbitrary M and V such a submanifold might not exist. Given K C, let AK denote the family of all subgroups of K. Then we have the following Proposition 5.4. (Rowlett) Let G be nilpotent, K G, and for every x E M let G K, and let K ker V. Then there is a C-invariant submanifold N C M dual to V. Furthermore, there is an JV-module homomorphism  i:iV(AK)  >  NG(AK)  of degree —k, such that ([M]) is represented by a submanifold dual to V. PROOF.  The proof can be found in [15].  •  Note that the statement of the proposition implies that any two submanifolds dual to V in M are bordant as AK-manifolds. We will abuse the notation slightly and denote any particular choice of such a submanifold by (M). Therefore, the bordism class ([M]) is represented by a manifold or equivalently ([M]) = [(M)]. The homomorphism L is usually referred to in the literature as the generalized Smith homomorphism. REMARKS:  40  5.2. Smith constructions, extensions, restrictions for G = Z r 2  Since we intend to generalize Proposition 5.4 to G-bordism of bundles, we will outline the key parts of the proof. Given a class [M] E the proof constructs a G-space U and shows that one can always select an equivariant map f : M —* U x V which is transverse regular to U in U x V. The manifold (M) is then defined to be f’(U). That this operation is well-defined with respect to bordism classes is demon strated as follows: suppose f:M —* U x V give (M) f (U) for i = 0 1 and 1. Also suppose W is such that W = 1 11M Then according to [15], 0 M . we can find some equivariant F: W —* U x V such that F is transverse reg ular to U, and furthermore FIM is isotopic to fj. Let G 1 :M x I Ux V denote these isotopies. Now glue M 0 x IUWUM 1 x I and denote the resulting manifold by W’. Also glue the maps Go U F U G 1 and denote the resulting equivariant map by F’. Therefore F’ : W’ — U x V and = f’(U) Ufj’(U) = (Mo) 1I(M ) by the construction of 1 W’ and F’. Therefore Z is well defined on bordism classes. —  r; Z 2 Now given a (Z r-i )-bundle , observe that the base M is a Z 2 r manifold 2 such that for every x E M, G = Z . Therefore [M] e .N (AZ 2 ). Take 2 V = L (i.e V R and the generator of Z r acts on R by multiplication by 2 (—1)) and define =  (5.1)  where (M) is the submanifold of M dual to R_. We show that this construction is well defined on the bordism classes of r (Z r 2 -i)-bundles. ;Z Let ebe a 2 r (Z _ i)-bundle ;Z over W such that and the base of each is M. Extend the gluing M 0 x IUWUM 1 xl to the equivariant clutching o x I U U i x I and denote the resulting r; Z 2 (Z _i)-bundle by ‘. Define F’ as before, and using the definition of 2 4j  41  5.2. Smith constructions, extensions, restrictions for G = Z r 2 we have =  = (o) T.1  =  Therefore (5.1) gives a well-defined homomorphism. Definition 5.5. Let  2 :*(Z r -i) ;Z  2 *(Z r -i) ;Z  be the homomorphism of .Af-modu1es which takes is the base manifold of  []  to  where M  .  ;Z 4 )-bundle case, we define the extension and the restriction 2 As in the (Z homomorphisms. Let T = e/’2r_1 be the generator of Z2r. Then given a ; {e})-bundle , let e (ii) be defined as 2 (Z e() =  r)/((—1) 2 (xZ  x T’)  and let the action on e () be given by 1 x T. Definition 5.6. The above construction defines a homomorphism of f modules  e” :(Z ; {e}) 2 which takes REMARK:  {i] to [e  r 2 *(Z r —i) ;Z  (n)].  