EQUIVARIANT ORIENTATION THEORY S.R. COSTENOBLE, J.P. MAY, AND S. WANER Abstract.We give a long overdue theory of orientations of G-vector bundl* *es, topological G-bundles, and spherical G-fibrations, where G is a compact * *Lie group. The notion of equivariant orientability is clear and unambiguous,* * but it is surprisingly difficult to obtain a satisfactory notion of an equivari* *ant orien- tation such that every orientable G-vector bundle admits an orientation.* * Our focus here is on the geometric and homotopical aspects, rather than the * *coho- mological aspects, of orientation theory. Orientations are described in * *terms of functors defined on equivariant fundamental groupoids of base G-space* *s, and the essence of the theory is to construct an appropriate universal t* *ar- get category of G-vector bundles over orbit spaces G=H. The theory requi* *res new categorical concepts and constructions that should be of interest in* * other subjects where analogous structures arise. Contents Introduction 2 Part I. Fundamental groupoids and categories of bundles 4 1. The equivariant fundamental groupoid 4 2. Categories of G-vector bundles and orientability 6 3. The topologized fundamental groupoid 8 4. The topologized category of G-vector bundles over orbits 9 Part II. Categorical representation theory and orientations 11 5. Bundles of Groupoids 11 6. Skeletal, faithful, and discrete bundles of groupoids 13 7. Representations and orientations of bundles of groupoids 15 8. Saturated and supersaturated representations 18 9. Universal orientable representations 22 Part III. Examples of universal orientable representations 25 10. Cyclic groups of prime order 25 11. Orientations of V -dimensional G-bundles 29 12. Complex Bundles and Odd-Order Groups 32 13. Abelian compact Lie groups 36 14. The universal orientable representation S VD6(2) 40 ____________ Date: March 8, 2001. 1991 Mathematics Subject Classification. Primary 55P91; Secondary 18B40, 20L* *15, 55N25, 55N91, 55P20, 55R91, 57Q91, 57R91. The second author was partially supported by the NSF. 1 2 S.R. COSTENOBLE, J.P. MAY, AND S. WANER Part IV. Refinements and variants of the theory 41 15. Fibrations over B and fibrant representations 41 16. Functoriality of universal orientable representations 43 17. Orientations and change of groups 45 18. Variant kinds of orientations 48 19. Categories of virtual G-bundles 51 Part V. The classification of oriented G-bundles 54 20. Introduction: classifying G-spaces 54 21. The classification of G-bundles and spherical G-fibrations 55 22. The classification of oriented G-bundles 57 23. The classification of oriented spherical G-fibrations 59 24. Moore loops and the classification of representations 62 References 66 Introduction What is an orientation of a smooth G-manifold, where G is a compact Lie group? Accepting the obvious answer "an orientation of its tangent G-bundle", what is * *an orientation of a G-vector bundle? Surprisingly, there is no satisfactory answer* * in the literature except under rather restrictive hypotheses. One of us began work on * *this question in a 1986 preprint [31], and the three of us took up the problem soon * *after. This paper is a revision of an undistributed 1989 preprint, and in the meantime a number of papers have appeared that are explicitly or implicitly based on that preprint [1, 3, 4, 5, 6, 7, 25]. The answer to the question is necessarily comp* *licated, and our present categorical framework is a significant improvement on our origi* *nal one. We give the idea by reviewing one approach to classical orientation theory. Nonequivariantly, an elaborately pedantic way of defining an orientation of an n-plane bundle p: E -! B runs as follows. We consider the category V (n) with one object Rn and the two morphisms given by the two homotopy classes of linear isometries Rn -! Rn. We have the inclusion S of the discrete subcategory S V (n) with just the identity morphism. We may think of S :S V (n) -! V (n) as obtained from SO(n) -! O(n) by passing to components. We choose and fix an isomorphism from each n-dimensional vector space_V to Rn, thereby obtaining an equivalence of categories from the category V (n) of all n-dimensional vector spaces to V (* *n). Using the bundle covering homotopy property (CHP),_we see that p induces a functor p* from the fundamental groupoid B to V (n) that sends a point b to the fiber Vb over b. Using our fixed equivalence of categories, this gives a fu* *nctor p*: B -! V (n). The functor p* fixes a choice of orientation of each fiber and describes how the orientations of fibers change as one traverses paths in the b* *ase space. The bundle p is orientable if and only if there is a lift of this funct* *or to S V (n), by which we mean a functor F :B -! S V (n) together with a natural isomorphism OE: S O F -! p*. A choice of such a lift is an orientation of p. He* *re the functor F is obviously unique if it exists, but there are then two choices * *of OE if B is path connected. We shall mimic this procedure equivariantly. We define the equivariant funda- mental groupoid G X and the equivariant analogue VG (n) of V (n) in x1 and x2, EQUIVARIANT ORIENTATION THEORY 3 where we also define p*: G B -! VG (n) for a G-vector bundle p: E -! B and explain what it means for p to be orientable. There are two main variants of the relevant categories, which coincide when G is finite. In x3 and x4, we show how to topologize G X and VG (n) so that the more commonly used variants are the respective homotopy categories of the variants most appropriate to our theory. The definition of the equivariant analogue S VG (n) of S V (n) turns out to be quite subtle. The idea is to find a functor S :S VG (n) -! VG (n) such that p is orientable if and only if p* factors through S VG (n) and such that S VG (n)* * is the "smallest" category with this property. To carry out this idea, we need the categorical notion of a bundle of groupoids over the orbit category OG , or over any category B with similar structure. We call these objects groupoids over B for short. This notion is defined in x5; G X and VG (n) are examples of groupoi* *ds over OG . To construct S VG (n), we need the restricted types of groupoids over B that are described in x6. These arise as quotients of G B through which p* factors when p is orientable. We introduce and explain a kind of representation theory of bundles of groupoids that allows us to define orientations of orienta* *ble G-bundles in x7. The construction and characterization of the "universal orient* *able representation" S :S VG (n) -! VG (n) used in the definition is carried out in * *x8 and x9. We obtain the following theorem. Theorem 0.1. A G-vector bundle p: E -! B of dimension n is orientable if and only if p*: G B -! VG (n) can be lifted to a functor F :G B -! S VG (n) together with a natural isomorphism OE: S O F -! p*. A choice of such a lift (F; OE) is * *an orientation of p. This notion correctly encodes the intuitive idea that an orientation should be a consistent set of orientations of the restricted bundles over orbits of B. H* *ere consistency entails consistency with all paths in all fixed point spaces in B. * *Since S VG (n) must allow for all possibilities, its construction is intrinsically co* *mplicated. The categorical representation theory that is involved may well have applicatio* *ns in other fields where formally analogous structures arise. We describe the universal orientable representation explicitly for cyclic gro* *ups of prime order in x10. We discuss G-bundles "of dimension V " for a representat* *ion V of G and illustrate the need for our theory with a simple example in x11. For G-bundles over a general compact Lie group, or even over a general finite group* * G, there seems to be no precursor to our theory in the literature. However, there * *is an evident naive notion of an orientation of a V -dimensional G-bundle. We show that this naive notion is insufficient to give a satisfactory theory. The noti* *on of orientability is straightforward and unambiguous. An obvious desideratum of a satisfactory theory is that every orientable G-bundle must admit an orientation, but this fails with the naive notion. In fact, the 2-sphere S2 with the circle * *group S1 or any of its cyclic subgroups acting by rotation gives an elementary example of an orientable G-manifold that admits no naive orientation. For cyclic groups* * of prime order, we display the orientations of S2 explicitly. We urge the reader w* *ho has not thought about equivariant orientation theory to consider that example f* *irst, since it well illustrates both the problem and our solution of it. We describe the universal orientable representation explicitly for any odd or* *der finite group G in x12. Here it turns out that a G-vector bundle is orientable i* *n the equivariant sense if and only if it is orientable in the nonequivariant sense, * *and then 4 S.R. COSTENOBLE, J.P. MAY, AND S. WANER equivariant orientations are uniquely determined by their underlying nonequivar* *i- ant orientations. The equivariant orientation describes additional fixed point * *space information that is implicit in the nonequivariant orientation. The nature of t* *his information is not obvious. In fact, there is a naive notion of an equivariant * *orien- tation of any G-vector bundle for a group of odd order. For V -dimensional bund* *les it coincides with the naive notion of x11, so the example there shows that not * *every oriented G-vector bundle can be naively oriented. We also describe the essentia* *lly trivial complex analogue of our theory in x12. We give a conjectural description of the universal orientable representation * *for an elementary Abelian 2-group in x13. We doubt that the conjecture is right, but with more work the ideas presented should lead to a correct description of the universal orientable representation for any Abelian compact Lie group G. As a first non-Abelian example, we display the universal orientable representation f* *or 2-dimensional G-vector bundles for the dihedral group G = D6 in x14. We return to the general categorical theory in x15 and x16, first showing how the theory of categorical fibrations gives an alternative way of thinking about* * ori- entations in x15 and then discussing the functoriality with respect to changes * *of the reference groupoid over B into which representations map in x16. In x17, we use this discussion to show that an orientation of a G-bundle p: E -! B induces orientations of the H-fixed point bundle over BH and of its complementary bundle over BH for all subgroups H of G. For an oriented smooth G-manifold M, this means that the fixed point manifolds MH and the normal bundles of the inclusions MH M inherit appropriate orientations. While our main focus is on G-vector bundles, the theory also applies to topol* *og- ical and PL G-bundles, to spherical G-fibrations, and to stable and virtual var* *iants of each of these. We explain this in x18 and x19. The discussion of functoriali* *ty in x16 allows comparisons among these versions of orientation theory. In x20 - x23, we describe classifying G-spaces and prove classification theor* *ems for oriented G-bundles and oriented spherical G-fibrations. We prove a related classification theorem for representations of fundamental groupoids in x24. Despite the length of this paper, we have by no means obtained a complete theory. Nonequivariantly, there are geometric and cohomological notions of orie* *n- tation, and the geometric theory coincides with the cohomological theory when we take ordinary cohomology with integral coefficients. That is a calculational f* *act that does not carry over to the equivariant context. While ideas here have been used in work towards the cohomological theory in [1, 3, 4, 5, 6, 7, 25], a more* * sys- tematic and unified theory remains to be developed. We intend to return to this matter elsewhere. Part I.Fundamental groupoids and categories of bundles 1.The equivariant fundamental groupoid We recall the definition and properties of the fundamental groupoid of a G-sp* *ace X. We understand spaces to be compactly generated (= weak Hausdorff k-spaces), and we let U denote the category of spaces. A topological category is a category enriched over U , so that its hom sets are spaces and composition is continuous* *. A functor between topological categories is continuous if it is continuous on hom* * sets. Recall that a category is a groupoid if all of its morphisms are isomorphisms. * *We EQUIVARIANT ORIENTATION THEORY 5 shall later be interested in topological groupoids, but we focus on the underly* *ing untopologized categories in this section and the next. Our ambient group G is a compact Lie group, and subgroups are understood to be closed. The orbit category OG is the topological category whose objects are * *the orbit G-spaces G=H and whose morphisms are the G-maps between orbits. The morphism set OG (G=H; G=K) is topologized as the subspace (G=K)H of G=K. The following definition is given by tom Dieck [10, 10.7]. We regard an eleme* *nt x 2 XH as the G-map G=H -! X that sends eH to x, going back and forth at will between the two interpretations, and similarly for paths, etc, in XH . Definition 1.1. Let X be a G-space. The (equivariant) fundamental groupoid G X of X is the category whose objects are the G-maps x: G=H - ! X and whose morphisms x -! y, y : G=K -! X, are the pairs (!; ff), where ff: G=H -! G=K is a G-map and ! is an equivalence class of paths x -! y O ff in XH . As usual, two paths are equivalent if they are homotopic rel endpoints. Composition is induced by composition of maps of orbits and the usual product on path class* *es. Let ss :G X - ! OG be the functor given by ss(x: G=H - ! X) = G=H and ss(!; ff) = ff. Lemma 1.2. A G-map f :X -! Y induces a functor f*: G X -! G Y . A G-homotopy h: f ' f0 induces a natural isomorphism h*: f* -! f0*. We write X for the nonequivariant fundamental groupoid of a space X. Remark 1.3. For a category B and an object b, we have the category B=b of objec* *ts a -! b over b. Taking X = G=H, the functor ss :G (G=H) -! OG factors through a functor G (G=H) -! OG =(G=H) that is surjective on objects and morphisms and is an isomorphism if G is finite. We record some properties of the fundamental groupoid that will later be ab- stracted to give the notion of a bundle of groupoids. For a functor ss :E - ! B, the fiber Eb over an object b 2 B is the subcategory of objects and morphisms of E that map to b and its identity morphism. Remarks 1.4. Let X be a G-space. (i)The fiber (G X)G=H is the nonequivariant fundamental groupoid XH . (ii)For an object y :G=K -! X in G X and a map ff: G=H -! G=K in OG , there is an object x: G=H -! X and a morphism (!; ff): x -! y. In fact, x = y O ff and the constant path ! give canonical choices for x and (!; * *ff). (iii)Let x: G=H - ! X, y :G=J - ! X, and z :G=K -! X be objects in G X. Suppose that we have maps (; fl): x -! z and (; fi): y -! z in G X and a map ff: G=H -! G=J such that fiff = fl: (!;ff) ss ff x6_6_ _ _//y _____// G=H>__________//G=J 66 >>> (;fl)6oeoe66(;fi) fl>>> fi z GOEOE=K: There is a unique map (!; ff): x -! y in G X such that (; fi)(!; ff) = (; fl), namely the one given by ! = (ff)-1. The existence and uniqueness of (!; ff) are encoded in the statement that the following diagram is a 6 S.R. COSTENOBLE, J.P. MAY, AND S. WANER pullback: G X(y; z) x G X(x; y)___O____//_G X(x; z) idxss|| |ss| fflffl| |fflffl G X(y; z) x OG(G=H; G=J)O(ssxid)//_OG(G=H; G=K): 2. Categories of G-vector bundles and orientability We need reference categories of G-vector bundles over orbits. By a G-bundle, we will understand a real G-vector bundle with orthogonal structure group. __ Definition 2.1. Let V G be the category whose objects are the G-bundles over orbits of G and whose morphisms are the equivalence classes of G-bundle maps between them. Here two maps are equivalent if they are G-bundle homotopic,_with the homotopy inducing the constant homotopy on base spaces. Let ss :VG -! OG be the functor that sends a G-bundle to its base space_and sends an equivalence class_of bundle maps to its map of base spaces. Let V G(n)be the full subcatego* *ry of V G consisting of the n-dimensional bundles. These categories are not small, but they have small skeleta. __ Definition 2.2. Let VG (n) be the full subcategory of V Gwhose objects are the * *n- plane G-bundles of the form GxH Rn -! G=H, where H acts on Rn through some representation : H -! O(n) and we choose one_such in each O(n)-conjugacy class. We obtain a retraction equivalence_V G(n) -! VG (n)by choosing a fixed isomorphism from each object in VG (n)to an object of VG (n), choosing the iden* *tity map if the object is in VG (n). Note that we still have functors ss :VG (n)-!_O* *G . Let VG be the disjoint union of the categories VG (n); it is equivalent to V G. We continue to write V for representations, even when we are thinking in terms of objects of VG . The following observations give a description of this catego* *ry. Lemma 2.3. Up to equivalence, a G-bundle over the orbit G-space G=H has the form G xH V -! G=H for some real representation V of H. A map "ff:G xH V -! G xK W of G-bundles over a map ff: G=H -! G=K has the form "ff(g; v) = (gg0; o(v)), where ff(eH) = g0K (hence g-10Hg0 K) and o :V - ! W is a linear isometry which is H-linear in the sense that o(hv) = (g-10hg0)o(v). Two maps "ff0and "ff1 over ff so determined by o0 and o1 are G-bundle homotopic over ff if and only if there is a path ot connecting o0 to o1 in the space of H-linear isometries V -!* * W . The skeletal nature of VG (n) implies the following useful observation. Lemma 2.4. If there is a map G xH V -! G xH W in VG (n), then V = W . Remark 2.5. The fiber VG (n)G=H is a groupoid that has one object V in each isomorphism class of representations of H in O(n) and has morphisms V -! V the homotopy classes of H-linear isometries; it has no morphisms V -! V 0if V 6= V * *0. By inspection of pullbacks, the evident analogues of Remarks 1.4(ii) and (iii) * *hold for the functor ss :VG (n)-! OG . EQUIVARIANT ORIENTATION THEORY 7 The following well known fact clarifies the structure of VG (n). Let OG (V )* * be the group of G-linear isometries of a representation V of G and let ss0(OG (V )* *) be its group of components. Lemma 2.6. The group ss0(OG (V )) is an elementary Abelian 2-group. Proof.Write V = Vi, where the Vi are the isotypical components of VQ, so that Vi~= Wi Rqifor some irreducible representation Wi. Then OG (V ) ~= iOG (Vi): Let Ki = Hom G (Wi; Wi). Each Ki is one of R, C, or H and Hom G(Vi; Vi) ~= Mqi(Ki). The corresponding subgroup of underlying real linear isometries is con- nected when Ki= C or H and has two components when Ki= R. The following basic construction is central to our work. Recall Lemma 1.2. Proposition 2.7. A G-bundle p: E -! B determines a functor p*: G B -! VG over OG . A G-bundle map (f"; f): p -! q, with f":E -! E0 the map of total spaces and f :B -! B0 the map of base spaces, determines a natural isomorphism "f*:p* -! q* O f* over the identity functor of OG . If ("h; h): (f"; f) ' (f"0;* * f0) is a G-bundle homotopy, then the following diagram commutes: p* F "f*yyyy FFFf"0*F yy FFF __yyy F"" q* O f*___q*Oh*___//_q* O f0*: In the last statement and later, we compose a functor (in this case q*) with a natural transformation (in this case h*) by applying the functor to the maps th* *at define the natural transformation; we often omit O in writing such composites. __ Proof.It suffices to work in V G, since we can then transfer information to the equivalent category VG . Pulling p back along G-maps x: G=H -! B, we obtain a system of G-bundles p*(x) -! G=H and bundle maps "x:p*(x) -! E. For a G-map ff: G=H -! G=K and a path w :x -! y O ff, the G-bundle covering homotopy property (G-bundle CHP) gives a homotopy "w:p*(x) x I -! E of "x that covers w. The map w"1covers y O ff and factors through a G-bundle map p*(w; ff): p*(x) -! p*(y) whose equivalence class depends only on the equivalen* *ce class ! of w. This constructs p*, and the remaining verifications are similar. With these definitions in place, we can define orientability precisely. Definition 2.8. The G-bundle p: E -! B is orientable if the functor p*: G B -! VG satisfies p*(!; ff) = p*(!0; ff) for every pair of morphisms (!; ff) and (!0* *; ff) with the same source and target and the same image in OG . That is, p*(!; ff) is independent of the choice of the path class !. For example, for a representatio* *n V of G, the projection B x V -! B is orientable. Remark 2.9. A G-vector bundle p is orientable if the defining condition holds w* *hen x = y, ff = id, and ! = id. Indeed, if (!; ff) and (!0; ff) are maps x -! y, th* *en, by Remark 1.4(iii), there is a map (; id) : x -! x such that (!0; ff) = (!; ff)(; * *id). If p*(; id) = id, then p*(!; ff) = p*(!0; ff). This gives the claimed implication.* * Thus orientability is a property of the restrictions of p over fixed point spaces BH* * . The following observation is immediate from Remark 1.3. 8 S.R. COSTENOBLE, J.P. MAY, AND S. WANER Proposition 2.10. If G is finite, any G-bundle over an orbit G=H is orientable. Example 2.11. This result fails for general compact Lie groups. For example, let L be the sign representation of the cyclic group H of order 2. Regarding H as a subgroup of S1, we can identify the open M"obius strip and its retraction to the circle as the S1-bundle S1 xH L -! S1=H. Clearly this is non-orientable. 3.The topologized fundamental groupoid When G is a general compact Lie group, we shall need topologies on the cate- gories G X and VG in order to define orientations of vector bundles. This secti* *on and the next deal with this issue and may be skipped by the reader who wishes to focus on finite groups. However, this material illuminates the structure of* * all of the categories that we have defined and should be of independent interest. T* *he following easy, but basic, observation appears to be new. Here and later, we use the term "bundle with discrete fibers" instead of "covering space" to emphasize that we are not assuming that the base spaces or total spaces are connected. In particular, we allow some fibers to be empty. Proposition 3.1. The category G X is a topological category such that, for obje* *cts x: G=H -! X and y :G=K -! X, ss :G X(x; y) -! OG (G=H; G=K) is a bundle with discrete fibers. Proof.For a simply connected open neighborhood U of a point ff 2 G=KH and a point fi 2 U, there is a unique path class ff;ficonnecting ff to fi in U. Compo* *sing with y gives a path class "ff;ficonnecting y O ff to y O fi. For ! :x -! y O ff* *, let U(!; ff) = {("ff;fi!; fi)|fi 2 U} G X(x; y): The U(!; ff) are the open sets of a basis for a topology on G X(x; y) such that ss :G X(x; y) -! ss(G X(x; y)) is a bundle with discrete fiber XH (x; y O ff) o* *ver ff. Indeed, if ff is in the image of ss, then ss-1(U) is the disjoint union of * *the U(!; ff) as ! ranges over the inequivalent classes of paths x -! y O ff. Corollary 3.2. Maps (!; ff); (; fi): x -! y, x: G=H -! X and y :G=K -! X, are homotopic if and only if there is a homotopy j :G=H x I -! G=K between ff and fi and a homotopy k :G=H x I x I -! X between paths w :G=H x I -! X and z :G=H x I -! X in the path classes ! and such that k(a; 0; t) = x(a) and k(a; 1; t) = yj(a; t) for a 2 G=H and t 2 I. Remark 3.3. Identifying homotopic maps, we obtain the homotopy category hG X and a functor ss :hG X -! hOG . By the corollary, hG X is tom Dieck's "discrete fundamental groupoid" [10, 10.9]. When G is finite, there is no distinction. Mu* *ch of our theory can be carried out in terms of homotopy categories, that being the approach taken in the original version (circa 1989) of this work. However, use * *of G X turns out to be preferable since it gives a closer relationship between G X and the XH and allows a more natural variant of representation theory. So far, G could have been any (locally simply connected) topological group. However, since G is a compact Lie group, we can give an explicit description of* * the quotient functor G X -! hG X. This depends on the following description of homotopies in OG [25, 1.1]. EQUIVARIANT ORIENTATION THEORY 9 Lemma 3.4. Let j :ff -! fi be a G-homotopy between G-maps G=H -! G=K. Then j is the composite of ff with a homotopy c: G=H x I -! G=H such that c(eH; t) = c(t)H, where c(0) = e and the c(t) specify a path in the identity co* *mpo- nent of the centralizer CG H of H in G. In particular fi = ff O c(1): G=H -! G=* *K. For a path c in CG H and a path w in XH , we obtain a path c . w in XH by setting (c . w)(s) = c(s)w(s). If ! is the path class of w, we write c . ! for * *the path class of c . w. Combining the notations and hypotheses of Corollary 3.2 and Lem* *ma 3.4, we obtain the following description of homotopies in G X. Proposition 3.5. Consider objects x: G=H -! X and y :G=K -! X of G X. Let (k; j) give a homotopy between maps (!; ff); (; fi): x -! y in G X, where j* * = ffOc and thus fi = ffOc(1). Then (!; ff) is homotopic to (c.!; ffOc(1)) (indepe* *ndent of ), and (c . !; fi) is equal to (; fi) in G X (independent of the homotopy j). Therefore hG X(x; y) is the quotient of G X(x; y) obtained by identifying (!; f* *f) with (c . !; ff O c(1)) for all paths c in CG H such that c(0) = e. Proof.Define h = h(w; c): I xI -! XH by h(s; t) = c(st)w(s), where w represents !. Then h: w ' c . w, h(0; t) = w(0), and h(1; t) = c(t)w(1). Interpreting in t* *erms of equivariant maps on orbits, this means that (h; j) gives a homotopy (!; ff) ' (c . !; fi). Regarding k as a map I x I -! XH , define a new map `: I x I -! XH by `(s; t) = c(s)c(st)-1k(s; t). Then ` is a homotopy rel endpoints between c .* * w and a representative z of . Remark 3.6. The functor ss :hG X -! hOG has properties similar to but less convenient than those of Remarks 1.4. The fiber (hG X)G=H is a quotient of XH , there is a noncanonical solution x and (!; ff) to the "source lifting" que* *stion in Remarks 1.4(ii), and there is a non-unique solution (!; ff) to the "divisibi* *lity" question in Remarks 1.4(iii). 4. The topologized category of G-vector bundles over orbits We have a topologization of the category VG that is precisely analogous to the topologization of G X in Proposition 3.1. We work with VG for simplicity of notation, but it will be the topological disjoint union of the VG (n). To empha* *size the analogy with G X, we write ! or (!; ff) for a morphism p -! q over ff, so t* *hat ! is an equivalence class of G-bundle maps "ffover ff. Proposition 4.1. The category VG is a topological category such that, for n-pla* *ne G-bundles p: D -! G=H and q :E -! G=K, ss :VG (p; q) -! OG (G=H; G=K) is a bundle with discrete fibers. Proof.As in the proof of Proposition 3.1, consider a simply connected open neig* *h- borhood U of a point ff 2 G=KH and the path classes ff;fiof paths vff;ficonnect* *ing ff to points fi in U. Let "ffbe a G-bundle map over ff. Applying the G-bundle C* *HP, we obtain a G-bundle homotopy "vff;fi("ff): D x I -! E of "ffthat covers vff;fi* *. Write "fiff;fi("ff): p -! q for the G-bundle map over fi obtained at the end of the h* *omo- topy. Further application of the G-bundle CHP shows that the equivalence class * *of "fiff;fi("ff) depends only on the equivalence class ! of "ffand the path class * *ff;fi. We write iff;fi(!) for the equivalence class of "fiff;fi("ff), and we define U(!; ff) = {iff;fi(!)|fi 2 U} VG (p; q): 10 S.R. COSTENOBLE, J.P. MAY, AND S. WANER The U(!; ff) are the open sets of a basis for a topology on VG (p; q), such that ss :VG (p; q) -! ss(VG (p; q)) is a bundle with discrete fibers. Indeed, if ff * *is in the image of ss, then ss-1(U) is the disjoint union of the sets U(!; ff) as ! range* *s over the equivalence classes of bundle maps p -! q over ff. Remark 4.2. We have a quotient category hVG obtained by identifying bundle maps p -! q over ff and fi if they are bundle homotopic over a homotopy ff ' fi. Again, if G is finite, there is no distinction. Passage to base spaces gives a * *functor ss :hVG -! hOG . The precise relationship between VG and hVG is analogous to that between G X and hG X described in Proposition 3.5. We see this by extending the last sentence of Lemma 2.3 to allow for homotopies on the base space level. Consider a path c in CG H and a bundle map "ff:G xH V -! G xK W over ff specified by "ff(g; v) = (gg0; o0(v)), where ff(eH) = g0K and o0: V -! W is an H-linear isometry. We obtain a bundle homotopy (4.3) c . "ff:(G xH V ) x I -! G xK W by setting (c . "ff)(g; v; t) = (gc(t)g0; o0(v)). If ! denotes the equivalence * *class of "ff, we let c(1) . ! denote the equivalence class of the map over fi given by settin* *g t = 1. Proposition 4.4. Let p: G xH V -! G=H and q :G xK W - ! G=K be G- bundles. Let ("j; j) give a homotopy between maps (!; ff), (; fi): p -! q in V* *G , where j = ff O c. Then ! is homotopic to c(1) . ! (independent of ) and c(1) . ! is equal to in VG (p; q) (independent of the homotopy j). Therefore hVG (p; q)* * is the quotient of VG (p; q) obtained by identifying (!; ff) with (c(1) . !; ff O * *c(1)) for all paths c in CG H such that c(0) = e. Proof.The bundle homotopy (4.3) gives ! ' c(1).!. We must show that c(1).! = . We are given a bundle homotopy "j:(G xH V ) x I -! G xK W over j from "ffto "fi, where "ffand "fiare in the homotopy classes ! and . We must define a bundle homotopy k :(G xH V ) x I -! G xK W over the constant homotopy at fi from c(1) . "ffto "fi. Observe that the H-action on W defined by hw = g-10c(t)-1hc(t* *)g0 is independent of t. We may write "j(g; v; t) = (gc(t)g0; ot(v)); where the ot * *specify a path in the space of H-linear isometries V -! W . We define k(g; v; t) = (gc(1)g0; ot(v)): We have a perhaps more natural alternative definition of orientability. We de* *fine it, but we then show that it agrees with the definition already given. Remark 4.5. Let p: E -! B be a G-bundle. Since the full strength of the G-bundle CHP allows us to vary maps on base spaces by homotopies, Proposition 2.7 remains valid if we replace the categories G B and VG over OG with the categories hG B and hVG over hOG . Thus we have a functor p*: hG B -! hVG over hOG . Definition 4.6. Thinking in terms of p*: hG B -! hVG , we say that a G-bundle p is h-orientable if p*(!; ff) = p*(i; fi) for every pair of morphisms (!; ff) * *and (; fi) of hG B with the same source and target and the same image in hOG . Proposition 4.7. A G-bundle p is orientable if and only if it is h-orientable. EQUIVARIANT ORIENTATION THEORY 11 Proof.The construction of both functors p* is by use of the G-bundle CHP, start* *ing from homotopies on the base space level. We see that p is orientable if it is * *h- orientable by starting with constant homotopies. Conversely, suppose that p is orientable. Then Proposition 3.5 implies that p is h-orientable. Indeed, with t* *he notation there, the fact_that p* in Remark 4.5 is well-defined implies that p*(* *!; ff) = p*(ffOc(1); c.!) in hV G. This can also be seen by direct verification using an* * evident cover of the homotopy h used in the proof of Proposition 3.5. Part II.Categorical representation theory and orientations 5. Bundles of Groupoids We abstract the properties of the equivariant fundamental groupoid to obtain * *the notion of a bundle of groupoids. We fix a topological category B, which the rea* *der should think of as OG . We assume that B is small, its morphism spaces are loca* *lly path connected, and every endomorphism of an object of B is an isomorphism. We call such a category a base category. If G is finite, we give all categories in* * sight the discrete topology. It is helpful to think in terms of 2-categories [19, XI* *Ix2], which have objects (the "0-cells"), morphisms between objects (the "1-cells"), * *and morphisms between morphisms or "homotopies" (the "2-cells"). Definition 5.1. A bundle of groupoids over B, or a groupoid over B for short, i* *s a small topological category E together with a continuous functor ss :E -! B that satisfy the following properties. (i)Each map ss :E (x; y) -! B(ss(x); ss(y)) is a fiber bundle with discrete* * fibers (possibly empty and varying over different components of the target). (ii)For each object b of B, the fiber Eb is a groupoid (possibly empty). (iii)(Source Lifting) For each object y of E and morphism ff: a -! ss(y) of B, there is a morphism ! :x -! y of E such that ss(x) = a and ss(!) = ff. (iv)(Divisibility) For objects x, y, z and morphisms :x -! z, : y -! z of E and a morphism ff: ss(x) -! ss(y) of B such that ss()ff = ss(), there is a unique morphism ! :x -! y of E such that ss(!) = ff and O ! = : x6_ _!_ _//y __ss_// ss(x)___ff___//_ss(y) 66 ::: 6666 ::: oeoe ss() :AEAE ss() z ss(z) Moreover, ! varies continuously with the data. The existence, uniqueness, and continuity are encoded by requiring the following diagram to be a pullback: E (y; z) x E (x;_y)O_____//E (x; z) idxss|| |ss| fflffl| fflffl| E (y; z) x B(ss(x); ss(y))O(ssxid)//_B(ss(x); ss(z)): We write ss generically for the projections of groupoids over B, and we often w* *rite E for ss :E - ! B. The groupoids over B are the 0-cells of a 2-category. The 12 S.R. COSTENOBLE, J.P. MAY, AND S. WANER 1-cells are the continuous functors F :E - ! F such that the following diagram commutes: E ____F____//99F 99 ss9oo99 ss B: We refer to these as functors over B. The 2-cells j :F - ! F 0are the natural transformations j :F -! F 0such that ss(j(x)) = idss(x)for all objects x of E .* * Since j(x) is then a morphism of the groupoid Fss(x), j must be a natural isomorphism. We refer to these as isomorphisms over B. Let B(E ; F ) denote the groupoid whose objects are the functors E - ! F over B and whose morphisms are the isomorphisms over B. Two groupoids E and F over B are equivalent if there are functors F :E -! F and F -1:F -! E over B whose composites are isomorphic over B to the respective identity functors; E and F are isomorphic if there are functors F and F -1over B whose composites are equal to the respective identity functors. Condition (ii) of the definition is redundant, being implied by unique divisi* *bility (see Remark 6.3 below). It is stated for emphasis. If we ignore the topology, t* *hen a groupoid over B is exactly a "categorie fibree en groupoides" over B, as defi* *ned by Grothendieck [15, pp. 165-166]; see also [8, p. 96]. It must be kept in mind* * that the convenient abbreviation "groupoid E over B" is an abuse of language, since E is not a groupoid. Remarks 1.4 and 2.5, Lemma 1.2, and Propositions 2.7, 3.1, a* *nd 4.1 are summarized in the following motivating examples. Proposition 5.2. For a G-space X, ss :G X -! OG is a groupoid over OG . For a G-map f :X -! Y , f*: G X -! G Y is a functor over OG . A G-homotopy h: f ' f0 induces an isomorphism h*: f* -! f0*over OG . Proposition 5.3. The functor ss :VG (n)-! OG is a groupoid over OG . For a G-bundle p: E -! B, p*: G B -! VG is a functor over OG . For a G-bundle map (f"; f): p -! q, "f*:p* -! q* O f* is an isomorphism over OG . The following diagram commutes for a G-bundle homotopy ("h; h): (f"; f) ' (f"0; f0): p* F "f*yyyy FFFf"0*F yy FFF __yyy F"" q* O f*___q*Oh*___//_q* O f0*: Divisibility implies the following result about the bundles of morphisms of s* *s. For an object x of E , let Aut(x) denote the (discrete) group of self-maps of x* * in the fiber Ess(x). Proposition 5.4. The map E (x; y) -! B(ss(x); ss(y)) is a principal Aut(x)-bund* *le onto its image. A map ! : x -! y in E determines a restriction homomorphism r : Aut(y) -! Aut(x) characterized by O ! = ! O r() for 2 Aut(y). EQUIVARIANT ORIENTATION THEORY 13 Proof.The pullback diagram given in (iv) restricts to the pullback diagram E (x; y) x Aut(x)__O___//E (x; y) p || |ss| fflffl| |fflffl E (x; y)__ss___//B(ss(x); ss(y)); where p is projection. This implies that Aut(x) acts freely and transitively on* * each nonempty fiber of the bundle ss :E (x; y) -! B(ss(x); ss(y)). The second statem* *ent is immediate from divisibility. 6. Skeletal, faithful, and discrete bundles of groupoids The following restricted kinds of bundles of groupoids play a major role in t* *he theory. Recall that a category is skeletal if each of its isomorphism classes o* *f objects consists of a single object and is discrete if all of its maps are identity map* *s. Recall that a functor is faithful if it maps morphism sets injectively. Definition 6.1. A groupoid ss :E -! B over B is skeletal or discrete if each fiber Eb is skeletal or discrete; it is faithful if the functor ss is faithful,* * in which case ss :E (x; y) -! B(ss(x); ss(y)) is an inclusion of a union of path components f* *or each pair of objects x and y of E . Warning 6.2. Observe that a discrete category can admit only the discrete topol* *ogy on its morphism sets and that the category E of a discrete groupoid over B need not be discrete in either the categorical or the topological sense. Henceforwa* *rd, the word "discrete" will be used only in the sense of categories or of bundles * *of groupoids; the context will make clear which is intended. Observe that the morphism space E (x; y) must be topologized as a subspace of B(ss(x); ss(y)) when E is faithful. For this reason, we need not pay much atten* *tion to the topology when studying faithful groupoids over B. The following basic observations are easily verified; they will be used heavily. Remarks 6.3. Let ss :E -! B be a groupoid over B. (i)If E is skeletal, then divisibility implies that the object x asserted t* *o exist in the source lifting property is unique. If E is both skeletal and fait* *hful, then the morphism asserted to exist in the source lifting property is al* *so unique. (ii)The fact that E (x; y) -! B(ss(x); ss(y)) is a principal Aut(x)-bundle i* *m- plies that ss is faithful if and only if every automorphism of every obj* *ect in every fiber Eb is an identity map. (iii)If ! :x -! y is a morphism of E such that ss(!) is an isomorphism, then ! is an isomorphism, as we see by an application of divisibility to the equality ss(!)ss(!)-1 = id. Since every endomorphism of any object of B is an isomorphism, every endomorphism of any object of E is also an isomorphism. (iv)We can construct a faithful groupoid E =ss over B with the same objects as E , but with (E =ss)(x; y) = Im(E (x; y) -! B(ss(x); ss(y))). The quo* *tient functor E -! E =ss over B is the universal functor from E into a faithful groupoid over B. 14 S.R. COSTENOBLE, J.P. MAY, AND S. WANER (v) We can construct a skeletal subgroupoid E 0 E over B by choosing a skeleton of each fiber Eb and taking the full subcategory of E whose obj* *ects are in the chosen skeleta of fibers. The inclusion E 0-! E is an equival* *ence of groupoids over B whose left inverse is a retraction over B. We call E* * 0a skeleton of E . (vi)By (iii), the passage from E to E =ss creates no new isomorphisms, so th* *at we can make the same choices of objects for E and for E =ss when forming skeleta. Then E 0=ss = (E =ss)0. This gives a canonical way of passing f* *rom any groupoid over B to an associated discrete groupoid over B. Discrete groupoids over B are central to our work. It is clear from the defin* *itions that if ss :E -! B is skeletal and faithful, then it is discrete. Remark 6.3(ii* *) implies the converse. Lemma 6.4. ss :E -! B is discrete if and only if it is skeletal and faithful. An obvious but useful observation is that the 2-category structure trivialize* *s for maps into discrete groupoids over B. Lemma 6.5. Let F be discrete. If F , F 0:E - ! F are functors over B and j :F -! F 0is an isomorphism over B, then F = F 0and j is the identity. In fact, discrete groupoids over B are actually quite simple and familiar obj* *ects. Lemma 6.6. The category of discrete groupoids over B and functors over B is equivalent to the category of continuous (= locally constant) set-valued contra* *variant functors on B and their natural isomorphisms. Proof.Given ss :E -! B, define a functor : B -! Setsby letting (b) be the set of objects of the fiber Eb. For a morphism ff: a ! b of B and an object y of Eb, let (ff)(y) be the unique object x of Ea that is the source of a map x -! y covering ff. Remark 6.3(i) implies that the inverse image in B(a; b) of a funct* *ion f = (ff) is the union of components ss(E (f(y); y)) and is thus open and closed. Conversely, given , define ss :E -! B as follows. The objects of E are the pairs (y; b) where b is an object of B and y 2 (b). A morphism (x; a) -! (y; b) is a morphism ff: a -! b of B such that (ff)(y) = x. The functor ss projects onto the second coordinate and restricts to an injection of E ((x; a); (y; b)) onto * *an open and closed subset of B(a; b); we give E ((x; a); (y; b)) the subspace topology.* * These constructions specify functors that give the claimed equivalence of categories. Although reassuring, this result is not useful to us because our theory focus* *es on a comparison between general groupoids over B and discrete ones. The germ of the comparison is the fact that the categories B=b of objects over b give di* *screte groupoids over B whose represented functors in the 2-category of groupoids over B detect the fiber groupoids of arbitrary groupoids over B. We use the following result to show this. Lemma 6.7. Let ss :E -! B be a groupoid over B, let b be an object of B, and let y be an object of E such that ss(y) = b. Consider the commutative diagram y E =y_____//_E ssy|| |ss| fflffl| fflffl| B=b _b___//B; EQUIVARIANT ORIENTATION THEORY 15 where b and y are the canonical functors and ssy is induced by ss. Then B=b is a discrete groupoid over B, E =y is a groupoid over B, and y and ssy are maps over B. The functor ssy has a section oe. If E is discrete, then oe is unique and ss* *y is an isomorphism of categories with inverse oe. Proof.The fiber (B=b)a is the discrete category whose objects are the maps a -!* * b in B, so that b is discrete. The rest of the first statement is straightforward* *. By the source lifting property, for each map ff: a -! b of B, there is an object x* *ffof E and a map oe(ff): xff-! y such that ss(oe(ff)) = ff. We may choose oe(idb) = * *idy. By the divisibility property, for maps ff0:a0- ! b and : a -! a0in B such that ff0O = ff, there is a unique map oe(): xff-! xff0in E such that ss(oe()) = and oe(ff0) O oe() = oe(ff). The pullback diagram in the divisibility property spec* *ializes to show that the resulting function oe :B=b(ff; ff0) -! E (oe(ff); oe(ff0)) is * *continuous, and the uniqueness of divisibility implies that oe is a functor. This gives the* * section oe, and it is the inverse isomorphism to ssy if ss is discrete by Remark 6.3(i). Proposition 6.8. Let E be a groupoid over B and let ": B(B=b; E ) -! Eb be the functor that sends functors F and isomorphisms j :F -! F 0over B to their evaluations on the object idbof B=b. Then " is an equivalence of groupoids. If E is discrete, then, for an object y of Eb, there is a unique functor "y:B=b -! E* * over B such that "y(idb) = y, and therefore " is an isomorphism of groupoids. Proof.Observe that a map ff: a -! b of B gives both an object ff of B=b and a morphism __ff:ff -! idbof B=b. The functor " is full and faithful by the divisi* *bility property of ss :E -! B. Indeed, for a morphism ! :F (idb) -! F 0(idb) of Eb, let j(ff): F (ff) -! F 0(ff) be the unique morphism of Ea such that F 0(__ff) O j(f* *f) = ! O F (__ff). The j(ff) give the unique morphism j :F -! F 0of B(B=b; E ) such * *that "(j) = !. By [19, p. 93], to prove that " is an equivalence of categories, it s* *uffices to show that for each object y of Eb, there is an object "y:B=b -! E of B(B=b; * *E ) such that "y(idb) = y, and we can take "y= y O oe for a section oe of ssy. If * *E is discrete, then "y= y O ss-1yis unique. 