T2’ = —1. Note that 1  In a completely analogous way to the Z 4 case (see Definition 4.10), we get Definition 5.7. Let  e’ :.M.(BZ ) 2  .N(BZ r 2 )  be the extension homomorphism of.Af-modules which sends a free involution r)/((—1) x (—1)), 1 x T]. 2 [Iv!, uj to [(M x Z  42  5.3. The bundle splitting for C = Z r 2 r to Z 2 Restricting the action from Z , induces a homomorphism 2 r_i) 2 *(Z r ;Z  —*  ;{e}) 2 *(Z  and a homomorphism r)—*jV*(Z 2 M(BZ ) , both of which will be denoted by rz . 2 Finally note the following Lemma 5.8. If [j  r_i) B*(Z r ;Z then e2 2a rz  = 0.  The proof is essentially the same as that of Lemma 4.6, since by definition z) = and the tubular neighborhood U of (M) in M is Z r-diffeomorphic to (M) x R_. Therefore M U again consists of two 2 parts exchanged under the action of the generator of Z r, and the rest of 2 the proof follows as before. • PROOF.  —  5.3  The bundle splitting for G =  r. Then Z 2 Let T = e”2r_1 be the generator of Z r acts on C by the 2 = (T2’) complex multiplication. Let K r with K as 2 , then G = Z 2 Z defined satisfies (Gi) and (G2). Also G/K Z r-i. 2 Similarly to the free Z -bordism case, we have 4 Proposition 5.9. (Katsube) r) is a free JV. -module with basis 2 .A/ (BZ  ,Tj, e([S 1 ” 2 {[S ,a]) 2 PROOF.  In  0}.  The proof of the proposition can be found in [11].  • 43  5.3. The bundle splitting for C  r 2 Z  As a consequence we obtain the following Proposition 5.10. If[A] E 2 i;Z i(Z _ i) and zS[A])  0, then  = [e()] for some (Z ; {e})-line bundle 2  .  The proof is similar to that of Lemma 6.1. By Lemma 2.6, r-i space for odd k. Let EZ 2 r = S(C), and ob 2 BOk(C°°) is a free Z serve that PROOF.  = BOi(C°°) by definition of BOi(C°°). As in the proof of Lemma 6.1, B0 1 (C°°) x EZ r-i is homotopy equivalent 2 r_i 2 Z  to  r/Z (EZ ) r _i = BZ BOi(C)/Z r 2 -i = 2 r. 2 Therefore,  2 ( 1 ’ Z r_i) 2 ;Z  jZr  1 (C°°)) (BO  BZ ) 2 . Denote this isomorphism by Then as in the proof of Lemma 6.1, l is given by the sphere-bundle construction ([A]) = [S(A)], where [A] E ;Z 2 r—l). The inverse, 2 is again given by the construction of the i (Z associated line bundle. .  Now suppose ([A])  0. By Lemma 5.8, 2 rz  0  i([A]) =  0,  (5.2)  44  5.3. The bundle splitting for G  =  r 2 Z  and therefore =  [F]e” ({S , a]) + >[Q 2 ’, T] 21 ] [S 3  and [F,] in N. This expansion is given on the free-Ark-module basis of M(BZ r) described by Proposition 5.9. 2 for  [Q]  Taking (5.2) and using OS( ) = S(( )), we get that rz 2 o l(([A]) = 0, and therefore that [Q ] = 0 for all j. Specifically, we have used here that 3 (1) ,T]) 21 ( 2 rz [S eD([S2i,a]) 2 (2) rZ  ,a] 21 [S  Z2r/Z  =  We can proceed to use Q to obtain line bundle ‘i. •  0 in 2 .AI(BZ ) , and  [S , 2 a]  ([AJ)  =  =  2r_1  [S , 2 a]  =  0.  [e (j)] for some (Z ; {e}) 2  This proposition, in combination with Theorems 5.2 and 5.3, allows us to generalize Proposition 4.15 to Z r. 2 First we must define the collection 2 £(Z r ;Z r_i). 2  Definition 5.11. Let bundle ‘ such that  r_i) .