7. Representations and orientations of bundles of groupoids We think of the ss :VG (n) -! OG as target groupoids over OG for a kind of representation theory, and we note that we have chosen these groupoids over OG to be skeletal. It is convenient to change our point of view on bundles of grou* *poids over B by focusing attention on a fixed target R for maps of groupoids over B. We adopt the following language. Remember that we write ss generically for the projections of groupoids over B. Definition 7.1. Fix a skeletal groupoid R over B and consider groupoids E and F over B. We define "representations", "maps", and "homotopies" that give the 2-category of representations in R. (i)A representation R of E in R is a functor R: E -! R over B. We denote a representation as a pair (E ; R) when R is understood. (ii)A map from a representation (E ; R) to a representation (F ; S) is a pair (F; OE), where F :E -! F is a functor over B and OE: S O F -! R is an 16 S.R. COSTENOBLE, J.P. MAY, AND S. WANER isomorphism over B: E/_____F______//_?F //?R??? S """ffiffi // ?? """ffiffi // ?OO""""ffiffiffi ss///R fssfiffi OE: S O F____//R: // | ffiffi //sffiffis| /fffiffiflffl| B The composite of (F; OE) and (K; ): (F ; S) -! (T ; T ) is (KOF; OEO(OF * *)). We say that (F; OE) is a strict map and write (F; OE) = F if OE is given* * by identity maps, so that S O F = R. (iii)A homotopy between maps of representations (F; OE) and (F 0; OE0) from (* *E ; R) to (F ; S) is an isomorphism j :F -! F 0over B such that the following diagram commutes: SOj S O F____________//_FFSwO F 0 FFF wwww OEFFF"" --wOE0ww R: If F and F 0are strict maps, this means that S O j = id: R -! R. (iv)We say that representations (E ; R) and (F ; S) are equivalent if there * *are maps (F; OE): (E ; R) -! (F ; S) and (F -1; ): (F ; S) -! (E ; R) whose composites are homotopic to the respective identity maps; (F; OE) is then called an equivalence. We say that (E ; R) and (F ; S) are isomorphic if there are maps (F; OE) and (F -1; OE-1) whose composites are equal to the respective identity maps. (v) A representation (E ; R) is skeletal, faithful, or discrete if the group* *oid E over B is skeletal, faithful, or discrete. (vi)A representation (E ; R) is orientable if R(!) = R(!0) for any pair of m* *aps !, !0:x -! y in E such that ss(!) = ss(!0); equivalently, by Remark 6.3(iv), R must factor through the faithful quotient E =ss. The following observation is easily verified. The analogue for skeletal repre* *sen- tations is not valid. Lemma 7.2. If one of two equivalent representations is either faithful or discr* *ete, then so is the other. Orientations will be maps into certain discrete representations, and we have * *the following immediate implication of Lemma 6.5. Lemma 7.3. Let j :(F; OE) -! (F 0; OE0) be a homotopy between maps of represen- tations (E ; R) -! (F ; S), where (F ; S) is discrete. Then (F; OE) = (F 0; OE0* *) and j is given by identity maps. Thus equivalent discrete representations are isomorp* *hic. We shall define an orientation of an orientable representation (E ; R) to be * *a map of representations from it to the "universal orientable representation" (S R; S* *). We shall construct (S R; S) in the following two sections. We give some intuiti* *on here. Since our interest now is in orientable representations and an orientable EQUIVARIANT ORIENTATION THEORY 17 representation on E factors through its faithful quotient E =ss, we focus on fa* *ithful representations. Since any bundle of groupoids over B is equivalent to a skelet* *al bundle of groupoids (see Remarks 6.3(v) and (vi)), we may as well focus on disc* *rete (= skeletal and faithful) representations. We seek a discrete representation th* *at has as many morphisms as possible, to increase the chance that other representations will map into it. We shall call such a representation saturated and will give a* * precise definition in the next section. This will allow the following definition and th* *eorem. Definition 7.4. Let R be a skeletal groupoid over B. A universal orientable representation (S R; S) in R is a saturated representation such that every fait* *hful representation (E ; R) maps into (S R; S). Theorem 7.5. Any skeletal groupoid R over B has a universal orientable repre- sentation (S R; S), and (S R; S) is unique up to isomorphism of representations. We shall prove the theorem in x9, where we give several characterizations of * *the representation (S R; S). Of course, if R itself is faithful and thus discrete, * *then (S R; S) = (R; Id) and the theory trivializes. We think of S :S R -! R as the best possible approximation of R by a discrete groupoid over B. Definition 7.6. Let (E ; R) be a representation in R. An orientation of (E ; R)* * is a map of representations (F; OE): (E ; R) -! (S R; S). Since S R is discrete and since any faithful representation maps into it, the following reassuring result is immediate from the definition. Corollary 7.7. A representation is orientable if and only if it has an orientat* *ion. We obtain the definition of an orientation of a G-bundle by specializing to t* *he case R = VG (n), starting with the following reinterpretation of Proposition 5.* *3. Proposition 7.8. For an n-plane G-bundle p: E -! B, (G B; p*) is a represen- tation in VG (n). For a G-bundle map (f"; f): p -! q, where q :E0- ! B0, (f*; "f*): (G B; p*) -! (G B0; q*) is a map of representations. For a G-bundle homotopy ("h; h): (f"; f) ' (f"0; * *f0) between G-bundle maps p -! q, h*: (f*; "f*) -! (f0*; "f0*) is a homotopy between maps of representations. This result suggests that there is a substantial analogy between the homotopy theory of representations and topological homotopy theory, and we shall say more about that point of view in x15. Definition 7.9. Let p: E -! B be an n-plane G-bundle. An orientation of p is an orientation of the representation (G B; p*) of G B in VG (n). That is, an orientation of p is a map of representations (F; OE): (G B; p*) -! (S VG (n); S): If (F 0; OE0) is an orientation of an n-plane bundle q :E0 -! B0, then a G-bund* *le map (f"; f): p -! q is orientation preserving if (F; OE) = (F 0; OE0) O (f*; "f* **); we then say that (F; OE) is the pullback of (F 0; OE0) along f and denote it by f*(F 0;* * OE0). In view of Lemma 7.3, our notion of an orientation is homotopy invariant. 18 S.R. COSTENOBLE, J.P. MAY, AND S. WANER Lemma 7.10. If ("h; h): (f"; f) ' (f"0; f0) is a G-bundle homotopy between maps p -! q of G-bundles and (F 0; OE0) is an orientation of q, then f*(F 0; OE0) = * *f0*(F 0; OE0). Proof.A little diagram chase shows that F 0O h* is a homotopy from f*(F 0; OE0)* * to f0*(F 0; OE0). Since its target is discrete, it must be the identity isomorphis* *m. 8.Saturated and supersaturated representations We fix a skeletal groupoid R over B throughout this section and the next. Representations will mean representations in R. We give some simple definitions and observations about groupoids over B before defining saturated representatio* *ns. Definitions 8.1. Let F :E -! F be a functor over B. (i)F is an injection or surjection if it is injective or surjective on both* * objects and morphisms. An injection is an inclusion if it is surjective on morph* *isms between pairs of objects, so that its image is a full subcategory of F . A retraction of groupoids over B is a functor over B left inverse to an inclusion. (ii)A map (F; OE): (E ; R) -! (F ; S) of representations is an injection, in* *clu- sion, surjection, or retraction if the underlying functor F :E -! F is an injection, inclusion, surjection, or retraction of groupoids over B. (iii)An injection is strict if (F; OE) is a strict map, so that OE = idand S * *O F = R. A strict inclusion F : (E ; R) -! (F ; S) specifies a subrepresentation;* * that is, F is the inclusion of a full subcategory E of F such that S|E = R. Lemma 8.2. Let F :E -! F be a functor over B. (i)If E is faithful, then F is a faithful functor. (ii)If E is skeletal and F :E (x; y) -! F (F (x); F (y)) is a surjection for* * all objects x and y of E , then F is injective on objects. (iii)If F is faithful, then any morphism !0:F (x) -! F (y) in F that is not in the image of F has the form !0= F (!)O, where ! :z -! y is a morphism in E , F (z) 6= F (x), and :F (x) -! F (z) is an isomorphism in F such that ss() is an identity map in B. (iv)If F is discrete, then the image of F is a full subcategory. Proof.Since ss O F = ss is faithful if E is faithful, (i) is immediate. For (i* *i), if F (x) = F (y), then E (x; y) contains a map covering the identity map of ss(x).* * This implies that x is isomorphic to y and hence, since E is skeletal, that x = y. F* *or (iii), source lifting gives a map ! :z -! y in E such that ss(!) = ss(!0). Here F (z) 6= F (x) since ss(F (!)) = ss(!0), F (!) 6= !0, and F is faithful. By div* *isibility, there is a map :F (x) -! F (z) such that F (!)O = !0and ss() is an identity ma* *p, so that must be an isomorphism. Part (iv) follows immediately from (iii). Definitions 8.3. We define saturated and supersaturated representations. (i)Fix a set O. Define a partial ordering on the collection of faithful rep* *resen- tations (E ; R) such that E has object set O by letting (E ; R) (E 0; R* *0) if there is a strict injection F :(E ; R) -! (E 0; R0) such that F is the i* *dentity on object sets. A faithful representation with object set O is supersatu* *rated if it is maximal with respect to this partial ordering. (ii)A saturated representation is a discrete supersaturated representation. (iii)A saturation of a faithful representation (E ; R) is a surjection (F; OE* *) from (E ; R) to a saturated representation (S ; S). EQUIVARIANT ORIENTATION THEORY 19 (iv)Two saturations (F; OE): (E ; R) -! (S ; S) and (F 0; OE0): (E ; R) -! (* *S 0; S0) are isomorphic if there is a map (K; ): (S ; S) -! (S 0; S0) of represen* *ta- tions such that (F 0; OE0) = (K; ) O (F; OE). It follows from Propositio* *n 8.14 below that (K; ) is then an isomorphism of representations. Supersaturated representations have all the maps they can hold while remain- ing faithful; saturated representations have no redundant objects. The followi* *ng analogous definition will not be very useful to us, but it helps to clarify ide* *as. Definition 8.4. We define replete and superreplete representations. (i)A faithful representation (E ; R) is superreplete if, for all objects x;* * y of E , ev- ery map ss(x) -! ss(y) in the image of ss : R(R(x); R(y)) -! B(ss(x); ss* *(y)) is also in the image of ss : E (x; y) -! B(ss(x); ss(y)). That is, every* * map in B that might possibly be the image under ss of a map in E is such an image. (ii)A replete representation is a discrete superreplete representation. (iii)A repletion of a faithful representation (E ; R) is a surjection (F; OE)* * from (E ; R) to a replete representation (F ; S). The following observations are immediate from the definition. Proposition 8.5. The following statements hold. (i)Any subrepresentation of a superreplete representation is superreplete. (ii)A representation is superreplete if and only if it has a replete skeleto* *n. (iii)If one of two equivalent representations is superreplete, then so is the* * other. (iv)Any superreplete representation is supersaturated. We would prefer to work with replete rather than saturated representations, b* *ut we have not been able to prove that enough of them exist for a workable theory. One might guess that the converse of (iv) holds. We state this guess formally, * *as a guide to future work, although we do not believe that it is true in general. We* * do think that it may hold under restrictive hypotheses on B and R. Conjecture 8.6. Any supersaturated representation is superreplete. We give some idea of the starting point towards a verification in special cas* *es. Remark 8.7. Suppose that, for all objects x; y of R, either R(x; y) is empty or ss : R(x; y) -! B(ss(x); ss(y)) is a surjection. By Lemma 2.3, this holds when R = VG (n) for an Abelian compact Lie group G. Then a representation (F ; S) is superreplete if and only if, for all objects x; y in F , either F (x; y) is * *empty or ss : F (x; y) -! B(ss(x); ss(y)) is a bijection. This condition on F is indepen* *dent of R. It implies that if (F; OE) : (E ; R) -! (F ; S) is a repletion, then, up* * to isomorphism, F must be the quotient category of E whose objects and morphisms are the equivalence classes of objects and morphisms of E , where objects x and x0 are equivalent if ss(x) = ss(x0) and x ~= x0 and morphisms ! : x -! y and !0 : x0 -! y0 are equivalent if x is equivalent to x0, y is equivalent to y0, a* *nd ss(!) = ss(!0). The functor F : E - ! F must send objects and morphisms to their equivalence classes, and ss : F -! B must be given by ss(F (x)) = ss(x) a* *nd ss(F (!)) = ss(!). These specifications give a well-defined discrete groupoid F* * over B and map F : E -! F of groupoids over B; F will satisfy the required bijectivi* *ty of ss on non-empty morphism sets if, for every pair of morphisms ff; fi : a -! * *b in 20 S.R. COSTENOBLE, J.P. MAY, AND S. WANER B, there is a morphism fl : a -! a such that ff O fl = fi. For example, this ho* *lds for OG if G is Abelian. However, further conditions are needed to ensure that R factors up to isomorphism through a functor S : F -! R. Saturated representations give the closest possible approximations to replete representations that can be constructed in general. The idea of their construct* *ion is to adjoin isomorphisms over identity maps in B, as suggested by Lemma 8.2(ii* *i), to expand any faithful representation to a supersaturated one. Unless Conjecture 8.6 holds, we cannot expect to expand all the way to a superreplete representat* *ion. We then take a skeleton to obtain a saturated representation. This makes sense in view of the following crucial result, which is the analogue for supersaturat* *ed representations of Proposition 8.5(i); it will imply the analogues of (ii) and * *(iii). Theorem 8.8. Any subrepresentation (E ; R) of a supersaturated representation (S ; S) is supersaturated. Proof.Suppose for a contradiction that (E ; R) < (E 0; R0). Then, by Lemma 8.2(* *iii), E 0is obtained from E by adjoining isomorphisms over identity maps of B. Let S 0 be the category obtained by adjoining these isomorphisms to S . Formally, the category S 0is the pushout of the inclusion E -! S and the injection E -! E 0: E ______//E 0 | | | | fflffl| fflffl| S ____//_S 0: The objects of S 0are the objects of S , and the morphism sets are constructed * *in a manner similar to the construction of amalgamated free products of groups. We do not assume familiarity with pushouts of categories such as this, and we will* * give an explicit construction of S 0below. By the pushout property, there is a unique functor ss :S 0-! B that restricts to the functors ss on S and E 0and a unique functor S0:S 0-! R over B that restricts to S on S and to R0 on E 0. We will show that S 0is a faithful groupoid over B. This will contradict the maximality* * of (S ; S) and complete the proof. We claim first that, assuming S 0exists, any morphism x -! y in it must admit a factorization of the form ! , where is an isomorphism in S over an identity map in B, is one of the adjoined isomorphisms of E 0over identity maps of B, and ! is a morphism in S . For any composite ! 0 in which and 0 are in E 0and ! is in S , the source and target of ! are in E and therefore, since E i* *s a full subcategory, ! is in E . Thus the composite is in E 0. This implies that w* *e can reduce the length of any word in morphisms of S and E 0to the form !0 00 with !0 and 0 in S and 0 in E 0. As in the proof of Lemma 8.2(iii), we can use the source lifting property in E and the divisibility property in S to write 0 = , where is a map in E and is an isomorphism in S over an identity map. By Lemma 8.2(iii), we can write 0 = !00 , where !00is a map in E and is one of the adjoined isomorphisms. With ! = !0!00, this gives the claimed factorization. We construct S 0by letting the morphisms x -! y be the equivalence classes of formal composites x -!s -! t !-!y as above, where ! is equivalent to !0 00 if ss(!) = ss(!0). This condition is equivalent to the existence of unique isomorphisms oe and o in E that make the EQUIVARIANT ORIENTATION THEORY 21 triangles commute in S and the rectangle commute in E 0in the following diagram: ?s_____//_t?? """| | ??!? """ | | ??? "" | | ?OO x ?? oe|| |o| ?y? ??? | | 0 ?? | | 0 ?Offlffl|fflffl|O! s0___0_//t0 Indeed, if oe and o make the diagram commute, they must be isomorphisms over identity maps of B since the 's and 's are isomorphisms over identity maps of * *B, and then ss(!) must equal ss(!0). Conversely, if ss(!) = ss(!0), then by divisi* *bility in S there is a unique map o making the right triangle commute. Since E is a fu* *ll subcategory of S , o is in E . By divisibility in E 0, there is then a unique m* *ap oe in E 0making the middle square commute. Then ss(oe) is an identity map by the diagram, and the left triangle commutes in E 0since E 0is faithful. But then oe* * is in E . Using the proof of the first claim above, it is an exercise to show that tw* *o such formal composites x -! y and y -! z can be spliced together to give a formal composite x -! z, well-defined up to equivalence. This makes S 0into a category with E and S as subcategories. The universal property required of a pushout is immediate from the diagram- matic description of equivalence between morphisms. For example, ss :S 0-! B can and must be defined by ss(! ) = ss(!), and it is immediate from the spec- ification of equivalence that ss is a faithful functor. Similarly, S0:S 0-! R c* *an and must be defined by S0(! ) = S(!)R0( )S(). Since the objects of S 0are those of S , source lifting is inherited from S . To show divisibility, suppose* * that = ! :x -! y and 0= !0 00:x0- ! y are morphisms in S 0, decomposed as usual, and suppose that ss() = ss(0)fi. By divisibility in S , there is a such* * that !0 = ! and ss() = fi, and then 0O ((0)-1( 0)-1 ) = in S 0. This completes the proof that S 0is a faithful groupoid over B. Corollary 8.9. A representation is supersaturated if and only if it has a satur* *ated skeleton. Proof.The theorem implies that a skeleton of a supersaturated representation is saturated, and the converse is obvious. Corollary 8.10. If one of two equivalent representations (E ; R) and (F ; S) is supersaturated, then so is the other. Proof.If (E ; R) is supersaturated, then (F ; S) is faithful, by Lemma 7.2, and* * (E ; R) and (F ; S) have equivalent and therefore isomorphic skeleta, by Lemma 7.3. Thus a skeleton of (F ; S) is saturated and (F ; S) is supersaturated. Corollary 8.11. The image of any map from any representation into a satu- rated representation is saturated. Therefore, any map from a faithful represent* *ation (E ; R) into a saturated representation factors through a saturation of (E ; R). Proof.This is now immediate in view of Lemma 8.2(iv) and the fact that a sub- category of a skeletal category is skeletal. 22 S.R. COSTENOBLE, J.P. MAY, AND S. WANER Now we can produce saturations of any faithful representation. Lemma 8.12. Any faithful representation (E ; R) has a saturation. Proof.An immediate application of Zorn's lemma shows that there exists a super- saturated representation (E 0; R0) (E ; R). A skeleton of (E 0; R0) is still s* *upersatu- rated, by Theorem 8.8, and is therefore saturated. The composite of the injecti* *on of (E ; R) into (E 0; R0) with a retraction of (E 0; R0) onto a skeleton is clearl* *y surjective on objects, hence on morphisms by Lemma 8.2(iv), so is a saturation. This is not the only way to produce a saturated representation from a faithful one. Here is another construction. Lemma 8.13. Any faithful representation (E ; R) contains a saturated subreprese* *n- tation that is maximal with respect to inclusion. Proof.This is just a direct application of Zorn's lemma to the set of saturated subrepresentations, partially ordered by inclusion. This set is nonempty becau* *se the empty representation is saturated, and it is easy to check that the union o* *f a chain of saturated subrepresentations is again saturated. It is interesting that neither of these two ways of producing a saturated rep* *re- sentation from a faithful one gives a unique result in general. In any case, we* * have plenty of saturated representations. They admit the following characterization. Proposition 8.14. Let (E ; R) be a faithful representation. If every map from (E ; R) to a faithful representation is an injection, then (E ; R) is saturated* *. If (E ; R) is saturated, then every map from (E ; R) to a faithful representation is an in* *clusion. Proof.We first show that if (E ; R) is not saturated, then there is a map from * *it to a faithful representation that is not an injection. If (E ; R) is not skelet* *al, then a retraction to a skeleton gives a map to a faithful representation that is not* * an injection. If (E ; R) is skeletal but not supersaturated, then a saturation of * *(E ; R) cannot be an injection since it would then be an isomorphism. Conversely, let (E ; R) be saturated, and let (F; OE): (E ; R) -! (F ; S) be * *a map into a faithful representation. By Lemma 8.2(i), F is faithful. Let E 0be the groupoid over B that has the same objects as E , but has morphism sets E 0(x; y* *) = F (F (x); F (y)). Certainly E 0is faithful. Extend R: E - ! R to R0:E 0-! R by letting R0 = R on objects and letting R0(!) = OE(y)S(!)OE-1(x) for a map ! :F (x) -! F (y). By the maximality of (E ; R), E 0can have no more maps than E , hence F :E (x; y) -! F (F (x); F (y)) must be a surjection for all x and y.* * By Lemma 8.2(ii), (F; OE) is an inclusion. 