(Z r ;Z if there exists some 2 r (Z r —i);Z e 2  r ;Z 2 (Z r_i) 2  , and the base of  ‘  r; Z 2 supports a (Z r-l )2  line bundle. Proposition 5.12. If r;Z_i), then 2 £(Z  [] e  Bk(Z2r;Z r 2 -1) is such that =  z([])  0 and  [e(C)]  ; {e})-bundle c. 2 for some (Z  We only need to make a few key group-specific observations. The reader is referred to the proof of Proposition 4.15 for notation and the general idea of the argument. PROOF.  45  r 2 5.3. The bundle splitting for G = Z We first argue for bundles of odd rank. Given [] e B2k+1(Z2r; Z ), by 1 _ 2 Theorem 5.2 K o ([]) is in 2 W*(BSO k +l) ® .M(BZr). Pulling back as r instead of EZ 2 before (except using EZ ), we get a (Z 4 r—i; 2 2 r_i)-bundle r; Z 2 and a (Z r_i)-line bundle A, such that [‘®)] = 2 But an analysis of the Kftnneth isomorphism again verifies that 7®A = ii x A’, where i is a (non-equivariant) orientable bundle, and A’ is some (Z r_i)-line bundle. 2 ;Z 2 Therefore (®A) = x (A’), and an application of the Proposition 5.10 gives the generalization.  [c].  )-bordism, if k is even, we let the class x of Theo 2 As is the case of (Z ;Z 4 rem 5.3 play the role of ( and the rest of the argument is completely analogous to the odd-case argument. •  [c]),  46  Chapter 6  Proofs of Bundle Results 6.1  Characteristic numbers  6.1.1  The proof of Theorem 3.1  We need to construct an isomorphism :Bk(Z4;Z2)  Af(BSOk  X  ) 4 BZ  for all odd k. PROOF. Given a rank k, (Z ;Z 4 )-bundle 2 (E, &, M, u), where k is odd, form A = (AE, A, M, ) which is a (Z ;Z 4 )-bundle since locally, on each 2 fiber, A = 2 (Aa) = (_1)A...A(_)=(_1)k.1A...A1=_1. ...Aa  Here we have used the fact that we can take some Z -invariant Riemannian 4 metric on ., giving that & IEX e 0(k), where E is the fiber above some 2 x e M. Now form ®A = (E®AE,®M,M,a) and this is a 2 ;Z (Z ) bundle since A is a 2 ; 4 (Z ) -bundle: Z explicitly, (6®A3) 2 = 2 ®(A&) = â (_i)k = (1)0 101 on the fibers. Note that ® A is orientable since A(®A)  4 E  non-equivariantly.  We show that the action of 60A& is orientation preserving. We can take UIE to be the length-preserving linear transformation from the fiber E to Denote L = E, V = E, and V’ = E(). Then L: V —* V’. Pick a  47  6.1. Characteristic numbers basis for V, say  {Vi,.  . . ,v}.  Then  Vi®(vlA  . .. A vk)  =  Wi  Vk®(V1A...AVk)Wk  is a basis of V®AV. Suppose we pick a basis with a different orientation ,... ,—Vi,.. . ,Vj}. 1 {v Then note that vi®(vi A... A(—v)A... AVk)  =  —Wi  —v®(vi A ...A(—v)A... AVk) =w  Vk®(V1A  But then  (—Wi) A  .. . A w A  ... A (—vi) A ...  A (—wk)  .  =  ..  A Vk)  1A w  =  ...  —Wk.  A wk since k is odd.  Therefore V®AV has a canonical choice of orientation, and likewise V’®AV’ has a canonical choice of orientation. But L(vi ),. . . , L(Vk) is a basis of V”, and the same argument as above applied to this basis shows that L ® AL  takes the canonical orientation of V® AV to the canonical orientation of V”®AV’. Therefore, â®A is orientation preserving. Therefore = ®/&®Ad is orientable, and we let the classifying map.  f : M/  —*  BSOk be  Now take S(A) = {v E A lvi = 1} (again, with respect to some Z 4 invariant Riemannian metric), and note that 0 = (A&) I S(A) gives a free Z4-action on the closed manifold S(A). By construction, S(A)/9 2 2 is 7Z diffeomorphic to (M, a) (the action on the quotient is induced by 0, which  48  6.1. Characteristic numbers covers o-), and S(A)/8 M/a. Let g : M/a -bundle S(A) — M/a. 4 Z Given  ), 2 ;Z 4 [] e B(Z  ([])  =  4 classify the principal BZ  define the homomorphism  :l3j(Z;  by  —*  ) 2 Z  >  .N(BSOk  X  BZi)  [M/u,f x g].  We first show that 4 is well-defined. We need to show that if ã = o U j, then ([o]) = ([]). This is fairly obvious: we simply apply the construction which defines , on , and note that 8A=A9,  (6.1)  D®AO  (6.2)  =  and that the Z 2 action on ®A preserves the boundary. Using this property, and applying (6.1) and (6.2), we get ØA/)  =  ®A)/  =  (o®Ao),/z  . 2 H(1®1)/z  And similarly  aS(A)  =  S(ö A )  =  S(AO)  =  S(Ao) U S(A ) 1  where 8S(A) stands for S(Ae)Iow with W being the base of The last step to do is the classification of these bundles into BSOk x BZ . We obtain 4 .  =  ([]) + ([]).  Next we show that  is an isomorphism by producing a homomorphism ) 4 I’:JV(BSOk x BZ  —*  Bk(Z4;Z2)  such that 4oP=id and ‘11o4=id.  49  6.1. Characteristic numbers  Let 7t BSOk x BZ 4 —* BSOk and 7r2 : BSOk x BZ 4 — BZ 4 be the projection maps. Also, let k denote the universal bundle over BSOk. Given [N, h] E oh)*(EZ and = (7r1oh)*(k). By definition 2 (lr ) ), let iS = 4 4 .N(BSOk xB7Z then, N is a principal Z -bundle over N, and we let denote the free Z 4 4 2, action on N. If we quotient N by we obtain the principal Z -bundle 2 N —* N/q , and we let q 2 5 denote the action induced by q on the quotient  We form the associated line bundle A, specifically let A = (N x x (—1)). This is a (Z ;Z 4 )-line bundle with the action given by q5 = x 1, 2 covering the action c q on the base (the twisting, induced by the (—1) action 5 on R_, ensures that N/ ; (jct/çj2)/q 2 = N denote the quotient map onto = ir*(), and note that ir*() C x i (by definition of the pull-back) has a Z -action induced by bq x 1. Denote this action by q. 2 Clearly, is a (Z )-bundle (with action given by (, q)). 2 ;Z 2 Now let N. Let  it:  ‘  Form 7®A, which is a (Z ;Z 4 )-bundle since, locally (q®q) 2 2 1®(—1) = —1®1. Define ‘([N,h]) = [®A].  =  ®(’A) (q5) 2  =  To show that ‘I’ is well defined, we can proceed as in the case of , and apply the construction to a manifold with boundary. We only need to observe that the operation of taking of boundary commutes with pulibacks: in symbols, thi*( )it*O( ). Now we show that 4 o I1  =  id.  and A to be as in the definition of , and form ®A. Note that, in what is to follow, all bundle isomorphisms are considered equivariant. Also, we remind the reader that Em denote the trivial, rank m, equivariant bundle (i.e. Em over some (M, a) is defined as (M x R, , x 1, M, a)). Take  ‘  50  6.1. Characteristic numbers  We have, A(®A) A®AA A®A. Note that A = A7r*() by definition, .*(A). But i is by definition orientable, therefore A and A1r*(i) *(El)  implying  =  . 