9. Universal orientable representations We are nearly ready to construct the universal orientable representation. It is not described in terms of the usual kind of universal property, but is inste* *ad characterized by one of several different equivalent properties. Definition 9.1. A faithful representation (E ; R) is an absolute retract if eve* *ry map from (E ; R) into a faithful representation has a left inverse. If (E ; R) is an absolute retract, then every map of it into a faithful repre* *sentation must be an injection, and hence, by Proposition 8.14, (E ; R) must be saturated. EQUIVARIANT ORIENTATION THEORY 23 Definition 9.2. A representation (E ; R) is said to be injective if, for any in* *clu- sion (F ; S) -! (F 0; S0) of discrete representations, each map (F ; S) -! (E ;* * R) extends to a map (F 0; S0) -! (E ; R). We shall construct the universal orientable representation by first construct* *ing a universal discrete representation that is characterized by a categorical univ* *er- sal property and is injective, but is not saturated. Its saturation will inheri* *t the injectivity and will be the representation we want. Proposition 9.3. There is a discrete representation (R"; D) in R such that any discrete representation (E ; R) in R lifts uniquely to a strict map "R:E -! "R.* * That is, (R"; D) is the universal discrete representation in R. Proof.We use the freeness property of the groupoids B=b over B given in Propo- sition 6.8. The objects of the category "Rare the functors F :B=b -! R over B. Define ss :"R-! B and D :"R-! R on objects by ss(F ) = b and D(F ) = F (idb). The morphisms F - ! F 0, F 0:B=b0 -! R, of "R are the strict maps of repre- sentations "ff:B=b -! B=b0, so that F 0O "ff= F . Define ss on morphisms by ss("ff) = ff: b -! b0, where ff = "ff(idb). Regarding ff as a map __ff:ff -! id* *0bin B=b0, define D on morphisms by D("ff) = F 0(__ff): F (idb) = F 0(ff) -! F 0(id0b): Composition in "Ris given by composition of functors. It is now easy to see tha* *t "R is a well-defined category, that ss and D are functors, and that ss = ss O D, s* *o that D is a map over B. By Proposition 6.8, the functor "ffis determined by ff, and * *this implies easily that "Ris a discrete groupoid over B. To check the universal property of D, let R: E -! R be a discrete representat* *ion and define "R:E -! "Ras follows. Since E is discrete over B, Proposition 6.8 sh* *ows that an object y of E with ss(y) = b determines a unique functor "y:B=b -! E such that "y(idb) = y. We let "R(y) = R O "y. If ! :y -! y0 is a morphism of E * *with ss(!) = ff: b -! b0, we let "R(!) = "ff:B=b -! B=b0, so that "y0"ff= "y. Then "* *Ris a well-defined map over B such that D O "R= R, and it is the only such map. Lemma 9.4. The representation (R"; D) is injective. Proof.Let (I; ): (S ; S) -! (S 0; S0) be an inclusion of discrete representatio* *ns and let (F; OE): (S ; S) -! (R"; D) be a map. Thus : S0OI -! S and OE: DOF -! S are isomorphisms over B. We claim that there is a representation S00:S 0-! R such that S00O I = D O F and an isomorphism OE0:S00-! S0 over B such that O (OE0O I) = OE. Taking F 0= "S00, this will give a map (F 0; OE0): (S 0; S0) * *-! (R"; D) such that (F 0; OE0) O (I; ) = (F; OE). To check the claim, note that S00and O* *E0 are already defined on the full subcategory I(S ) of S 0. Define S00(y) = S0(y) and* * let OE0(y) be the identity map for an object y of S 0that is not in S . For a morph* *ism ! :x -! y of S 0, we can and must define S00(!) = OE0(y)-1 O S0(!) O OE0(x): We can now construct the universal orientable representation in R. Theorem 9.5. The following conditions on a faithful representation (S ; S) in R are equivalent. Moreover, there exists a representation (S R; S) satisfying th* *ese conditions, it is unique up to isomorphism, and any map from it to itself is an isomorphism. This representation is called the universal orientable representat* *ion. 24 S.R. COSTENOBLE, J.P. MAY, AND S. WANER (i)(S ; S) is a maximal saturated subrepresentation of (R"; D). (ii)(S ; S) is a saturation of (R"; D). (iii)(S ; S) is a saturated retract of (R"; D). (iv)(S ; S) is an absolute retract. (v) (S ; S) is a saturated injective representation. (vi)(S ; S) is saturated, and any faithful representation maps into it. (vii)(S ; S) is saturated, and any saturated representation maps into it. Proof.We understand the first statement to mean that if (S ; S) satisfies one of the conditions and (S 0; S0) satisfies another, then (S ; S) is isomorphic to (* *S 0; S0). The proofs will show how to obtain the required isomorphisms. (i) () (ii) and (ii) () (iii): By Lemma 8.13, there exists an (S ; S) satisfying (i). By Lemma 8.12, there also exists a saturation (F; OE): (R"; D) -! (S 0; S* *0). By Proposition 8.14, the restriction of (F; OE) to (S ; S) is an inclusion. By* * the injectivity of (R"; D) and Proposition 8.14, the inclusion of (S ; S) in (R"; D* *) extends to an inclusion of (S 0; S0). By maximality, the inclusion of (S ; S) in (S 0; * *S0) is an isomorphism. This already proves that (i) is equivalent to (ii) and that the* *se conditions imply (iii); (iii) implies (i) trivially. (iii)=)(iv): Let (S ; S) -! (E 0; R0) be a map into a faithful representation a* *nd let (E 0; R0) -! (E 00; R00) be a retraction onto a skeleton. By Proposition 8* *.14, the composite of these maps is an inclusion. By the injectivity of (R"; D), th* *is inclusion extends to a map of (E 00; R00) into (R"; D). Composing with a retrac* *tion (R"; D) -! (S ; S), we see that (S ; S) is a retract of (E 0; R0). (iv)=)(iii): There is a map (S ; S) -! (R"; D) by the universal property of (R"* *; D). This map admits a retraction since (S ; S) is an absolute retract. (iii)=)(v): A retract of an injective representation is injective. (v)=)(vi) and (vi) () (vii): Any faithful representation retracts to a skeletal* * and thus discrete representation. For discrete representations, condition (vi) is j* *ust the special case of the injectivity condition (v) in which the given domain represe* *ntation is empty. Obviously (vi)=)(vii), and (vii)=)(vi) by Corollary 8.11. (vi)=)(i): By (vi) and Proposition 8.14, there is an inclusion of any given max* *imal saturated subrepresentation (S 0; S0) of (R"; D) in (S ; S). By the injectivit* *y of (R"; D) and Proposition 8.14, this inclusion extends to an inclusion of (S ; S)* * in (R"; D). By the maximality of (S 0; S0), the first inclusion must be an isomorp* *hism. The argument of the last paragraph also gives the uniqueness, up to isomorphi* *sm, of (S ; S). Varying the argument by composing the initial inclusion of (S 0; S0* *) with any self-map of (S ; S), and noting that such a self-map is an inclusion, we se* *e that any self-map of (S ; S) is an isomorphism. Criterion (vi) is the one on which our definition of an orientation is based.* * Cri- terion (ii) is the one most useful for the actual construction of specific exam* *ples. One implication of the theorem is that the set of self-maps of (S R; S) is a gr* *oup under composition. This is not true for a general saturated representation, as* * is shown by the following example. Example 9.6. Take B to be the "unit interval" category I . It has two objects, 0 and 1, and one non-identity morphism I : 0 -! 1. Clearly functors ss :E -! I EQUIVARIANT ORIENTATION THEORY 25 are determined by the images of objects. Let R be the groupoid over I with a single object "0over 0, a single object "1over 1, and morphism sets R("0; "0) =* * Z, R("0; "1) = Z, and R("1; "1) = {id}. Composition is defined so that R("0; "0) =* * Z as a group and R("0; "0) acts on R("0; "1) by addition. (i)Let E be the groupoid over I with a single object x over 0, infinitely many objects yi, i 1, over 1, and a map !i:x -! yi for each i as its only non-identity maps. Define a representation R: E - ! R by sending !i to i 2 R("0; "1). Clearly (E ; R) is discrete and replete, hence satu* *rated. There is a map (F; OE): (E ; R) -! (E ; R) with F (x) = x, F (yi) = yi+1* * for each i, OE(x) = 1 : "0-! "1, and OE(yi) = id"1. The map (F; OE) is not * *an isomorphism. (ii)Unusually, in this example (R"; D) is its own saturation. Its descriptio* *n is similar to that of (E ; R), except that there is now an object yi for ev* *ery integer i. Its group of self-maps is Z. Of course, there may be many different orientations (F; OE): (E ; R) -! (S R;* * S) of a given orientable representation (E ; R). We have the following observation* *s. Proposition 9.7. Let = (S R; S) be the group of automorphisms of the uni- versal orientable representation (S R; S) and let (E ; R) be the set of orienta* *tions of an orientable representation (E ; R). Then acts on (E ; R) by composition. This action has the following properties. (i)If (E ; R) is saturated, then acts transitively on (E ; R), hence (E ; * *R) is isomorphic as an -set to =, where is the isotropy group of any chosen orientation of (E ; R). (ii)If (E ; R) is faithful, then any orientation (E ; R) - ! (S R; S) factors through a saturation of (E ; R). Therefore, a (E ; R) ~= (E 0; R0) [E 0;R0] as -sets, where the union runs over one representative from each isomor- phism class of saturations (E ; R) -! (E 0; R0) of (E ; R). Proof.This follows from Corollary 8.11, Proposition 8.14, and the fact that (S * *R; S) is an injective representation whose self-maps are all automorphisms. Our explicit examples of universal orientable representations will be replete* *. In such cases, Conjecture 8.6 certainly holds. Indeed, the following result is imm* *ediate from Propositions 8.14 and 8.5. Proposition 9.8. If the universal orientable representation (S R; S) is replete, then every supersaturated representation (E ; R) is superreplete. Part III.Examples of universal orientable representations 10.Cyclic groups of prime order We here determine (S VG (n); S) explicitly for G = Z=p. We first insert the following observation. Recall that OG =(G=K) ~=G (G=K) when G is finite. Proposition 10.1. If G is finite, then the groupoids "VG(n) and S VG (n) over OG have only finitely many objects and finitely many morphisms. 26 S.R. COSTENOBLE, J.P. MAY, AND S. WANER Proof.It suffices to consider "VG(n). Since "VG(n) is faithful over OG and OG* * is finite, it suffices to show that "VG(n) has finitely many objects. The objects * *are the representations F :OG =(G=K) -! VG (n). Such functors are consistent families of homotopy classes of maps G xH V -! G xK W = F (idG=K) of G-vector bundles, where W is a fixed K-representation and H runs over the subconjugates of K. We are working in the skeletal category VG (n), and there are only finitely many c* *hoices for W , finitely many H, finitely many choices of V for each H, and finitely ma* *ny choices of maps for each H and V . In the proof just given, W actually determines all of the representations V . However, it does not determine the maps G xH V -! G xK W , since we can precompose a given map with any map G xH V -! G xH V over the identity map of G=H. This variability is crucial to understanding "VG(n). Postcomposition wi* *th maps G xK W -! G xK W over the identity map of G=K leads to natural trans- formations fi -! fi0between objects of "VG(n) with the same W . This variabilit* *y is crucial to understanding the passage from (V"G(n); D) to its saturated retract,* * the universal orientable representation (S VG (n); S). Let G = Z=p, where p is a prime, and let t be a generator of G. The orbit category OG has only the two objects G = G=e and P = G=G. The maps G -! G form a copy of G, there is a unique quotient map q :G -! P , and the only map P -! P is the identity. We consider the cases p = 2 and p > 2 separately. In bo* *th cases, it turns out that (S VG (n); S), and therefore every saturated represent* *ation, is replete. For this reason, the description of (S VG (n); S) can easily be gue* *ssed. However, for illustrative purposes, we follow the outline of the general theory. Example 10.2. p = 2. In this example, we write fl generically for a self-map of* * a representation that reverses (non-equivariant) orientation and satisfies fl2 = * *1. (a) The category VG (n). (i)VG (n) has a single object U = G x Rn over G. This object has four self-maps, namely 1 and fl (that is, 1 x fl) over the identity map o* *f G, and t (that is, action by t) and tfl = flt over t: G -! G. (ii)For 0 k n, let Vk = Rn-k Lk denote the sum of n - k copies of the trivial one-dimensional representation R and k copies of the non-trivial one-dimensional representation L. These are the objects * *of VG (n) over P . There are no maps from Vk to Vk0 unless k = k0, in which case there are either the two maps 1 and fl if k = 0 or n, or there are the four maps 1, 1 fl, fl 1, and fl fl if 0 < k < n; fl* * fl is the composite of 1 fl and fl 1 in either order. (iii)There are two maps U -! Vk, namely the action of G on Vk and fl. We have (fl 1) = fl = (1 fl). More interestingly, t = if k is even, but t = fl if k is odd. (b) The category "VG(n). (i)The objects over G are the representations G G -! VG (n), and G G has two objects over G connected by an isomorphism t over t: G -! G. There are two such representations, U+ and U- ; the first sends t to t and the second sends t to tfl. There are no maps between these objects, and each has a self-map "tover t. (ii)The objects over P are the representations OG = G P - ! VG (n). For each k there are two representations, (Vk)+ and (Vk)- , sending P EQUIVARIANT ORIENTATION THEORY 27 to Vk. The first sends q to and the second sends q to fl. In both cases, t must be sent to t if k is even and to tfl if k is odd. There a* *re no non-identity maps among these objects. (iii)There is a map "q:U+ -! (Vk) over q if and only if k is even; there is a map "q:U- -! (Vk) over q if and only if k is odd. (c) Saturations of "VG(n). The only maps that we may add to "VG(n) are maps between (Vk)+ and (Vk)- . For each k we have a choice of how to send such an added map to VG (n): we may send it to 1fl or to fl1. There is only one choice, f* *l, when k = 0 or n. Thus there are 2n - 2 possible supersaturations. We obtain a saturation by passing to a skeleton, which means throwing out one of (Vk)+ or (Vk)- for each k. For definiteness, we discard (Vk)- . Changing notation, we have the following description of the universal orientable representation. (d) The universal orientable representation (S VG (n); S). (i)S VG (n) has two objects, u+ and u- , over G, and S sends each of them to U = G x Rn. Each has a self map t over t: G -! G; S sends t: u+ -! u+ to t and sends t: u- -! u- to tfl. (ii)S VG (n) has n + 1 objects vk, 0 k n, over P , and S(vk) = Vk. (iii)There is a map m: u+ -! vk over q :G -! P if and only if k is even; there is a map m: u- -! vk if and only if k is odd; S(m) = in both cases. Pictorially, the universal orientable representation looks like this: t@AGFEDfflffl|t@AGFEDfflffl|GFBCEDtfflffl|@AGFtEDfflffl|GFBCEDtflfflffl| _G u+_ u- _ _U ________U _ | | | | | | ooss_ | | _S__//_ | | q || m|| m || || || | | | | | | | | | | fflffl| fflffl| fflffl| fflffl| fflffl| P v2k v2k+1 V2k V2k+1 OG S VG (n) in VG (n) (e) The group n = (S VG (n); S). n is an elementary abelian 2-group of order 2n+1. Since (d)(iii) rules out the possibilities A(u+ ) = u- or A(u- ) = u+ , every automorphism (A; ff) has A = id:S VG (n) -! S VG (n), hence is specified by giving the isomorphisms ff(x): S(x) -! S(x) for objects x in S VG (n). As generators for n we can take the collection {ff+ ; ff- ; ff1; : :;:ffn-* *1}, where, specifying only the behavior of each of these on objects not assigned identity maps, we have ff+ (u+ ) = fl and ff+ (v2k) = fl 1 (or fl if 2k = 0 or n), ff- (u- ) = fl and ff- (v2k+1) = fl 1 (or fl if 2k + 1 = n), and ffk(vk) = fl fl: (f) An example of non-unique saturation. 28 S.R. COSTENOBLE, J.P. MAY, AND S. WANER We exhibit two saturations of the representation (G G; U+ ) given in (b)(i) above that are not isomorphic by displaying two maps, (F; OE) and (F 0; OE0), from (G G; U+ ) into (S VG (n); S) whose images do n* *ot differ by an automorphism in n. In fact, let F send both objects of G G to u+ , and let OE be the identity; let F 0send both objects to * *u- , and let OE0send both to the map fl :G x Rn -! G x Rn. Observe that the category S VG (n) has two components, although VG (n) has the initial object U and is therefore connected. Observe too that the order of n increases as n increases. As we shall see, neither of these phenomena can occur* * for a finite group of odd order. Example 10.3. p odd. In this example we write fl generically for an orientation reversing map of Rk, k 1, with fl2 = 1. A representation that contains no triv* *ial summands admits no orientation reversing self G-map. (a) The category VG (n). (i)VG (n) has a single object U = G x Rn over G. This object has 2p self maps, namely 1 and fl (that is, 1 x fl) over the identity map of G, * *and ti (that is, action by ti) and tifl = flti over ti:G -! G. (ii)G has the trivial irreducible representation R and (p - 1)=2 non-tri* *vial irreducible two-dimensional representations. The objects over P are the distinct n-dimensional sums of these. Write V for a typical such object; it has a non-identity map fl if and only if it contains a tr* *ivial summand. (iii)There are two maps and fl from U to each V . We have t = and, if V contains a trivial summand, fl = fl. (b) The category "VG(n). (i)The objects over G are the representations G G -! VG (n), and G G has p objects {e1; : :;:ep} over G cyclically permuted by isomorphis* *ms tPover t: G -! G. For each sequence ffl = (ffl1; : :;:fflp), ffli= 0* * or 1 and ffli even, there is a representation Uffl:G G -! VG (n) that sends t: ei- ! ei+1to tflffli. We write U0 when all ffli= 0. There is a se* *lf-map "tover t of U0. If t acts on the sequences ffl by cyclic permutation* *, there are maps "t:Uffl-! Utfflover t. We also have the iterates "ti. (ii)The objects over P are the representations OG = G P - ! VG (n). For each V , there are two such representations, V+ and V- , that se* *nd P to V . The first sends q to and the second sends q to fl; both send t to t. There are no non-identity maps among these objects. (iii)For each V , there are maps "q:U0 -! V over q, and there are no other maps over q. (c) Saturations of "VG(n). When V contains a trivial summand, we may add an isomorphism between V+ and V- that maps to fl in VG (n). We may also add an isomorphism over the identity map of G between U0 and any other Uffl. This isomorphism may be chosen to map to either the identity or fl in VG (n). This choice then determines isomorphisms between U0 and the other Uffl0such that ffl0 is a cyclic permutation of ffl. We pas* *s to a saturation by discarding V- when V contains a trivial summand, and discarding all Ufflwith ffl 6= 0. Changing notation, we have the fol* *lowing description of the universal orientable representation. EQUIVARIANT ORIENTATION THEORY 29 (d) The universal orientable representation (S VG (n); S). (i)S VG (n) has one object u over G, and S(u) = U = G x Rn. u has self-maps ti over ti:G -! G, and S(ti) = ti. (ii)For each representation V that contains a trivial summand, S VG (n) has one object v over P with S(v) = V . For each representation V that contains no trivial summand, S VG (n) has two objects v+ and v- over P , and S(v ) = V . (iii)For each V that contains a trivial summand, there is a map m: u -! v over q with S(m) = . For each V that contains no trivial summand, there are maps m+ :u -! v+ and m- :u -! v- over q; S(m+ ) = and S(m- ) = fl. Pictorially, the universal orientable representation looks like this: t@AGFiEDfflffl| ti@AGFEDfflffl| t@AGFiEDfflffl| _G _u 3 _ U1 | fififi|333 flfl|111 | ooss_ fifi| 33 __S_//_ flflfl|11 q|| m fifi|m+|3m-33 flflfl||1fl11 | fififi| 33 flfl | 11 | fi | 33 flfl | 11 fflffl| fifi fflffl|ssss fl fflffl| P v v+ v- V V V (V G 6= 0) (V G = 0) OG S VG (n) inVG (n) (e) The group n = (S VG (n); S). n = Z=2. The non-trivial automorphism (A; ff) has A(u) = u and ff(u) = fl; A(v) = v and ff(v) = fl if V contains a trivial summand; and A(v+ ) = v- , A(v- ) = v+ , and ff(v+ ) = ff(v- ) = idV if V does not contain a trivial summand. 11. Orientations of V -dimensional G-bundles Let p: E -! B be a G-bundle. Our general theory has been set up to deal with the problem that the fiber representations of p can vary. The fiber Fx over a point x 2 B with isotropy group Gx is only a Gx-representation Vx, and the bundle over the orbit G . Fx ~=G=Gx is isomorphic to G xGx Vx. The universal orientable representation gives a universal model for describing all possible w* *ays that such local data can be oriented consistently. It is sometimes possible to * *restrict to G-bundles with a single type of fiber representation. Definition 11.1. Let V be a representation of G. A G-bundle p: E -! B has dimension V if each of its fiber representations Vx is isomorphic to V , regard* *ed as a representation of Gx. This means that p*: G B - ! VG factors through VG (V ), where VG (V ) is the full subgroupoid over OG of VG whose objects are * *the G-bundles isomorphic to G=H x V ~=G xH V for some H G. Thus (G B; p*) is a representation in VG (V ). For example, p must be V -dimensional for some V if B is G-connected, in the sense that all of its fixed point spaces are non-empty and path connected. One * *might think that our theory becomes trivial or unnecessary for G-bundles of dimension* * V but, as we now explain, that is not the case. 30 S.R. COSTENOBLE, J.P. MAY, AND S. WANER Define an orientation of a fiber Fx to be a choice of a homotopy class OE(x) * *of Gx-linear isometries Fx -! V . This directly generalizes the nonequivariant idea that an orientation of a fiber is a choice of one of the two homotopy classes of isomorphisms with a fixed copy of Rn. Define a naive orientation of the G-bundle p to be a compatible collection {OE(x)} of orientations of the fibers Fx. Here* *, by "compatible", we mean that the following two conditions hold. First, noting that Ggx = gGxg-1 for x 2 B and g 2 G and that g-1 maps Fgx to Fx, the OE(x) are G-invariant in the sense that OE(gx) = gOE(x)g-1. Second, if x 2 BH , y 2 BK a* *nd (!; ff) is a map from x to y in G B, then OE(x) = OE(y) O "!, where "!::Fx ! Fy is the homotopy class of H-linear isometries determined by (!; ff). This defini* *tion can be reformulated representation theoretically as follows. To avoid pedantry,* * we ignore the choice of isomorphism between G=H x V and the unique isomorphic G-bundle in the skeletal OG -groupoid VG in what follows. Definition 11.2. Define I :I VG (V ) -! VG (V ) to be the injection of the sub- groupoid over OG that contains all of the objects of VG (V ) but only the maps * *that are of the form ff x id:G=H x V - ! G=K x V for a map ff: G=H -! G=K in OG . A naive orientation of a V -dimensional G-bundle p: E -! B is a map of representations (F; OE): (G B; p*) -! (I VG (V ); I): As in the nonequivariant case, the functor F here is uniquely determined. It is an instructive exercise to verify that our two definitions of a naive orient* *ation coincide; that is, a naive orientation as defined in the previous definition de* *termines and is determined by a set of compatible orientations of fibers. However, if we insist that every orientable G-bundle must have an orientation, then the naive definition of an orientation of a V -dimensional bundle is inade* *quate, at least when V G = 0. Since the representation (I VG (V ); I) is clearly disc* *rete, it maps into the universal orientable representation (S VG (V ); S) in VG (V ).