1 E  Therefore A’  =  A,  A(®A)  and so =  (7®A)®A(7®A)  (®A)®A  As observed in Section 2.3, this suffices to show that sake of clarity, we will show this explicitly. Note that over the same base:  [‘] = [fl]. But for the 7’, A’, 7, and A are all  \C7 and therefore we can glue  f  x I to  ‘  x I along  37’  37’  2 ; 4 (Z ) Z  37, and  x {O} by  1 at  equivariant clutching (once the restrictions of 37’ x {1 } and base are glued by an identity diffeomorphism). Therefore  37  x {1}  similarly A’ 2 ; 4 (Z ) Z  37 x  {O} to the  A, and both bordisms are  over the same base N/ 2 x I. Applying S( ) to the latter bordism gives 8(A). Therefore S(A’) ‘([N, h]) = ([37®A]) = [N, h]. o  ‘-  4 ;Z (7Z ) 2  We finally show that ‘I’ o = id. We use the notation as in the definition of . Let r : M —+ M/o denote the principal Z -bundle of the quotient. 2 Then by the definition of ØA/&®A, the pullback along ir gives The line bundle A associated to S(A) —* M/o is by definition A, therefore as 2 ; 4 (Z ) -bundles. Z As before, this shows ‘ = (®A)®A “.  and we have ‘I’ o 4([])  =  [a’] [c]. =  •  2 ; 4 (Z ) Z  51  6.1. Characteristic numbers  6.1.2  The proof of Theorem 3.3  We need to show that if [] e 4 ;Z 4 £(Z ) , then we can k 2 B ; 7Z2) (Z and E 2 assign a class x e .N(B02k x BZ ) to 4 such that if x = 0 then [j = 0.  [c],  PROOF. take [] E B2k(Z4; Z ) such that 2 ;Z 4 )-bundle ‘ such that 2 exists a (Z  E  ;Z 4 £(Z ); in other words, there 2 , and the base of ‘ supports  e’ (Z ) 2 4 ;Z  some 4 (Z Z ; )-line bundle, which we denote by A. Let 2 and A = (L, L, M’, u’).  ‘ =  (E’, &‘, M’, u’)  As in the proof of Theorem 3.1, ‘®A is now a (Z ;Z 2 )-bundle: the action 2 is given by the bundle map a’®dL. This action, however, is not necessarily orientation preserving (neither is ‘ ® A necessarily orientable). Let f : M’/’ —* BO classify the quotient (‘ØA)/(6’®L), and let g: M’/r’ —* BZ 4 classify the principal 7Z -bundle S(A) —* M’/cr’. Combine 4 these maps to obtain the class [M’/a’, f x g] in .M(BO2k x BZ ), which 4 we denote by x. This construction is not well-defined with respect to the (Z Z ; 4 )-bordism relation (see the Remark after the proof). Moreover, the 2 choice of A is not canonical, therefore the assignment of the class x is not canonical either: Specifically, A 2 could be another (Z ;Z 4 )-line bundle over 2 the base of ‘, but possibly 2 e’®A ‘®A thereby producing a different x• 4 However, if  x = 0 in d(BO2k  x BZ ) then 4  {]  =  0.  To see this, suppose (W, F) is such that O(W, F) = (M’/’, fxg). Using a re verse construction similar to the one defining “Ti in the proof of Theorem 3.1, we pull back this bordism, to obtain a (Z )-bundle bordism. Let ir be the 2 ;Z 4 quotient map of the principal Z -bundle SA)/Z 2 2 —* W, ir 1 the projection onto B02k, 7r2 is the projection onto B7Z , and 72k the universal bundle over 4 ; 4 (Z ) -bundle Z ØA, where = lr*((iri OF)*Qy2k)), and B02k. We obtain a 2 ). 4 A is the line bundle such that S(A) (7r2 o F)*(EZ  52  6.2. Splitting  But now a(’® A) ). Therefore 2 B2k(Z4; Z  OA  []  =  []  (‘ 0 A) 0 A I = 0.  ‘,  showing that  [c’]  =  0 in  Here is an example showing why the assignment of x does not produce a well-defined homomorphism. Let be a (Z )-bundle such that 2 ;Z 4 = II ‘, and let (W, 0) denote the base of Then, as it is clear from obstruction theory, it is not necessarily true that A extends over (W, 0): for example, if H 1 (W; Z ) = 0. For a specific example, take the non-trivial line 2 bundle 1 over RP’ and let Z 2 act by multiplication by (—1) on the fibers. Form 12 = (7Z x1)/((—1) x (—1)) which is a (Z 4 ;Z 4 )-bundle, the action being 2 induced by i x 1. Note that [121 is not 0 in i(Z ;Z 4 ). The base however 2 is RP 1 x Z , which is a boundary of a free Z 2 -manifold: a(D 2 2 x Z ) = 2 x Z . (To see that [12] 2 0, note that 1 does not extend over D , but 2 1 x I, id x (—1)) is not a free Z also (IRP -manifold.) 