* * We may view a map (I VG (V ); I) -! (S VG (V ); S) as an orientation of (I VG (V )* *; I). However, I VG (V ) has too few morphisms for this orientation to be an isomorph* *ism of representations in VG (V ). Therefore, a naive orientation gives an orienta* *tion, but not conversely. When V G 6= 0, a V -dimensional G-bundle does admit a naive orientation, but it may have more orientations than naive orientations. To display an orientable V -dimensional G-bundle that has no naive orientatio* *n, let G be the circle group S1 and let G act on S2 by rotation about the axis thr* *ough the poles. Let V be the tangent representation at the north pole n. We can view* * S2 as the one-point compactification of the standard representation of the unit ci* *rcle of complex numbers on C, taking the origin as the north pole and the point at 1 as the south pole. Thus V is just a copy of C with its standard action by S1. The tangent representation at the south pole s is isomorphic to V , and the tan* *gent bundle o of S2 is V -dimensional. Moreover, o is obviously orientable since the* *re is only one path class connecting any two objects of the equivariant fundamental groupoid G S2. However, o does not have a naive orientation. There is only one homotopy class of G-linear isomorphisms from V to itself, so there is only one * *choice of equivariant orientation at both the north and the south poles. These orienta* *tions are not compatible since any nonequivariant path from the south to the north po* *le induces a map of tangent planes that reverses orientation. Now let G be a cyclic group of order p embedded as usual as a subgroup of the circle group. By restriction, we can regard the S1-bundle just displayed as EQUIVARIANT ORIENTATION THEORY 31 a G-bundle. It is still V -dimensional and orientable, and it still admits no n* *aive orientation. In the following example, we use the universal orientable G-bundl* *es constructed in the previous section to display orientations of this bundle. Example 11.3. Let G = Z=p, let V = L2 = V2 if p = 2, and let V be an irreducible two-dimensional representation if p is odd. Consider the sphere SV obtained by one-point compactification of V . A skeleton 0G(SV ) of the fundamental groupoid has one object x over G, and two objects n and s, the north and south poles, ov* *er P . The representation o* :0G(SV ) -! VG (2) induced by the tangent bundle of SV sends x to U = GxR2 and both n and s to V . There are maps x -! n and x -! s over q, and they are not sent to the same map in VG (2) due to the incompatibil* *ity phenomenon that we have observed. Rather, one is sent to and the other is sent to fl. There are two orientations (F; OE): (0G(SV ); o*) -! (S VG (2); S). If p* * = 2, they both have F (x) = u+ and F (n) = v2 = F (s); one has OE(x) = 1 and the oth* *er has OE(x) = fl. If p > 2, they both have F (x) = u, one has F (n) = v+ and F (s* *) = v- , and the other has F (n) = v- and F (s) = v+ ; both have OE(n) = id= OE(s), while one has OE(x) = 1 and the other has OE(x) = fl. There is a still more naive notion of an orientation of a G-vector bundle, na* *mely a nonequivariant orientation such that G acts by orientation preserving maps. T* *his notion is widely used in the literature when G is a finite group of odd order. * *Here the following observation shows that the requirement that the action preserve t* *he orientation is a negligible restriction on nonequivariantly orientable G-bundle* *s. Lemma 11.4. Let G be finite of odd order and let p : E -! B be a nonequivariant* *ly orientable G-vector bundle. (i)If B is path connected, then G acts on p by orientation preserving maps * *for either of the orientations of p. (ii)If B=G is path connected, then p admits two orientations such that G acts on p by orientation preserving maps. (iii)In general, p admits orientations such that G acts by orientation preser* *ving maps. Proof.For (i), an orientation of p is given by a Thom class in "Hn(T ; Z) ~=Z, * *where T is the Thom space of p, and action by an odd order group element must preserve this class. For (ii), pick a path component B0 of B with isotropy group G0 and * *fix an orientation of the restriction of E over B0. This is a G0-bundle, and G0 act* *s by orientation preserving maps by (i). By translation by group elements, we obtain orientations on the rest of the path components of B such that the action by G * *is orientation preserving, and this is the only way that such orientations can ari* *se. Part (iii) follows, since B is the disjoint union of G-spaces with path connect* *ed orbit spaces. Remark 11.5. In view of (i), the action of Z=p S1 on S2 in Example 11.3 shows that a nonequivariantly oriented V -dimensional G-bundle with an orientation pr* *e- serving action by G need not be naively G-oriented in the sense of Definition 1* *1.2. However, if G is finite of odd order and B is G-connected, then any nonequivari* *antly oriented V -dimensional G-bundle p : E -! B does have a naive G-orientation, as is easily deduced from Lemma 12.1 below. Example 11.3 shows that the G- connectivity of B is essential to the conclusion. 32 S.R. COSTENOBLE, J.P. MAY, AND S. WANER 12. Complex Bundles and Odd-Order Groups There is of course an analog of our theory of orientations in which we restri* *ct attention to complex representations and bundles throughout. However, the re- sulting theory is quite trivial. Define UG (n) in the same way as VG (n), using complex n-dimensional bundles over G-orbits. Then UG (n) is discrete over OG since there is at most one homotopy class of complex G-bundle maps covering a given map in OG . This implies that the identity functor of UG (n) is itself th* *e uni- versal orientable complex representation. In fact, this functor is obviously re* *plete, and thus saturated, and any representation R: E -! UG (n) specifies a strict map (E ; R) -! (UG (n); Id). Moreover, it is also clear that there are no nontrivia* *l au- tomorphisms of (UG (n); Id). Therefore, every complex representation has a uniq* *ue complex orientation. Notice that we need not assume that the underlying groupoid E is faithful. We conclude that complex G-bundles over arbitrary G-spaces admit unique complex orientations. ___ __ We have an evident "realification" functor r :U G(n) -! V G(2n) between the categories of all complex and all real G-vector bundles over orbits. Using the chosen retraction equivalences from these categories to UG (n) and VG (2n), we obtain a realification representation r :UG (n) -! VG (2n). Nonequivariantly, t* *his just corresponds to choosing an identification of Cn with R2n. We consider an n- dimensional complex representation (E ; R) as a 2n-dimensional real representat* *ion by composing it with r, and we orient an n-dimensional complex bundle as a 2n- dimensional real bundle by composing its complex orientation with a fixed choic* *e of a real orientation (UG (n); r) -! (S VG (2n); S). There exists such an orientat* *ion since r is an orientable representation in VG (2n). We turn now to another simple case: orientations of real representations when* * G is a finite group of odd order. We work with a fixed given odd order finite gro* *up G in the rest of the section. We can generalize the description of the universal ori* *entable representation for Z=p, p odd, given in Example 10.3 to obtain a description in the general case. The essential point is that if V is a representation of G w* *ith V G = 0, then V admits a complex structure, and so its group of orthogonal G- linear isometries is connected. This observation can be codified as follows. Lemma 12.1. Let f : V -! V be an orthogonal G-map. Then the following three statements are equivalent. (i)f is linearly homotopic to the identity. (ii)f is G-linearly homotopic to the identity. (iii)fG : V G -! V G is linearly homotopic to the identity. Proof.Let VG be the orthogonal complement of V G. Then f = fG fG , and fG : VG -! VG is G-linearly homotopic to the identity. Since fG is linearly homotopic to the identity if and only if it is G-linearly homotopic to the iden* *tity, the conclusion follows. Therefore, for objects G xH V and G xK W of VG (n), there are at most two morphisms G xH V - ! G xK W in VG (n) over a given map G=H -! G=K in OG . If there is one such morphism, then whether there are one or two morphisms is entirely determined by whether or not V H is empty. Again, it turns out that (S VG (n); S), and thus every saturated representation, is replete. EQUIVARIANT ORIENTATION THEORY 33 Construction 12.2. Let G be a finite group of odd order. We can describe the universal orientable representation S :S VG (n) -! VG (n) as follows. The categ* *ory S VG (n) has one object v = v(H) for each object G xH V in VG (n) such that V H 6= 0; it has two objects v+ = v+ (H) and v- = v- (H) for each G xH V such that V H = 0. Of course, ss :S VG (n) -! OG and S :S VG (n) -! VG (n) send these objects to G=H and G xH V . For each object G xH V of VG (n), choose a map : G x Rn -! G xH V over the quotient map G -! G=H. We think of as specifying an orientation of V . Suppose that there exists a G-bundle map G xH V -! G xK W over a G-map ff: G=H -! G=K, where ff(eH) = gK. Write "ff+for a map G xH V -! G xK W covering ff and satisfying "ff+O = O g, if such exists, where g :G x Rn -! G x Rn is given by right multiplication by g on the G coordinate. Write "ff-for a map covering ff and satisfying "ff-O = O fl O g, * *if such exists, where, as usual, fl is the orientation reversing map on G x Rn. We thin* *k of "ff+as preserving orientation, and "ff-as reversing it. If V H 6= 0, then both * *maps occur. If V H = 0, then only one of them occurs. If W K 6= 0 and therefore V H * *6= 0, then S VG (n) has a map m+ :v -! w. If W K = 0 and V H 6= 0, then S VG (n) has maps m+ :v -! w+ and m- :v -! w- . Finally, if V H = 0, then if "ff+exists, there are maps m+ :v+ -! w+ and m+ :v- -! w- , while if it is "ff-that exists, there are maps m- :v+ -! w- and m- :v- -! w+ . Whenever composites are defined, they satisfy the generic rules m+ O m+ = m+ ; m+ O m- = m- ; m- O m+ = m- ; and m- O m- = m+ : The functor ss sends any of these maps to the underlying map ff of G-orbits tha* *t is used to define it, while the functor S sends m+ to "ff+and m- to "ff-. It is ea* *sy to check that S VG (n) is a well-defined OG -groupoid and S is a well-defined func* *tor over OG . Intuitively, the last is just the observation that the composite of * *two orientation preserving maps preserves orientation, the composite of two orienta* *tion reversing maps preserves orientation, and the composite of an orientation prese* *rving and an orientation reversing map reverses orientation. The structure of (S VG (n); S) perhaps becomes clearer if we notice that the * *group of automorphisms of any object maps isomorphically to the group of automorphisms of G=H, and that S maps the automorphism over any ff: G=H -! G=H to "ff+. One only encounters orientation reversing maps between representations that do not contain trivial summands when a change of subgroups is involved. However, there is no getting around such maps by relabeling, as the following example sh* *ows. Example 12.3. Let G = H x K, where H and K are cyclic of order 3 with generators s and t. Let V be the two-dimensional representation of G on which both s and t act by rotation by 2ss=3; let W be the two-dimensional representat* *ion of G on which s acts by rotation by 2ss=3 and t acts by rotation by -2ss=3. The* *n V and W restrict to the same representation Y of H and to the same representation* * Z of K. The category S VG (2) has corresponding objects v , w , y , and z . W* *e can choose orientations so that there are maps y+ -! v+ , y+ -! w+ , and z+ -! v+ in S VG (2), but then we are forced to have a map z+ -! w- . Moreover, no relabeli* *ng will result in only orientation preserving maps in S VG (2). The following theorem validates Construction 12.2. Theorem 12.4. If G has odd order, then the representation (S VG (n); S) is the universal orientable representation. 34 S.R. COSTENOBLE, J.P. MAY, AND S. WANER Proof.It is clear from the construction that S VG (n) is replete and thus satur* *ated. By Theorem 9.5, it suffices to show that any faithful representation (E ; R) ma* *ps into (S VG (n); S). Since (E ; R) factors through a skeleton, we may assume wit* *hout loss of generality that E is skeletal and thus discrete. We must construct a functor F :E -! S VG (n) over OG and an isomorphism OE: S O F -! R over OG . Let ss-1(G=e) be the full subcategory of E containing * *all objects x with ss(x) = G=e. For any object x 2 ss-1(G=e), we must take F (x) = * *r. Choose an object x0 in each component of ss-1(G=e), and let OE(x0) be the ident* *ity map of G x Rn. This initial choice will determine F and OE. By source lifting a* *nd discreteness, there is a unique map !g;x:x -! x0 over g :G -! G for each g 2 G; divisibility implies that every x in the component of x0 is the domain of such a map !g;xand that every map in the component is a composite of such maps and their inverses. We must take F (!g;x) to be the unique map r -! r over g; we let OE(x) = idif R(!g;x) = g and OE(x) = fl if R(!g;x) = g O fl. This is the unique* * choice such that OE(x0) O S(F (!g;x)) = R(!g;x) O OE(x). This specifies the restrictio* *ns of F and OE to ss-1(G=e). Now let y be an object of E such that R(y) = G xH V with H 6= e. By source lifting and discreteness, there is a unique map ! :x -! y over the quotient map G -! G=H. In the naturality relation OE(y) O S(F (!)) = R(!) O OE(x), OE(x) and R(!) are already specified, and F (!) will be determined once we specify F (y).* * If V H 6= 0, then we must let F (y) = v. Here, if OE(x) = id and R(!) = , or if OE(x) = fl and R(!) = O fl, we must take OE(y) = id; in the remaining two case* *s we must take OE(y) = fl, the unique non-identity map of G xH V . On the other hand, if V H = 0, then OE(y) must be the identity map of G xH V . Here, if OE(x) = id and R(!) = , or if OE(x) = fl and R(!) = O fl, we must take F (y) = v+ ; in the remaining two cases we must take F (y) = v- . Finally, suppose given a map :y -! z in E over ff: G=H -! G=K, where ff(eH) = gK. Let ! :x -! y and !0:x0 -! z be the unique maps over the quotient maps G -! G=H and G -! G=K. By divisibility and discreteness, there is a unique map i :x -! x0over g :G -! G such that !0i = !. A diagram chase (comparing preservation and reversal of orientations) shows that in all cases t* *here is a unique map F (y) -! F (z) over ff in S VG (n); we can and must take F () t* *o be this map. It is then clear that F is a well-defined functor. The naturality rel* *ations for i, !, and !0 imply that the naturality relation OE(z) O S(F ()) = R() O OE(* *y) holds when precomposed with S(F (!)) and therefore holds as written. In the proof just given, we could instead have chosen OE(x0) = fl. This would force changes from OE(y) = idto OE(y) = fl and vice-versa when R(y) = G xH V with V H 6= 0, and from F (y) = v+ to F (y) = v- and vice-versa when V H = 0. Moreover, we have an independent such choice for each component of ss-1(G=e). Said another way, making a choice for each component fixes a saturation of (E ;* * R), and making the opposite choice in each component gives an isomorphic saturation. Proposition 9.7 gives the following interpretation. Corollary 12.5. Let G have odd order and let (E ; R) be a faithful representati* *on in VG (n) such that the subcategory ss-1(G=e) of E is connected. (i)The group (S VG (n); S) is cyclic of order two. (ii)(E ; R) has a unique isomorphism class of saturations. EQUIVARIANT ORIENTATION THEORY 35 (iii)(E ; R) has exactly two orientations, and an orientation (F; OE) is dete* *rmined by OE(x0) for any chosen object x0 over G=e. This has the following implication for orientations of G-vector bundles. Corollary 12.6. If G has odd order and B=G is path connected, then an orientable G-vector bundle p over B admits exactly two orientations, and an orientation of* * p is completely determined by a choice of nonequivariant orientation of the restr* *iction of p to any path component of B. Thus our equivariant orientations encode information implicit in nonequivaria* *nt orientations. The following complementary observation generalizes and makes pre- cise the folklore result (see for example [11, p. 16]) that, for actions of odd* * order groups, nonequivariant orientability implies equivariant orientability. Theorem 12.7. Let G be finite of odd order. A G-vector bundle p : E -! B is equivariantly orientable if and only if it is nonequivariantly orientable. * *The equivariant orientations of p are in bijective correspondence with the nonequiv* *ariant orientations on which G acts by orientation preserving maps. Proof.Clearly equivariant orientability implies nonequivariant orientability. S* *up- pose that p is nonequivariantly orientable. Let (; id) : x -! x be a map in G B and thus in (BH ). Regarding as a path in B and working nonequivariantly, we have p*() = idby the nonequivariant orientability of p. By Lemmas 2.3 and 12.1, we conclude that p*(; id) = idequivariantly. By Remark 2.9, this implies that p* * is equivariantly orientable. The second statement follows from the first together * *with Lemma 11.4 and Corollary 12.6. In Definition 11.2 we defined naive orientations of V -dimensional G-bundles * *for any G. Using our explicit description of S VG , we can define naive orientation* *s of arbitrary G-bundles when |G| is odd. The notions coincide when both are defined. Definition 12.8. Let I VG be the subcategory of S VG obtained by omitting the objects v- and the maps to and from them. Deleting subscripts + from the notation, the category I VG has an object v = v(H) for each G xH V and a map m: v -! w over ff: G=H -! G=K if there is an orientation preserving G-bundle map G xH V -! G xK W over ff. Let I :I VG -! VG be the restriction of S. A naive orientation (F; OE) of a G-bundle p: E -! B is a functor F :G B -! I VG over OG and an isomorphism OE: I O F -! p* over OG . Remarks 12.9. We compare naive orientations and orientations for |G| odd. (i)Clearly, we may view a naive orientation (F; OE) as a restricted kind of orientation. Therefore, when p admits a naive orientation, its orientati* *ons and naive orientations determine one another. (i)We can interpret Example 12.3 as showing that I VG need not be a groupoid over OG because it need not satisfy the source lifting property. Thus it* * is only the genuine orientations that fit naturally into our definitional f* *rame- work. (ii)The full subcategories of I VG and S VG with one object v = v(H) for each G xH V such that V H 6= 0 coincide, hence naive orientations and or* *i- entations coincide for G-bundles all of whose fiber representations cont* *ain trivial summands. In particular, this holds stably. 