2 REMARK:  .  6.2 6.2.1  Splitting The proof of Proposition 4.15  We will need the following Lemma 6.1. If[A] e 2 ;Z and z([A]) 4 i(Z )  ([A1)  =  0, then  [e()]  for some (Z ; {e})-line bundle r. 2 Let S°° (S(C°°), i), here Z 4 acts on C by multiplication by the 2 by definition of BOk(C°°) (as complex number i. Then BO 1 (C°°) = S00/i 0 = U S(C’’) giving S°°/i applied to the case k = 1). More carefully, S 2= 2 S(Cc)/i is diffeomorphic Uk(S(C)/i) by the topology of a union. Each 2 = BO to the space of lines in C’s’. Therefore S00/i (C°°) by the definition 1 of the latter, and the free Z -action is induced by i; we denote this action 2 by r. PRooF.  53  6.2. Splitting  The quotient B0 (C°°) 1 BOi(C°°)/r is therefore a principal 7Z -bundle. 2 Let EZ 2 = (5(R°°), (—1)) be the top space of the universal principal Z 2 bundle. The projection map K of the fibration (BOi (C°°) x EZ ),//( 2 ><  BOl(C°°)/,  (—1))  is a homotopy equivalence, since the typical fiber EZ 2 is contractible. But by definition S = EZ 4 giving that BOi(C°°)/r = BZ . 4 Therefore we can conclude 2 ; 4 i(Z ) Z i’f(BOi(C°°)  )<  ) 2 EZ  (C°°)/Z 1 f(BO ) 2 .Ar(BZ). Therefore, the bordism of (Z ;Z 4 )-bundles of rank 1 is isomorphic to the 2 principal Z bordism. Denote the isomorphism by 4 2 ( 1 h ; 4 ) Z Z  .Af(BZ ) 4 .  We are only interested here in the .A1 -module structure, so for us the iso4 morphism is an .N-modu1e isomorphism. Let {).] e i(Z ;Z 4 ), and if we 2 denote A = (L, L, M, o) then 1 is given explicitly by  [S(A),âLIs(,)=us]. The inverse is given by the construction of the line bundle associated to the principal Z -bundle S(A) —* S(A)/6. 2 By Lemma 4.6, rz 2 o  =  0; i.e. there exists a (Z ; {e})-bundle 2  such  54  6.2. Splitting that OE  =  (z(A)). 2 rz  But then rz2  by the definition of  (z([?L]))  since S(E)  ,  =  =  0  (6.3)  S(AQj).  By Proposition 4.13, the N-modu1e basis of V(BZ ) is given by the ele 4 ments e([S2Th,a]), where a denotes the antipodal action, and [S ’,T], 2 where 21 = S(C°°) and T is induced by the multiplication by i on C. Therefore  [P] e ([52z, a]) +  =  for [Pt] and  [Q]  [Q]  .  t2+1, 5 [  T]  in J.  But by (6.3), 0  =  ((z([A]))) 2 rz  =  =  ,a]) + 2 2 oe([S rz  [Q]  2o2 r [S ’ ,T]  [Qj]. [S ,a]. 21  2i, 5 The last equality follows since rz a]) = 2 [S ) 2 ’, a] = 0 in .M 2 2 o e ([ . (BZ 1 by Lemma 4.12, and since rz ’, T] = [S2i+l, a] by the definition ofT. 2 2 o[S i+ a] are non-trivial in .N(BZ 2 [S , Note that 1 ) (see [27]), therefore [Q 2 ]=0 3  for allj. Hence, 1(([A])) = e([N]) for some [N] e .M(BZ ), and by constructing 2 the line bundle associated to the principal 2 -Z bundle e(N) we get z ([A]) = (e ([N])) = [e (n)], where is the (Z ; {e})-line 2 bundle associated to the principal Z -bundle N —* N/7Z 2 . U 2  55  6.2. Splitting  Finally, we use Lemma 6.1 to prove Proposition 4.15. We need to show that if ;Z 4 £(Z ) , then E 2  e  ) 2 ;Z 4 [] e 13*(Z  0 and  [eQ)}  =  for some (Z ; {e})-bundle 2  is such that ([j)  C.  