36 S.R. COSTENOBLE, J.P. MAY, AND S. WANER Remark 12.10. For a general compact Lie group G, there is an essentially similar generalization of Construction 12.2 that applies when VG is replaced by its full subcategory whose objects are those G xH V such that the group of H-linear isometries of VH is connected, where VH is the orthogonal complement of V H. Th* *at is, we only take the representations Rq W such that W contains no irreducible summands of real type, only summands of complex and quaternionic type. 13.Abelian compact Lie groups We explore our theory for Abelian compact Lie groups G and give a conjectural description of universal orientable representations when G is an elementary Abe* *lian 2-group (Z=2)k. More precisely, we first construct a replete, hence saturated, * *rep- resentation (S (n); S) for any Abelian compact Lie group G and any n 1. For G = (Z=2)k, we then show that (S (n); S) is universal for replete representatio* *ns. Thus it is universal if Conjecture 8.6 is true for R = VG (n). However, Remark * *8.7 is relevant, and we doubt that the conjecture is true even in this simple case. Since G is Abelian, The orbit category OG has a very simple form. The function that sends gH to the G-map g : G=H -! G=H is an isomorphism of Lie groups G=H -! OG (G=H; G=H): We regard it as an identification. We have a subcategory QG of OG whose objects are the orbits G=H and whose morphisms are the quotient maps q : G=H -! G=K associated to inclusions H K. There is a map ff : G=H -! G=K in OG if and only if H K, and then ff is the composite of q and a map g : G=H -! G=H. The relations in the category are generated by the obvious commutative diagrams G=H __g__//G=H q|| q|| fflffl| fflffl| G=K __g__//G=K: The skeletal category VG (n), n 1, also admits a simple description since G * *is Abelian, as we see by use of Lemmas 2.3 and 2.6. Consider the set of maps ! : G xH V -! G xK W: This set is non-empty when H K and there is an H-linear isometry : V -! W . Any other H-linear isometry V -! W is a composite of and an H-linear isometry V -! V . We deduce that any map ! is the composite of a self-map of GxH V and the map "q: G xH V -! G xK W over q : G=H -! G=K that is obtained from by passage to orbits. Note that ss0(OH (V )) = Aut(G xH V ) and, by Proposition 5.4, a map ! : G xH V -! G xK W induces a restriction homomorphism ss0(OK (W )) -! ss0(OH (V )): Taking ! = "q, this map is the evident composite r : ss0(OK (W )) -! ss0(OH (W )) ~=ss0(OH (V )): It is independent of the choice of and thus of the choice of "qover q since in* *ner H-linear automorphisms of V are homotopic to the identity. Any self-map of G xH V is a composite of a map over the identity map of G=H induced by an element of ss0(OH (V )) and the map given by multiplication by g, which covers g : G=H - ! G=H. These two types of maps commute with one EQUIVARIANT ORIENTATION THEORY 37 another. An element h 2 H gives an H-linear isometry h : V - ! V . By passage to homotopy, this gives a homomorphism V : H -! ss0(OH (V )). Via Lemma 2.3, we deduce an isomorphism of groups VG (G xH V; G xH V ) ~=G xH ss0(OH (V )); where H acts on ss0(OH (V )) via V . Moreover, viewing G as G xH H, we see that V extends to a homomorphism "V : G -! VG (G xH V; G xH V ) that induces a map of extensions p 1 ________//_H________________//G_________________//_G=H_________//_1 V || |"V| |~=| fflffl| fflffl| fflffl| 1 ____//_ss0(OH (V_))_//VG (G xH V; G xH V_)ss//_OG (G=H; G=H)___//1; where p : G -! G=H is the quotient homomorphism. Now let E be a discrete groupoid over OG and consider a representation R of E in VG (n). We consider general features of the functor R : E -! VG (n). Let x be an object of E with ss(x) = G=H and R(x) = G xH V . Since E is faithful, ss : E (x; x) -! OG (G=H; G=H) ~=G=H is an inclusion. We are especially interested in the case when ss is an isomorp* *hism, so that E (x; x) ~=G=H. Let Gx = p-1E (x; x) G and Vx = ss-1E (x; x) VG (R(x); R(x)): The map of extensions displayed above restricts to a map of extensions p 1_________//H________//_Gx___//_E (x;_x)__//1 V || |"x| |||| fflffl| fflffl| || 1_____//ss0(OH (V_))_//_Vx_ss//_E (x;_x)_//_1: Since ss O R = ss, the bottom extension is split by R : E (x; x) -! Vx. Define aex : Vx -! ss0(OH (V )) by aex(!) = ! . R(ss(!))-1. Then the homomorphism ox = aex O "x: Gx -! ss0(OH (V )) extends V . We can turn this observation around. Clearly Vx ~=Gx xH ss0(OH (V )); and it follows that an extension ox : Gx -! ss0(OH (V )) of V determines a spli* *tting R : E (x; x) -! Vx. Explicitly, R is given by R(q(g)) = (g; ox(g)-1) for g 2 Gx. Looked at another way, R gives the inclusion of the first coordinate of a split* *ting Vx ~=E (x; x) x ss0(OH (V )) under which "x has coordinates p and ox. We now consider maps between distinct objects of E . Taking x as above, let ! : x -! y be a map such that ss(!) = ff : G=H -! G=K and R(y) = G xK W . Then H K and there is an H-linear isomorphism : V - ! W . By divisibility, for any map : y -! y, there is a map : x -! x such that O ! = ! O covering each map : G=H - ! G=H such that ff O = ss() O ff. In particular, when 38 S.R. COSTENOBLE, J.P. MAY, AND S. WANER H = K and thus V = W , we see that the existence of a map x -! y implies that Gx = Gy and ox = oy. When H 6= K, unique source lifting (see Remark 6.3(i)) and divisibility imply that each factorization ff = q O , : G=H -! G=H, is uniquely covered by a factorization ! = "qO ". That is, ! is a composite of a map ": x -* *! x0 over and the unique map "qy: x0 -! y with target y over q : G=H -! G=K. When E (x; x) ~=G=H, Lemma 8.2(ii) implies that x0= x, so that every morphism x -! y is a composite of a self-map of x and a canonical map "qy: x -! y. In particular, (E ; R) is replete in the sense of Definition 8.4 when E (x; x) ~=G* *=H for all H and all x over G=H. In general, since for any map : y -! y there is a map : x -! x such that O "q= "qO , we see that Gy Gx. By a diagram chase from the functoriality of R, we find that the following diagram commutes: Gy _____________//Gx oy|| |ox| fflffl| fflffl| ss0(OK (W ))r__//ss0(OH (V )): Thus we can think of ox as an extension of r O oy. When (E ; R) is replete, we conclude that the functor R is determined by maps R("q) that give a functor over QG OG together with homomorphisms ox that extend the V and make these diagrams commute. The point is that the commutativity of these diagrams ensures that relations Oq"= "qO as above are carried to relations R()OR("q) = R"q)OR(). This gives an inductive description of replete representations. We can describe maps (F; OE) : (E ; R) -! (F ; S) between replete representat* *ions in terms of these data. Since VG (n) is skeletal, for there to be an isomorphi* *sm OE : S O F -! R over OG , we must specify F on objects so that S(F (x)) = R(x) * *for all x 2 E . Since (F ; S) is replete, we must then specify F on non-empty morph* *ism sets to be the composite -1 E (x; y)ss_//OG (ss(x);_ss(y))ss//_F (F (x); F (y)): To have SOF = R on sets of endomorphisms, we must have ox = oF(x)for all objects x of E . Finally, we must choose isomorphisms OE : S(F (x)) = R(x) -! R(x) that make the evident naturality diagrams over QG commute: (13.1) S(F (x))_OE__//R(x) S(F("q))|| |R("q)| fflffl| fflffl| S(F (y))OE__//R(y): Here R("q) = S(F ("q)) O q for some q : R(x) -! R(x), and the diagram can be rewritten by replacing the top arrow with qO OE and the right arrow with S(F (* *"q)). This relates the construction of the maps OE to the behavior of restriction maps r : ss0(OK (W )) -! ss0(OH (W )). The following observation is relevant. Proposition 13.2. Let K be a subgroup of G that is not topologically cyclic and let W be a K-space. (i)Restriction maps induce an isomorphism ss0(OK (W )) -! limHss0(OH (W )), where the limit is taken over the proper subgroups H K. EQUIVARIANT ORIENTATION THEORY 39 (ii)Assume given a homomorphism oH : G -! ss0(OH (W )) for each proper subgroup H of K such that oH extends V : H -! ss0(OH (V )) and oH restricts to oJ, oJ = rHJO oH , when J H. Then there is a unique homo- morphism oK : G -! ss0(OK (W )) that extends K and restricts to oH for each H K. Proof.Let SOH (W ) be the identity component of OH (W ). Since the closure of the subgroup generated by any element of K is a proper subgroup, we have OK (W ) = \H OH (W ) = limHOH (W ) and SOK (W ) = \H SOH (W ) = limHSOH (W ): Here ss0(OK (W )) ~=limOH (W )=SOK (W ) since ss0(OK (W )) = OK (W )=SOK (W ). Since lim1 vanishes on countable systems of compact groups, the countably many short exact sequences 1____//_SOH (W )=SOK (W_)__//OH (W )=SOK (W_)__//_ss0(OH (W_))//_1 give an exact sequence on passage to limits. This gives (i), and (ii) follows b* *y letting oK be the homomorphism obtained from the oH by passage to limits. Of course, we can replace the OH (W ) in this result by the spaces of H-linear isometries VH -! W for any choices of VH that are H-isomorphic to W . We now construct the promised replete representation (S (n); S). Construction 13.3. Let S (n) have objects all pairs x = (GxH V; ox), where H is* * a (closed) subgroup of G, V is an H-space such that GxH V is an object of VG (n),* * and ox : G -! ss0(OH (V )) is a homomorphism that extends V : H -! ss0(OH (V )). The functors ss : S (n) -! OG and S : S (n) -! VG (n) send x to G=H and to G xH V . Let y = (G xK W; oy). There are no morphisms x -! y unless H K, V is H-isomorphic to W (without a specified choice of isomorphism), and oy restri* *cts to ox. When these conditions hold, we require ss : S (n)(x; y) -! OG (G=H; G=K) to be a bijection. Composition is dictated by the functoriality of ss. In parti* *cular, ss : S (n)(x; x) -! OG (G=H; G=H) ~=G=H is an isomorphism, Gx = G, and the discussion above shows that the homomor- phism ox determines a homomorphism S : S (x; x) -! VG (n)(S(x); S(x)) such that ss O S = ss. These homomorphisms specify S on self-maps in S (n). For each object y = (G xK W; oy) and each proper subgroup H of K we have a unique H-space VH such that G xH VH_is_in VG (n) and VH is H-isomorphic to W . The retraction equivalence from V G(n) to VG (n) chosen in Definition 2.2 fixes a c* *hoice of isomorphism H : VH - ! W . Let y|H denote the object (G xH VH ; rKHO oy). We define S on the unique morphism "q: y|H - ! y that covers the quotient map q : G=H -! G=K to be the map S(y|H ) -! S(y) determined by H and passage to orbits. Clearly this specification of maps is functorial over QG . By the discu* *ssion above, this completes the specification of the functor S. It is immediate that (S (n); S) is replete, hence saturated. Theorem 13.4. Let G = (Z=2)k. Then every replete representation maps into (S (n); S). Therefore (S (n); S) is a universal orientable representation in VG* * (n) if and only if Conjecture 8.6 holds for R = VG (n). 40 S.R. COSTENOBLE, J.P. MAY, AND S. WANER Proof.The second statement follows from the first by Theorem 9.5(vi) and Propo- sition 9.8. To prove the first, let (E ; R) be replete. We must construct a m* *ap (F; OE) : (E ; R) -! (S (n); S). By the discussion above, the construction is l* *argely tautological. For an object x of E with ss(x) = G=H and R(x) = G xH V , we must define F (x) = (R(x); ox) on objects x, and it remains only to specify isomorph* *isms OE : S(F (x)) = R(x) -! R(x) that make the diagrams (13.1) commute. We proceed by induction on the rank of H. If H = {e}, then R(x) = G x Rn and we choose OE arbitrarily. Suppose next that H is of rank one. There is a unique map "q: x0- * *! x over q : G -! G=H. Since r : ss0(OH (V )) -! ss0(O(n)) is an epimorphism (see Example 10.2(ii)), we can choose OE : G xH V - ! G xH V making the relevant diagram (13.1) commute. Now assume that H is of rank s and OE has been specified for all objects over all orbits G=J with rank J < s. We deduce from Proposition 13.2(i) that there is a unique way to specify OE : S(F (x)) -! R(x) such that a* *ll of the relevant diagrams (13.1) commute. 14. The universal orientable representation S VD6(2) We give one non-Abelian even order example to illustrate ideas. Let G = D6 be the dihedral group of order 6. It contains a normal subgroup K of order 3 and three conjugate subgroups H1, H2, H3 of order 2. Let J = G=K. Schematically, with doubleheaded arrows indicating conjugations, the orbit category OG looks l* *ike _ @AGF|EDfflffl| G=e GSSSWWWWW xx GGGSSSSSSWWWWWWWWW _ xxx GGG SSSSSSWWWWWWWW @AGF|E--xxxDfflffl| ##G SSS)) WWWWWWW++ G=K F G=H1 oo___//G=H2koo__//G=H3:ggg FF www kkkkkkggggggggg FFF wwwkkkkkkggggggg F##F --wwuukkkkkssggggggggg G=G The group G has three irreducible real representations, namely the trivial repr* *esen- tation R, the representation L obtained by pullback of the non-trivial irreduci* *ble representation of J, and a representation V whose restriction to K is isomorphi* *c to the standard representation of K on C. The restriction of V to each Hi is the s* *um of a trivial and a non-trivial representation. We describe S VG (2), which has features of both cases G = Z=2 and G = Z=p from x10. Recall from Definition 11.1 that an n-dimensional representation V of G determines the full subgroupoid VG (V ) of VG (n) with objects isomorphic to * *the G xH V . Here G xH V is isomorphic to G=H x V since G acts on V . The reader will appreciate that examples where subgroups have representations that are not restrictions of representations of G will be substantially more complicated. We have the four two dimensional representations R2, R L, L2, and V, the first three of which are trivial as representations of K. The elements of orde* *r 2 act in an orientation preserving way on R2 and L2 and in an orientation reversi* *ng way on R L and V. The category VG (2) has two components. For the first three representations U, there is a copy of the orbit category in S VG (2) that maps * *to the subcategory of VG (2) that is depicted schematically as follows, where down* *ward EQUIVARIANT ORIENTATION THEORY 41 pointing arrows restrict to the identity map on representations. __ @AGFEDfflffl| pG x R2MMVVVVVYYYYYYYYY ____ ppp MMMMVVVVVVVVYYYYYYYYYYYY ppp MMMM VVVVVV YYYYYYYY @AGF|EDfflxxpppffl| M&& VV++V YYYYYYYY,, G=K x R2 G=H1 x U oo__//_G=H2 x Uoo__//G=H3exeU: MM qq hhhhh eeeeeeee MMM qqqq hhhhhh eeeeeeeee MMM qqq hhhhheeeeeeee MM&& xxqqsshhhhhrreeeeeeeee G=G x U One component is generated by the two subcategories with U = R2 and U = L2, with the parts of the two categories with objects labelled by R2 rather than by U identified; in this component, all maps by elements of order 2 are orientation preserving; that is there is no twisting by maps fl of degree -1 on representat* *ions. The other component is generated by the subcategory with U = R L and another subcategory that maps to the subcategory of VG (2) generated by V in a manner that we depict schematically as follows: __ @AGFEDfflffl| G x R2 UU " AAAPPPPPUUUUU """ || AAA PPPPUUUUUUU """"fl| AAA PPPPP UUUUUUU "" | AAA PPPPP UUUUUUU ___ """ ___ || AAA PPPPP UUUUUUU @AGF|EDf"""""flffl|@fflffl|AGF|EADfflffl| PP(( UUU** G=K x@V G=K x V@ G=H1 x V oo___//G=H2 x Voo___//G=H3 x V: nn " iiinn @@@ @@@@ """" fl| nnnn""fl" iiiiiinnnn @@@ @@ "" |nnnnn ""iiiiiinnnnfln @@ @@@"""nn||nn i"""iiiiinnnn @@@ """ @@@nnn|nni"""iiiiiinnnnn @@@ ""nnnnii@@ii|ii""nnnn @OO """"wwnntt@iifflffl|""""ivvnn G=G x V G=G x V The copies of G x R2 in these two subcategories are identified, and all elements of order 2 act in an orientation reversing way on the resulting object and also* * on G=K xR2 in the subcategory for U = RL and on the two copies of G=K xV in the last subcategory (compare Example 10.2(d)). Vertical arrows involve twistings fl as indicated by the labels of arrows. It is clear that this does specify a well* *-defined groupoid S VG (2) over OG together with a representation S : S VG (2) -! VG (2). This representation is replete, hence saturated, and it is not hard to check as* * in x10 that it is universal. Part IV. Refinements and variants of the theory 15. Fibrations over B and fibrant representations The analogy between topological and categorical homotopy theory, in particular the categorical notion of a fibration, is illuminating to the categorical repre* *sentation theory of Part II. Although we shall not make essential use of this material, we 42 S.R. COSTENOBLE, J.P. MAY, AND S. WANER here show how to use fibrations to express orientations in terms of strict maps* * of representations. To begin with, we return to the framework of x5. With I as in Example 9.6, an isomorphism over B between functors E -! F over B can be regarded as a map j :E x I - ! F . In fact, B x I is a base category, E x I is a groupoid over B x I , and j is a functor over the projecti* *on B x I -! B. Here we must take B x I and not B as the base of E x I since otherwise the source lifting and divisibility properties would fail. Using thi* *s, we can mimic topological definitions and constructions. Definition 15.1. Let F and R be groupoids over B. A functor S :F -! R over B is a fibration over B if it satisfies the categorical CHP: for maps R: E - ! R and F :E -! F of groupoids over B and an isomorphism OE: S O F -! R over B, there is a functor F 0:E -! F over B such that R = S O F 0and an isomorphism "OE:F -! F 0over B such that OE = S O "OE. In terms of diagrams of functors, th* *is takes the following familiar form, in which R = OE O i1 and F 0= "OEO i1: E __F___//_F;; "OEww i0|| w w |S| fflffl|w fflffl| E x I __OE//_R: Since objects in the category of representations in R are maps into R in the category of groupoids over B, we think of fibrations as fibrant representations. Definition 15.2. A representation (F ; S) in R is fibrant if S :F -! R is a fib* *ra- tion over B. This means that, for any map (F; OE): (E ; R) -! (F ; S) of repres* *en- tations, there is a strict map F 0:(E ; R) -! (F ; S) and a homotopy "OE:F -! F* * 0. As in topology, a functor over B is a fibration if and only if it satisfies t* *he path lifting property (PLP). Let RI be the functor category of maps I - ! R over identity maps of B. Its objects are maps = (I): (0) -! (1) in R, and must be an isomorphism since ss() = id. A map (ae0; ae1): -! i is a pair of ma* *ps aei:(i) -! i(i) such that the evident diagram commutes. We have the pullback F xR RI of S :F - ! R and p0: RI -! R. The universal property gives a functor : F I- ! F xR RI whose projections are p0: F I- ! F and SI :F I- ! RI . Lemma 15.3. A functor S :F -! R is a fibration if and only if it satisfies the categorical PLP: there is a functor : F xR RI - ! F I such that O = Id. This criterion and the path lifting property for (Hurewicz) fibrations of G-s* *paces imply the following relationship between topological and categorical fibrations. Proposition 15.4. If p: E -! B is a fibration of G-spaces, then G p: G E -! G B is a fibration of groupoids over B. We can replace general functors over B with equivalent fibrations over B. EQUIVARIANT ORIENTATION THEORY 43 Construction 15.5. Let R: E - ! R be a functor over B. We construct the associated fibration R0:E 0-! R via the following commutative diagram, in which the right square is a pullback of categories and the lower triangle specifies R* *0: E ___J__//E_0P___//E 0 ---| | R ||R--- | |R fflffl|""--fflffl|fflffl| R op1o_RI _p0_//_R: The objects of E 0are the pairs (x; ), where x is an object of E and is an isomorphism with (0) = R(x) and ss() = idss(x). A map (!; ae1): (x; ) -! (y; i) consists of maps ! :x -! y in E and ae in R such that (R(!); ae1) is a map -! * *i; that is, R(!) = i-1ae1. The functor J sends x to (x; id) and ! to (!; id). It* * is straightforward to verify the following claims. (i)E 0is a groupoid over B, so that (E 0; R0) is a representation in R. (ii)The map R0:E 0-! R is a fibration over B, so that (E 0; R0) is fibrant. (iii)R = R0OJ, so that J :(E ; R) -! (E 0; R0) is a strict map of representat* *ions, and :R O P -! R0is a map over B, so that (P; ): (E 0; R0) -! (E ; R) is a map of representations, where (x; ) = :(R O P )(x; ) = R(x) = (0) -! (1) = R0(x; ): (iii)P O J = Id:E -! E and the maps (id; ): (x; id) -! (x; ) specify a homotopy J O P -! Id, so that the representation (E ; R) is a deformation retract of the fibrant representation (E 0; R0). In view of Corollary 8.9 and the categorical CHP, we have the following alter* *na- tive description of orientations. Recall Definition 7.1(iii). Corollary 15.6. Let (S R; S) be the universal orientable representation and let (S 0R; S0) be an equivalent fibrant representation. Then (S 0R; S0) is supersat* *urated (but not saturated), and orientations of a representation (E ; R) are in biject* *ive correspondence with homotopy classes of strict maps F : (E ; R) -! (S 0R; S0). 16. Functoriality of universal orientable representations We have started with a fixed skeletal groupoid R over B. There are variants of our theory that deal with other types of G-bundles and with G-fibrations, and s* *till other variants that deal with special types of G-bundles, for example V -dimens* *ional ones for a fixed representation V of G. To make comparisons, we must understand the functoriality of our constructions with respect to changes of R. Proposition 16.1. Let ae: R -! R0 be a functor over B, where R and R0 are skeletal groupoids over B. (i)There is a unique map "ae:"R-! R"0of universal discrete representations covering the map ae, D0O "ae= ae O D. (ii)There is a map oe :S R -! S R0 of universal orientable representations covering ae; it is unique up to isomorphism and can be chosen to cover "* *ae. (iii)If ae0:R0 -! R00is another functor over B, then we can choose a map oe0:S R0- ! S R00covering ae0so that oe0O oe covers ae0O ae. 44 S.R. COSTENOBLE, J.P. MAY, AND S. WANER (iv)By composition with oe, an orientation (F; OE) of a representation (E ; * *R) in R induces an orientation of (E ; R) in R0: E _____F______//_@@S_R___oe_____//zSx0R0 @@ zzz xxx R @@OO@""zSzzzae --xS0xxx R _______________//DDR0 DD yyyy ssDD""DD__ssyyyy B: Proof.The essential point is that the notions of skeletal, discrete, and faithf* *ul representations are specified entirely in terms of underlying functors over B. * *Thus (i) is immediate from the universal property of D0:R"0 -! R0. For (ii), it is convenient to apply the characterization of Theorem 9.5(i). We take S = D O I, where I :S R -! "R is the inclusion of a maximal saturated subrepresentation. The image of "aeO I is a saturated subrepresentation of R"0. We expand the image to obtain an inclusion I0:S R0- ! "R0of a maximal saturated subrepresentation, and we take S0= I0O D0. With this construction, oe is given by "aeO I and we ha* *ve S0O oe = ae O S. This can be interpreted as giving a strict map of representati* *ons in R0. Taking this construction as the starting point for the analogous constructi* *on of oe0, part (iii) is clear. Part (iv) is also clear. To study change of groups or to restrict to the study of G-spaces of restrict* *ed isotropy types, we must also change the base category B. Proposition 16.2. Let A and B be base categories, let : A -! B be a con- tinuous functor, and let ss :R -! B be a groupoid over B. Define *R to be the pullback displayed in the diagram *R ____//_R || |ss| fflffl| fflffl| A _____//_B: Then *R is a groupoid over A that is skeletal or discrete if R is skeletal or d* *iscrete. If is a faithful functor and R is faithful over B, then *R is faithful over A . Proof.It is elementary to check that : *R -! A satisfies conditions (i)-(iv) of Definition 5.1. Note for this that the fibers of are copies of fibers of ss, * *a fact that also explains why *A is skeletal or discrete if ss is so. The last stateme* *nt is a standard property of pullbacks of categories. Remark 16.3. Assume the hypotheses of the proposition. (i)Lemma 6.4 implies that if R is discrete, then *R is discrete and therefo* *re faithful. This holds whether or not is faithful. (ii)If is an injection, then *R just restricts R to the subcategory given by the objects and maps that ss sends to objects and maps in the image of A* * . EQUIVARIANT ORIENTATION THEORY 45 (iii)Suppose given a groupoid Q over A and a commutative diagram ae Q _____//_R || |ss| fflffl||fflffl A _____//B: Then ae factors uniquely through a functor _ae:Q -! *R over A . If R and Q are skeletal over B, then *R is skeletal over A and Proposition 16.1 applies to compare the universal orientable representations of Q and *R. 17. Orientations and change of groups So far, we have restricted attention to a fixed ambient group G. Various natu* *ral constructions on equivariant bundles lead one to consider what happens on passa* *ge to subquotient groups. For example, let H G, let NH be the normalizer of H in G, and let W H = NH=H. For a G-bundle p: E -! B, the restriction p|BH is an NH-bundle that has a subbundle pH :EH - ! BH , which is a W H-bundle, together with a complementary NH-subbundle pH , so that pH pH ~=p|BH . We here describe how orientations of p give rise to orientations of these related * *bundles. In particular, if a smooth G-manifold is oriented, then its orientation, which * *is an orientation of its tangent G-bundle, determines an orientation of the W H-manif* *old MH and an orientation of the normal NH-bundle of the inclusion of MH in M. Note that the dimensions of fixed`point bundles can vary over components and recall that we write VG for VG (n), and similarly for S VG . We first consider subgroups and then consider quotient groups, making heavy use of the observations of the previous section. Let i: H -! G be an inclusion. We have the functor i*: OH -! OG given by i*(H=K) = G xH (H=K) ~=G=K on objects and i*(ff) = G xH ff on morphisms. For a groupoid ss :E -! OG over OG , let i*E denote the pullback of E along i*. By Proposition 16.2, i*E is a groupoid over OH that is skeletal, fait* *hful, or discrete over OH if E is skeletal, faithful, or discrete over OG . The funct* *or i* is an injection, and i* just restricts E to those orbits G=K such that K is a subg* *roup of H. We have the following observations. Proposition 17.1. For a G-space X, i*G X is isomorphic over OH to H X. For any n, i*VG (n) is isomorphic over OH to VH (n). Proof.We do mean isomorphism and not just equivalence of categories over OH . The result for fundamental groupoids is clear since, for K H, a K-fixed point of X can be viewed as either an H-map H=K -! X or a G-map G=K -! X, and similarly for paths. We obtain inverse isomorphisms between our categories * *of G-bundles by extending H-bundles H xK V to G-bundles GxK V and restricting G- bundles GxK V to their H-subbundles HxK V . We are using the skeletal categories specified in Definition 2.2, and the respective composites are easily verified * *to be identity functors. Proposition 17.2. Let p: E -! B be a G-bundle and let p|H denote p regarded as an H-bundle. The representation (p|H)*: H B -! VH is isomorphic to i*p*: H B ~=i*G B -! i*VG ~=VH : 46 S.R. COSTENOBLE, J.P. MAY, AND S. WANER Proof.Again, we mean isomorphism and not just equivalence of representations. With the notations of Definition 7.1(iv), we set F = F 0= Id:H B -! H B and seek an isomorphism OE: (p|H)* -! i*p* of functors H B -! VH over OH ._Recall the proof of Proposition 2.7, remembering the use of the equivalence V G -! VG described in Definition 2.2. For a morphism (!; ff): x -! y of H B, ff: H=K -! H=L, (p|H)*(!; ff) is the composite H(x)-1 * "!1 *H(y) H xK V ____//_(p|H) (x)__//(p|H) (y)__//H xL W; where H (x) and_H (y) are two of the chosen isomorphisms used in defining the equivalence V H -! VH and "!1is obtained from the H-bundle CHP. We may extend x: H=K -! B to a G-map G=K -! B, and similarly for y, and we may view (!; ff) as a morphism in G B. Then "!1in our description of (p|H)*(!; ff) * *can be taken to be the restriction of a map "!1obtained by use of the G-bundle CHP. The map i*p*(!; ff) is the restriction to H-subbundles of the composite G(x)-1 * "!1 * G(y) G xK V ____//_p (x)__//p (y)__//_G xL W Here G (x) and_G (y) are two of the chosen isomorphisms used in defining the equivalence V G - ! VG . The required isomorphism OE is specified by OE(x) = G (x)H (x)-1, where G (x) is restricted to the canonical H-subbundles. The point is that we obtain an isomorphism of representations even though we must make independent choices of the G (x) and H (x). Let (S VG ; SG ) denote the universal orientable representation in VG . The r* *ep- resentation (i*S VG ; i*SG ) in VH is not universal, but it is orientable. Ther* *efore it admits an orientation (17.3) (FHG; OEGH): (i*S VG ; i*SG ) -! (S VH ; SH ): We think of (FHG; OEGH) as specifying the universal way to orient the underlying H-bundle of an oriented G-bundle. That is, given an orientation (F; OE): (G B; p*) -! (S VG ; SG ) of a G-bundle p: E -! B, the induced orientation of p|H is the composite (17.4) (FHG; OEGH) O (i*F; i*OE): Here we use Propositions 17.1 and 17.2 to interpret the domain of (i*F; i*OE) as (H B; (p|H)*). The dimensions of bundles are unchanged under these construc- tions, so that we can work equally well with VH (n) and VG (n). To study bundles associated to a subquotient group W H = NH=H, we can use the observations above to first restrict down from G to NH. Thus, thinking of the normal subgroup H of NH with quotient group W H, we change notations and consider a normal subgroup N of a group G with quotient group J. Let q :G -! J be the quotient homomorphism. We have the functor q*: OJ -! OG given by q*J=K = G=H on objects, where H = q-1K. Since q induces an isomor- phism of G-spaces from G=H to J=K regarded as a G-space via q, we can define q* on maps by regarding a J-map of orbits as a G-map of orbits. For a groupoid ss :E -! OG over OG , let q*E denote the pullback of E along q*. By Proposition 16.2, q*E is a groupoid over OJ that is skeletal, faithful, or discrete over OJ* * if E EQUIVARIANT ORIENTATION THEORY 47 is skeletal, faithful, or discrete over OG . The functor q* is an injection, an* *d q* just restricts E to those orbits G=H such that H contains N. We also have the functor q*: OG -! OJ that sends a G-orbit G=H to the J-orbit J xG G=H ~= G=HN ~= J=K, where K = HN=N. In contrast to our previous functors between orbit categories, q* is not an injection. Observe that the composite q*q*: OG -! OG sends G=H to G=HN. The quotient G-maps fl :G=H -! G=HN specify a natural transformation fl :Id- ! q*q* of functors OG -! OG . For a groupoid F over OJ, let q*F denote the pullback of F along q*. By Proposition 16.2, q*F is a groupoid over OG that is skeletal or discrete over OG if F is skeletal or discrete over OJ . We sha* *ll be interested mainly in the case F = q*E , where E is a groupoid over OG . An obje* *ct of q*q*E is a pair (G=H; x), where x is an object of E such that ss(x) = G=HN; a morphism (G=H; x) -! (G=L; y) is a pair (ff; !), where ff: G=H - ! G=L, ! :x -! y, and ss(!) is the map G=HN -! G=LN induced by ff. For a G-space X, XN is a G-space with action factoring through J. Proposition 17.5. For a G-space X, q*G X is isomorphic over OJ to JXN and q*q*G X is isomorphic over OG to G XN . Proof.If H N and K = q(H), then XH = (XN )K . An H-fixed point of X can be viewed as either a G-map G=H - ! X or a J-map J=K - ! XN , and similarly for paths. This gives the first isomorphism. For a general subgroup H* * of G, (XN )H = XHN . An H-fixed point of XN can be viewed either as G=H together with a G-map G=HN -! X or as a G-map G=H -! XN , and similarly for paths. This gives the second isomorphism. However, q*VG is not isomorphic to VJ since q*VG is the groupoid over OJ of G-bundles over N-fixed orbit spaces and the total space GxH V of such a G-bundle need not be N-fixed. Letting K = q(H) and taking N-fixed points, we obtain the J-bundle (G xH V )N ~=J xK V N over J=K. Since this gives fixed point bundles of varying dimensions, we must work with VG and VJ rather than working one dimension at a time. Let N :q*VG -! VJ denote the functor obtained by passage to N-fixed bundles. The orthogonal com- plement VN of V N is an H-subrepresentation of V , and we also have the functor N :q*VG -! q*VG that sends G xH V to G xH VN . Finally, we shall need the functor : q*q*VG -! VG that sends an object (G=H; GxHN V -! G=HN) to the pullback GxH V -! G=H along the quotient G-map fl :G=H -! G=HN. Proposition 17.6. Let p: E -! B be a G-bundle and let pN be the complementary G-bundle to the N-fixed point J-bundle pN over BN , so that pN pN ~=p|BN as G- bundles. The representation (pN )*: JBN - ! VJ is isomorphic to the composite q*p* N JBN ~=q*G X _____//q*VG___//_VJ: 48 S.R. COSTENOBLE, J.P. MAY, AND S. WANER The representation (pN )*: G BN - ! VG is isomorphic to the composite q*q*p* * q*N * G BN ~=q*q*G X ____//_q q*VG___//q q*VG___//VG : Proof.The proof is precisely analogous to that of Proposition 17.2, but with pa* *ssage to N-fixed point bundles and to complementary G-bundles replacing restriction to H-subbundles. With an evident notation, the isomorphism OE from (pN )* to the displayed composite is given on objects x: J=K -! BN by OE(x) = G (x)N J(x), and similarly for (pN )*. The representations (q*S VG ; N Oq*SG ) in VJ and (q*q*S VG ; Oq*N Oq*q*SG ) in VG are not universal, but they are orientable. Therefore they admit orientat* *ions (17.7) (FJG; OEGJ): (q*S VG ; N O q*SG ) -! (S VJ; SJ): and (17.8) (FGN; OENG): (q*q*S VG ; O q*N O q*q*SG ) -! (S VG ; SG ): We think of (FJG; OEGJ) and (FGN; OENG) as specifying the universal way to orie* *nt the N-fixed J-bundle pN and the complementary G-bundle pN of an oriented G-bundle p. That is, given an orientation (F; OE): (G B; p*) -! (S VG ; SG ) of a G-bundle p: E -! B, the induced orientation of pN is the composite (17.9) (FJG; OEGJ) O (q*F; N O q*OE) and the induced orientation of pN is the composite (17.10) (FGN; OENG) O (q*q*F; O q*N O q*q*OE): Here we use Propositions 17.5 and 17.6 to interpret the domains of (q*F; N O q** *OE) and (q*q*F; O q*N O q*q*OE) as (JB; (pN )*) and (G B; (pN )*). 18.Variant kinds of orientations Nonequivariantly, there is only one sensible definition of an orientation of * *a vector bundle, but this is a calculational fact that does not extend to the equivariant setting. The point is that Z2 ~=ss0(O(n)) ~=ss0(P L(n)) ~=ss0(T op(n)) ~=ss0(F (n)) for all n 1, including n = 1. Nothing like this holds equivariantly. There are (at least) eight different reasonable orientation theories for G-vector bundles* * corre- sponding to the linear, piecewise linear, topological, and homotopical categori* *es and their stable variants. Similarly, there are six orientation theories for PL G-b* *undles, four for topological G-bundles, and two for spherical G-fibrations. We proceed * *to make this precise, the crucial point being that our categorical framework is su* *f- ficiently general to set up and compare all of these variants with little addit* *ional work. The following diagram displays the skeletal groupoids over OG that serve * *as the target categories for the relevant representations. EQUIVARIANT ORIENTATION THEORY 49 VG (n)_____//PLG (n)_____//_TopG (n)____//FG (n) (18.1) || || || || fflffl| fflffl| fflffl| fflffl| sVG (n)____//_sPLG (n)____//sTopG (n)___//sFG (n) We shall write Cat as shorthand for any one of V , PL ,_Top,_or F . To form the top row of (18.1), we take skeleta of the categories CatG(n) of G-vector bu* *ndles over orbits, with morphisms being, respectively, the homotopy classes of maps of G-vector bundles, PL G-bundles, topological G-bundles, and spherical G-fibratio* *ns. In the last case, we identify objects with their fiberwise one-point compactifi* *cations and understand maps to mean maps that preserve the resulting section. We are requiring our fibers to be equivalent, in the appropriate sense, to linear repr* *esenta- tions since we are interested in locally linear bundle and fibration theories. * *There are further variants in which more general fibers are allowed, for example based spaces homotopy equivalent to spheres in the fibration case. In all four cases,* * there is an analogue of Lemma 2.3 that gives a complete description of maps in CatG (* *n) in terms of the homotopy theory of CatG (n)-maps between representations. The source lifting and divisibility properties required of a groupoid over OG are e* *asily checked for these categories. The arrows of the top row of (18.1) are obtained * *by neglect of structure, together with choices of isomorphisms of representative o* *bjects as we pass from more rigid to less rigid types of bundles. The bottom row of (18.1) is the stabilization of the top row. To control the relevant colimits, we let U be the direct sum of countably many copies of each * *irre- ducible representation of G; that is, U is a "complete G-universe". By a subspa* *ce of U, we understand a finite dimensional sub inner product space. Observe that U is also a complete H-universe for any H G since any representation of H extends to a representation of G on a possibly larger vector space. We stabilize over U* * as follows. Consider objects G xH V and G xK W in CatG (n). A stable map between these objects over a given base map ff: G=H -! G=K is the image in the colimit of a homotopy class of CatG (n)-bundle maps "ff:G xH (V Z) -! G xK (W Z) over ff, where Z is a K-subspace of U. The maps of the colimit system are given by "suspension": for Z Z0, we suspend "ffto a map G xH (V Z0) -! G xK (W Z0) by taking its product with the identity map of the trivial bundle Z0-Z -! *, wh* *ere Z0- Z is the orthogonal complement of Z in Z0. We emphasize that denotes_ external direct sum in the previous two displays. Using_the_objects of CatG(n) * *but replacing maps by stable maps, we obtain a groupoid sCatG (n) over OG . Taking a skeleton, we obtain the category sCatG (n). Choosing isomorphisms between objec* *ts of CatG (n) and the chosen representatives of their stable equivalence classes * *and sending maps to their stable equivalence classes, we obtain a functor CatG (n) * *-! sCatG (n). We obtain the arrows in the bottom row of (18:1) similarly. The four squares then commute up to natural isomorphisms, which are again determined by the chosen isomorphisms between representative objects. The CatG (n) analogs of Proposition 2.7 are valid. A CatG (n) G-bundle or fib* *ra- tion p: E -! B determines a functor p*: ssG (B) -! CatG (n) over OG . The proof 50 S.R. COSTENOBLE, J.P. MAY, AND S. WANER is based on the appropriate bundle covering homotopy property. Using the functor CatG (n) -! sCatG (n), we obtain the stable analogue. Definition 2.8 then gives us eight notions of orientability corresponding to these eight choices for the * *target category. Similarly, Theorem 7.5 gives us eight universal orientable representa* *tions. If R denotes any of the eight target categories, we have a corresponding univer* *sal orientable representation (S R; S). By varying the target groupoid, and so the corresponding universal representation, Definition 7.9 gives us the eight diffe* *rent notions of an orientation of a G-vector bundle, and similarly for PL G-bundles * *and so on. By Proposition 16.1, if ae: R -! R0 is any one of the functors over OG displayed in (18.1), then there is a map oe :S R -! S R0 of groupoids over OG that covers ae. This allows us to compare our various kinds of orientations. We give a few examples. Observe first that the functor VG (n) -! sVG (n) can * *be taken to be the identity on objects, since if two representations are stably eq* *uivalent, then they have the same characters and are therefore equivalent. This is implied by the fact that RO(H) is the free group on the irreducible representations, and it does not carry over to our other categories. Moreover, we see from the proof* * of Lemma 2.6 that the functor VG (n) -! sVG (n) is faithful. Example 18.2. Let G = Z=2. There are two stable self-maps of V0 and Vn in sVG (n) that are not in VG (n); that is, the exceptional cases in parts (a)(ii)* * and (c) of Example 10.2 are not exceptional stably. We see that SsVG (n) = S VG (n) and that S : SsVG (n) -! sVG (n) is the composite of S : S VG (n) -! VG (n) and the stabilization functor VG (n) -! sVG (n). The main change is that the group * *of automorphisms (SsVG (n); S) is an elementary abelian two group of order 2n+3. Example 18.3. Let G be a finite group of odd order and again consider sVG (n). The stable analogue of Construction 12.2 is similar but simpler since, stably, * *one need not consider the case V H = 0 separately. The category SsVG (n) has one object v over G=H corresponding to each bundle G xH V in VG (n). There is a map m: v -! w over ff: G=H - ! G=K whenever there is a stable G-bundle map G xH V -! G xK W covering ff; the functor S carries m to the unique stable bundle map preserving chosen orientations. While, as in Proposition 16.1, we can construct a functor oe : S VG (n) -! SsVG (n) that covers the functor ae : VG (n) -! sVG (n), it is more natural to let oe send both v+ and v- , when present, to v, and send maps labeled m, m+ , or m- to m. We then have a natural isomorphism OE: ae O SVG(n)-! SsVG(n)O oe that is the identity on objects v or * *v+ , and is the orientation reversing stable self map of G xH V on objects v- . Probably the most interesting variant of orientation theory is the one concer* *ning stable spherical G-fibrations, which fits naturally into equivariant stable hom* *otopy theory. Here the Burnside rings A(H) of the subgroups of G come into play. In f* *act, it is immediate from the definitions and the standard isomorphism between A(H) and the zeroth stable H-homotopy group of S0 that the self maps in sSphG (n) of any object G xH SV form a copy of the group of units in A(H). (A more general result may be found in [9, 10.2.2].) There are fewer objects in sSphG (n) than in VG (n) since inequivalent representations can have stably homotopy equivalent spheres. For finite groups, an analysis of when this happens may be found in [9, x9.1]; by a result of Traczyk, the case of general compact Lie groups reduc* *es to the case of finite groups [27]. Since, as in the nonequivariant world, equiv* *ariant EQUIVARIANT ORIENTATION THEORY 51 cohomology theories detect only stable homotopy type, stable spherical orientat* *ions are the appropriate ones to relate to ideas of cohomological orientation. 19.Categories of virtual G-bundles As said in the introduction, we defer comparison between geometric and coho- mological oriention theory to later work. However, as is made clear in [4, 5], * *for such work it is useful to have four more variants of the basic theory, in which* * sta- ble bundles are replaced by virtual bundles. In fact, the diagram (18.1) can be extended by adding another row of skeletal groupoids over OG . sVG (n)____//_sPLG (n)____//sTopG (n)___//_sFG (n) (19.1) || || || || fflffl| fflffl| fflffl| fflffl| vVG (n)____//_vPLG (n)____//vTopG (n)___//_vFG (n) The bottom row, which is also defined for negative values of n, is obtained f* *rom the top row_by_passage to virtual bundles. Thus vCatG (n) is a skeleton of the category vCatG (n) of virtual G-vector bundles of virtual dimension n and virtu* *al CatG -maps. Here a virtual G-vector bundle of virtual dimension n is a pair of bundles (E; F ) = (G xH V1; G xH V2) over the same G-orbit, with |V1| - |V2| = * *n. We think of (E; F ) as a formal difference E-F . Intuitively, a virtual map is * *a stable pair of maps, but we must be careful with the definition. Define a virtual map * *from (GxH V1; GxH V2) to (GxK W1; GxK W2) over a map of orbits ff: G=H -! G=K to be the equivalence class of a pair of CatG -maps fi:G xH (Vi Z) -! G xK (Wi Z0) over ff, where Z is an H-subspace of U and Z0 is a K-subspace of U such that |V1| + |Z| = |W1| + |Z0| and thus also |V2| + |Z| = |W2| + |Z0|. The equivalen* *ce relation is generated by two basic relations, the first being G-bundle homotopy. The second relation is as follows. Let k :G xH T - ! G xK T 0be a CatG -map over ff, where T is an H-subspace of U orthogonal to Z and T 0is a K-subspace of U orthogonal to Z0. Then the pair (f1; f2) is equivalent to the "suspension" (f1 k; f2 k), where fi k :G xH (Vi (Z T )) -! G xK (Wi (Z0 T 0)) is the obvious fiberwise direct sum of maps. Note that, in Z T , the sum is int* *ernal in U. Composition is defined by suspending until the morphisms can be composed as pairs of bundle maps. The following easily verified observation, applied to* * U regarded as an H-universe, implies that this gives a well-defined category. Lemma 19.2. Let U be a complete G-universe, let V , V 0, W , and W 0be G- subspaces of U, and suppose given CatG -maps h: V -! V 0and k :W - ! W 0. Then there exist G-subspaces Z and Z0 of V such that V Z, W Z, V 0 Z0, and W 0 Z0 together with CatG -maps j :(Z - V ) -! (Z0- V 0) and `: (Z - W ) -! (Z0- W 0) such that h j ' k `: Z -! Z0 as CatG -maps. If V = V 0, we can take j ' id. 52 S.R. COSTENOBLE, J.P. MAY, AND S. WANER It is not hard to check that vCatG (n) is a groupoid over OG . It is also not* * hard to check that the set of isomorphism classes of objects over G=H in vVG (n) is * *in bijective correspondence with the set of n-dimensional elements of RO(H); objec* *ts of the other categories admit similar descriptions. For n 0, we define a map sCatG (n) -! vCatG (n) by passing to skeleta from the evident functor that sends the G-bundle G xH V to the pair (G xH V; G xH 0). This functor is an inclusion * *of groupoids over OG . Using this functor, we obtain a theory of virtual orientati* *ons of virtual G-vector bundles, by taking pairs of orientations in the evident fas* *hion. To relate equivariant orientation theory to equivariant classifying spaces, i* *t is convenient to have small categories that, although not skeletal, are defined in* * terms of the complete universe U and are equivalent to the skeletal categories CatG (* *n), sCatG (n), and vCatG (n). Definition 19.3. Fix a complete G-universe U. ____ (i)Define CatG (n; U) to be the full category of CatG(n) whose objects are * *the G-vector bundles of the form GxH V for any H G and any n-dimensional H-subspace V of U. (ii)Define vCatG (n; U) to be the colimit over the G-subspaces V of U of the categories CatG (n + |V |; V U), where the colimit runs over the functo* *rs CatG (n + |V |; V U) -! CatG (n + |W |; W U) that are obtained by adding W - V W to objects and maps, where V W . (iii)Define sCatG (n; U) to be the image of CatG (n; U) in vCatG (n; U) obtai* *ned by setting V = {0}in the colimit system. ____ It is clear_that_the composite of the inclusion CatG (n; U) CatG(n) and the retraction CatG(n) -! CatG (n) is an equivalence of categories. The analogue for virtual bundles is less obvious. Proposition 19.4. The categories vCatG (n) and vCatG (n; U) are equivalent, and the equivalence restricts to an equivalence between sCatG (n) and sCatG (n; U). Proof.For each H-representation V such that G xH V is an object of CatG (n), choose an H-space V U and an H-linear isomorphism iV : V -! V . These choices determine a functor CatG (n) - ! CatG (n; U) that is an equivalence of categories. Also, choose an H-space V ? U such that V ? is orthogonal to V and V V ?is a G-space, and choose a G-isomorphism jV : U -! (U - (V V ?)). We first define a functor J : vCatG (n; U) - ! vCatG (n). An object X of vCatG (n; U) is represented by an object GxH V1 of some CatG (n+|V2|; V2U), and we let_J(X)_be the object of vCatG (n) isomorphic to the object (GxH V1; GxH V2) of vCatG (n). That is, we think of X as the virtual G-bundle G xH V1- G xH V2. * *If G xK W1 in CatG (n + |W2|; W2 U) represents a second object Y and OE : X -! Y is a map in vCatG (n; U), then OE is represented by a CatG -map f1 : G xH (V1 Z) -! G xK (W1 Z0) in CatG (n + |T |; T U), where Z and Z0 are G-subspaces of U such that Z is orthogonal to V2, Z0 is orthogonal to W2, and V2 Z = W2 Z0= T , say. We let J(OE): J(X) -! J(Y ) be the map represented by the pair (f1; f2), where f2 : G xH (V2 Z) -! G xK (W2 Z0) EQUIVARIANT ORIENTATION THEORY 53 is induced by the base map G=H -! G=K of f1 and the identity map on T . We next define a functor R : vCatG (n) -! vCatG (n; U). Consider an object X = (G xH V1; G xH V2) of vCatG (n). Embed the external direct sum V2? V1in the universe V2 V2? U by including V2?in V2 V2?and including V1in U. This allows us to view GxH (V2?V1) as an object of CatG (n+|V2V2?