First we consider the case of bundles of odd rank. Pick [] e ) such that ({j) 0 and the base of is of dimension n. (Since 2 B2k+1(Z4; Z the rank of the bundle is odd, A is a (Z )-bundle, so is automatically 2 ;Z 4 in £(Z ;Z 4 ).) Then ([]) e ffl(BSO2k+1 x BZ 2 ), where 4 is as defined in 4 Theorem 3.1. By the Künneth isomorphism K on the .A/-homology functor, PROOF.  ,g 3 [Q,f]®[M  Ko([])= i+j=Th  for some  {Q,f]  in 2 ,g in .A/(BZ [M ] ). 4 A/(BSO k +l) and some 3  We can argue on each term of the sum separately (for the sake of clarity, we omit the indices). We have o  K([Q, f]®[M,g])  x M, f x g]) by the definition of K  = =  where, by the definition of is a (Z ; Z2)-lme bundle, and 4  [y®1, the bundle  [®l  =  [1  )-bundle, while .A 2 is a 2 (Z Z ; ). 2 2k+1(Z4; Z ‘  Let lrl,1r2, and ‘Y2k+1 be as in the proof of Theorem 3.1. Then the line bundle ). is determined by the property that  S)  (7r2 o  (f  ) 4 x g))*(Ez  =  Q  x  ) 4 g*cEz  =  Qx  Specifically, A is the line bundle associated to the principal Z -bundle 2 56  6.2. Splitting g*(EZ ) . If we M —* M/Z , where we let M denote the free Z 2 4 manifold 4 denote the (Z ;Z 4 )-line bundle associated to M —* M/ 2 2 by A’, we then have A=QxA’.  = f*(2k+1) x M, then Let , = (r o (f x 7 is given by the pullback of the following diagram (note that the action on Q is trivial by definition):  =  f*() x M  QxM,  S(AzQxM,,  where ir is the principal 2 -Z bundle defined by the quotient M/Z 4 This gives *() = f*() x M. But then =  The base of ®A is  f*() x A’.  M.  (6.4)  , and so 2 Q x M/Z x M/Z ) 2  since the action on  =  =  ), 2 Q x (M/Z  (6.5)  Q is trivial. Therefore by (6.4), =  f*() x  i(A’).  (6.6)  57  6.2. Splitting Now note that z([])  [()} = [(®A)]. However, {f*() x (A’)] by (6.6)  [(7®)j  =  [(A’)] by the definition of the product  =  [f*(2k+1)1 e([A]) by Lemma 6.1, for some (Z ; {e})-line bundle A. Now take 2 conclusion of the theorem follows.  x A, and the  =  Now suppose that the rank of is even, and that e C(Z ). Then by 2 ;Z 4 Theorem 3.3, we can assign a class x in .Af*(B02k x BZ ) to The even-rank 4 argument is analogous to the odd-rank case: the only difference is that we let x play the role of ( and the reverse construction discussed in the proof of Theorem 3.3 is used in place of • .  [a]),  58  Bibliography [1] M.F. Atiyah, K-theory, Lecture notes by D. W. Anderson, W. A. Ben jamin, Inc., New York-Amsterdam, 1967.  [2] R.P. Beem, The action of free G-bordism on G-bordism, Duke Math. J. 42 (1975), 297—305. [3]  On the bordism of almost free Soc. 225 (1977), 83—105.  [4]  The structure of 7Z 4 bordism, Indiana Univ. Math. J. 27 (1978), no. 6, 1039—1047.  ,  k 2 Z  actions, Trans. Amer. Math.  ,  [5] R.P. Beem and R.J. Rowlett, The fixed point homomorphism for maps k, Indiana of period 2 Univ. Math. J. 30 (1981), no. 4, 489—500. [6] R.P. Beem and E.R. Wheeler, The image of unitary bordism in unori ented bordism the equivariant case, Proc. Amer. Math. Soc. 45 (1974), 445—449. -  [7] M. 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