|; V2V2? U), and we let R(X) be the image of this object in the colimit vCatG (n; U). For a virtual map OE : (G xH V1; G xH V2) -! (G xK W1; G xK W2) represented by CatG -maps fi: G xH (Vi Z) -! G xK (Wi Z0); define R(OE) as follows. Using Lemma 19.2, we can find an H-space T in U that is orthogonal to both V2 V2?and Z and a K-space T 0in U that is orthogonal to both W2 W2? and Z0 together with a G-map k : G xH (V2? T ) -! G xK (W2? T 0) (where the sums are external) such that (i)Z T and Z0 T 0are G-subspaces of U. (ii)V2 V2? jV2(Z T ) = W2 W2? jW2 (Z0 T 0) = S, say, and (iii)The CatG -map G xH S -! G xK S induced by f2 k and our chosen isomorphisms is the map induced by the base map G=H -! G=K of f1 and the identity map on S. We let R(OE) be the map in vCatG (n; U) that is represented in CatG (n + |S|; S* * U) by the map G xH S -! G xK S induced by f1 k and our chosen isomorphisms. Here, in the domain and target respectively, we are thinking of S as V2 V2? jV2(Z T ) and W2 W2? jW2 (Z0 T 0): It is laborious, but straightforward, to check that J and R are inverse equiv* *a- lences of cotegories. The categories just defined are`closely related to standard models for equiva* *riant classifying spaces. We let C = nC (n) for any of the families of groupoids C * *(n) over OG that we have introduced. Definition 19.5. Let U be a complete G-universe. Define BOG (n; U) to be the Grassmann G-space of n-dimensional subspaces of U; G acts via restriction of its action on U. Of course, BOG (n; U) is the colimit of the Grassmann G-manifolds of n-planes in G-spaces V U. Similarly, let BOG (U) = colimV U(q nBOG (n; V U)); where the colimit runs over the G-spaces V U and the system of maps BOG (n; V U) -! BOG (n + |W - V |; W U) given by addition of the plane W - V for V W . It is standard [14, 18] that BOG (n; U) classifies n-dimensional G-vector bun* *dles and that BOG (U) classifies virtual G-vector bundles over finite G-CW complexes* * X. For the latter, this means that KOG (X) = [X; BOG (U)]G . A direct comparison of definitions gives the following relationship between categories of bundles a* *nd fundamental groupoids. Proposition 19.6. As groupoids over OG , VG (n; U) ~=G (BOG (n; U)) and vVG (U) ~=G (BOG (U)): 54 S.R. COSTENOBLE, J.P. MAY, AND S. WANER Part V. The classification of oriented G-bundles 20.Introduction: classifying G-spaces The main purpose of this part is to display classifying G-spaces for oriented G-bundles and G-fibrations. We take the opportunity to explain several related classifying spaces and classification theorems. We shall focus on G-vector bund* *les, but the arguments apply equally well to topological G-bundles and to spherical G-fibrations. The case of PL G-bundles requires the use of analogous simplicial techniques and will be left to the interested reader. Our construction of classifying G-spaces relies on the two-sided categorical * *bar construction of [21, x12]. Modulo a slight change of language, B assigns a G- space B(Y ; D; X ) to each triple consisting of a small topological category D,* * a continuous covariant functor X :D ! GU , and a continuous contravariant functor Y :D ! U . The G-space B(Y ; D; X ) is the geometric realization of the simplic* *ial G-space whose space Bn(Y ; D; X ) of n-simplices consists of tuples y[fn; : :;:* *f1]x, where fi:di-1 ! di is a map in D, x 2 X (d0), and y 2 Y (dn); B0(Y ; D; X ) is the disjoint union over the objects of D of the spaces Y (d) x X (d). The action of G on Bn(Y ; D; X ) is induced by the action of G on the X -coordinate. The construction is functorial in all three variables. The G-space B(Y ; D; X ) is a fattened up homotopical version of the coend Z d2D Y D X = Y (d) x X (d); and there is a canonical map " : B(Y ; D; X ) -! Y D X : We sometimes also write " for its composite with a given G-map Y D X -! Z: We think of Y as a D-shaped diagram and call it a D-space. We think of X as a fixed functor used to induce a G-action on a coalescence of Y to a single space. For example, Elmendorf [12] applied this construction to construct a G-space T associated to an OG -space T . Here T = B(T; OG ; ), where : OG -! GU is the evident functor that sends an orbit G=H, regarded as an object of the categ* *ory OG , to the orbit G-space G=H. Passage to fixed point spaces XH associates an OG -space X to a G-space X, and there is a natural spacewise weak equivalence T -! T . It provides the counit of an adjunction (20.1) [X; T ]G ~=[X; T ]OG for G-CW complexes X [12], [13, V.3.2]. The unit X -! X of the adjunc- tion is a G-homotopy equivalence. Formally, this adjunction arises from a Quill* *en equivalence of model categories [26], [20, IIIx1]. Passage to components gives a natural discretization map ffi from an OG -space T to a discrete (or set-valued) OG -space ss0(T ). In particular, we have ffi :* *X ! ss0(X). We say that X is homotopically discrete if ffi is a spacewise weak equi* *va- lence, so that each component of each fixed point space XH is weakly contractib* *le. Examples of homotopically discrete G-spaces T that are constructed from dis- crete OG -spaces T are central to the classification of G-bundles. In particula* *r, recall that a family F of (closed) subgroups of a topological group G is a set of subg* *roups closed under passage to conjugates and subgroups and observe that F determines a EQUIVARIANT ORIENTATION THEORY 55 discrete OG -space TF that takes G=H to a point if H 2 F and to the empty space if H =2F . The G-space TF is denoted EF and called the universal F -space. We are concerned with categories D over OG ; that is, we are given a fixed continuous functor ss : D -! OG . We compose this functor with : OG -! GU to obtain a functor O ss :D -! GU . The following definition fixes notations f* *or the G-spaces of greatest interest to us. Definition 20.2. Define the classifying G-space of a category D over OG to be BD = B(*; D; O ss); where * is the constant functor that takes each object of D to a point. More generally, for a contravariant functor Y : D -! U , define B(Y ; D) = B(Y ; D; O ss): Since (G=K)H = OG (G=H; G=K), B(Y ; D)H = B(Y ; D; OG (G=H; ss(-))): In the following four sections, we explain how to use the bar construction to classify G-bundles and G-fibrations, oriented G-bundles, oriented G-fibrations,* * and representations of fundamental groupoids in a given groupoid R over OG . The methods and some of the results are due to Waner [29, 30], who gave the equivar* *iant generalization of the nonequivariant classification theory developed by May [21* *, 22]. Since our methods are quite similar to those in the cited references, we will n* *ot give complete details of all proofs. 21. The classification of G-bundles and spherical G-fibrations Our theory of orientations is based on the bundles of groupoids over OG that * *are described in xx1-4. Here the term "bundle" refers to bundles with discrete fibe* *rs, in accordance with the definitions of x5. These groupoids over OG are obtained by passage to equivalence classes of maps from much richer categories over OG . For example, when G = e, V (n) is just ss0(O(n)), regarded as a category with a single object Rn. To classify bundles, we need to use O(n) itself. The equivari* *ant situation is similar. We will use Roman letters for categories that correspond* * in this way to some of the categories specified by script letters in x18. We conti* *nue to write ss for the functors from these larger categories to OG . Definition 21.1. Define VG (n) to be the category over OG whose objects are the objects of VG (n) but whose morphisms G xH V -! G xK W are the maps of G- vector bundles, topologized with the function space topology. Passage to homoto* *py classes of maps gives a functor ! : VG (n) -! VG (n) over OG . Define T opG (n)* * and FG (n) similarly, using maps of topological G-bundles and of spherical G-fibrat* *ions. The functor ss : VG (n) -! OG is given by passage to base spaces, and it is continuous. Since ! O ss = ss, it is clear that ! : VG (n) -! VG (n) is also co* *ntinuous, where VG (n) is topologized as in Proposition 4.1. In fact, by compactness, it follows that the topologies on morphism spaces specified there must coincide wi* *th the quotient topologies induced from the topologies on the morphism spaces of VG (n). Similarly, the functors ! : T opG (n) -! TopG (n) and ! : FG (n) -! FG (n) 56 S.R. COSTENOBLE, J.P. MAY, AND S. WANER are continuous, although in these cases we do not know whether or not the quoti* *ent topology coincides with the topologies defined as in Proposition 4.1. The following theorem gives the relevant special cases of Waner's general cla* *ssifi- cation theorems [29] for equivariant bundles and fibrations. In particular, it * *implies that BVG (n) is equivalent to the Grassmann classifying G-space BOG (n; U) spec* *i- fied in Definition 19.5. Theorem 21.2. The G-space BVG (n) classifies G-vector bundles of dimension n. That is, for G-CW complexes X, [X; BVG (n)]G is in natural bijective correspon- dence with the set of equivalence classes of G-n-plane bundles over X. Similarly BT opG (n) classifies locally linear topological G-bundles of dimension n and B* *FG (n) classifies locally linear sectioned spherical G-fibrations of dimension n. Here the term "locally linear" refers to our restriction to fibres homeomorph* *ic to G xH V or fiber homotopy equivalent to G xH SV for some subgroup H of G and H-representation V . The term "sectioned" refers to our use of fibrations w* *ith based fibers whose basepoints give a canonical section. There is an earlier and perhaps more conceptual proof of the classification t* *heo- rem that applies to G-vector bundles and topological G-bundles but not to spher* *i- cal G-fibrations. We describe it in the case of G-vector bundles. For a topolog* *ical structure group , such as = O(n) or = T op(n), a principal (G; )-bundle is a -free (Gx)-space. We think of G as acting from the left and as acting from the right, the two actions commuting. Just as in nonequivariant bundle theory, every n-dimensional G-vector bundle p : E -! X has an associated principal (G; O(n))- bundle ss : P -! X, and equivalence classes of G-n-plane bundles over X are in bijective correspondence with equivalence classes of principal (G; O(n))-bundle* *s. The analogue for topological G-bundles also holds. There is a standard construction of a universal principal (G; )-bundle ss : EG () -! BG () in terms of the universal F -space for a well chosen family F of subgroups of G x . Definition 21.3. Let FG () be the family of subgroups of G x such that \ = {e}. Such a subgroup has the form {(h; (h)) | h 2 H} for some subgroup H G and homomorphism : H ! . Define EG () = EFG () and BG () = EFG ()= and let ss : EG () -! BG () be obtained by passage to orbits. The G-map ss is the universal principal (G; )-bundle; see [13, VIIx2], for ex* *am- ple. Theorem 21.4. For G-CW complexes X, [X; BG ()]G is in natural bijective cor- respondence with the set of equivalence classes of principal (G; )-bundles over* * X. Remarks 21.5. (i) When = O(n), we view a homomorphism : H -! O(n) as specifying an action of H on Rn, and we denote this representation by V (). Thus groups in FG (O(n)) correspond to objects p() : G xH V () -! G=H of the category VG (n). This is the beginning of a direct comparison between our two w* *ays of classifying G-n-plane bundles. (ii) When = T op(n), we must modify the definition of FG () to account for our restriction to linear fibers. Viewing subgroups of G x O(n) as subgroups of G x T op(n), we here take FG (T op(n)) to be the family of subgroups of G x T o* *p(n) EQUIVARIANT ORIENTATION THEORY 57 that is generated under conjugation by the subgroups in the family FG (O(n)). S* *ee [17, x1] for background on the relevant G-bundle theory. 22. The classification of oriented G-bundles In this section, we give a classification theorem for oriented G-bundles, sta* *rting from Theorem 21.4. We treat spherical G-fibrations in the next section. We assu* *me throughout that our base G-spaces X have the G-homotopy types of G-CW com- plexes. For definiteness, fix a representation S :S -! VG (n). We could use oth* *er of the target categories displayed in (18.1) or (19.1). We have in mind the uni* *versal orientable representation (S VG (n); S). However, (S ; S) need not be universal* * for the theory of this section, and the generality is likely to have applications i* *n the study of restricted types of G-bundles. We have the notion of an orientation (F; OE) : (G B; p*) -! (S ; S) of a G-n- plane bundle p : E -! B. We know what it means for a map of G-bundles to be orientation preserving, and we can pull back orientations along G-bundle maps; * *see Definition 7.9 and Lemma 7.10. We shall classify equivalence classes of oriente* *d G- bundles under the relation given by orientation preserving equivalence of bundl* *es. That is, we shall construct a G-vector bundle p: EG (O(n); S) -! BG (O(n); S) together with an orientation = (F; OE): (G BG (O(n); S); p*) -! (S ; S) such that (p; ) is universal in the sense that pullback of bundles and orientat* *ions along G-maps specifies a bijection from the set [X; BG (O(n); S)]G of G-homotopy classes of G-maps X -! BG (O(n); S) to the set of equivalence classes of orient* *ed G-n-plane bundles over X. We begin by giving a diagrammatic description of orientations. Recall Definition 21.3 and Remark 21.5(i). Definition 22.1. We define a discrete OGxO(n)-space TS. Let TS(G x O(n)=) be empty if =2FG (O(n)) and be the set of orientations of G xH V () -! G=H in (S ; S) if 2 FG (O(n)). As is easily checked from Lemma 2.3, a (G x O(n))- map from G x O(n)= to G x O(n)=0determines and is determined by a map of G-vector bundles from G xH V () -! G=H to G xH0 V (0) -! G=H0. Thus, by pullback of orientations, TS is a well-defined contravariant functor on OGxO(n). Remark 22.2. Taking S = S VG (n), Proposition 2.9 and Example 2.10 show that TS(G x O(n)=) is non-empty if G is finite and 2 FG (O(n)), but that this fails for general compact Lie groups G. We have the following basic observation. See for example [16, Thm. 12] for the analysis of the fixed point spaces of principal (G; O(n))-bundles and their* * base spaces which is used in its proof. Lemma 22.3. Let p: E -! B be a G-n-plane bundle and let ss :P - ! B be its associated principal (G; O(n))-bundle. An orientation (F; OE) of p in (S ;* * S) determines and is determined by a map of OG -spaces :P -! TS. Proof.We have E = P xO(n)Rn and B = P=O(n). Let x 2 BH . If ss(y) = x, then y 2 P for some = {(h; (h)) | h 2 H}; and thus are determined by x only up to O(n)-conjugacy. There is a unique choice such that the corresponding bund* *le p() : G xH V () -! G=H is in the skeletal category VG (n), and p*(x) = p(). Regarding x as a G-map G=H -! B, the specification of p* on morphisms depends 58 S.R. COSTENOBLE, J.P. MAY, AND S. WANER further on a fixed choice of isomorphism between the pullback of p along x and * *the bundle p(). Such an isomorphism is given by a bundle map "x:G xH V () -! E over x, and the choice of such a map is equivalent to the choice of y 2 P . Two choices of y differ by right action by an element o in the centralizer O(n) of* * , and the corresponding maps "xdiffer by precomposition with the bundle automorphism of G xH V () determined by o. The choice of retraction VG (n) -! VG (n) in Definition 2.2 fixes "xand thus y. A map :P -! TS of OG -spaces is given by a natural choice of maps constant on components from the spaces P to the sets of orientations of the bundles p(). Write (Fy; OEy) for the orientation assigned to the component of y. Thus Fy is* * a functor G (G=H) -! S over OG and OEy is an isomorphism S O Fy -! p()* over OG . We specify the corresponding orientation (F; OE) of p by letting the func* *tor F :G B - ! S be given on objects x = ss(y) 2 BH by F (x) = Fy(idG=H). The specification of F on morphisms and the specification of the isomorphism OE: SOF -! p* over OG are similarly determined by the choices above together wi* *th the functors Fy and isomorphisms OEy specified by . The details of the verifica* *tion that this gives a bijective correspondence are tedious but straightforward. Theorem 22.4. There exists a universal oriented bundle (p; ). The G-n-plane bundle p : EG (O(n); S) -! BG (O(n); S) is characterized as follows. If p: E -!* * B is a G-n-plane bundle with associated principle (G; O(n))-bundle ss :P -! B, th* *en p admits an orientation in (S ; S) such that (p; ) is universal if and only if* * P is homotopically discrete and the diagram ss0(P ) is isomorphic to TS. Proof.Define PG (O(n); S) = TS and BG (O(n); S) = PG (O(n); S)=O(n), and let EG (O(n); S) = PG (O(n); S) xO(n)Rn: With the evident projections p and ss, this gives a G-n-plane bundle whose asso- ciated principal (G; O(n))-bundle has the prescribed fixed point structure. By * *the lemma, the discretization map ffi : PG (O(n); S) -! TS gives p an orientation . Moreover, PG (O(n); S) is a (GxO(n))-CW complex. The adjunction (20.1) and use of (G x O(n))-CW approximation imply that any other principal (G; O(n))-bundle P with the stated fixed point structure is weakly G-equivalent to PG (O(n); S),* * and the lemma implies that its associated G-n-plane bundle has a compatible orien- tation. To check universality, suppose given an oriented G-n-plane bundle over a G-CW complex X and let Q be its associated principle (G; O(n))-bundle. By the lemma, the orientation is given by a map : Q -! TS of OG -spaces. By the cited adjunction, there results a (G x O(n))-map Q -! PG (O(n); S). Passage to base G-spaces from this map of principal (G; O(n))-bundles gives the required classi* *fying G-map X -! BG (O(n); S), and the rest is routine. Corollary 22.5. There is a G-map f : BG (O(n); S) -! BG (O(n)) that represents the forgetful functor from oriented G-n-plane bundles to G-n-plane bundles. Proof.Since there is a unique function from any set to a point, there is an evi* *dent natural map of discrete OG -spaces TS -! TFG(O(n)). We obtain f by first applyi* *ng and then passing to orbits over O(n) Remark 22.6. We can obtain the analogous results for locally linear topological G-bundles in exactly the same fashion, replacing O(n) by T op(n) and interpreti* *ng FG (T op(n)) as in Remark 21.5(i). EQUIVARIANT ORIENTATION THEORY 59 23. The classification of oriented spherical G-fibrations We now turn to the case of spherical G-fibrations, where we follow the methods of [21, 22, 29, 30]. As in the cited references, slight modifications give alte* *rnative treatments of the cases of G-vector bundles and topological G-bundles. We fix n and a representation S :S -! FG (n); we could instead use the corresponding category of stable or virtual fibrations over orbits. Define a sp* *here space to be a G-map p: E -! B such that, for each b 2 BH , p: Gp-1(b) -! Gb is a sectioned G-fibration that is fiber G-homotopy equivalent to G xH SV for some subgroup H of G and representation V of H. A map (f"; f): p -! p0 of sphere spaces is a pair of maps f :B -! B0 and "f:E -! E0 such that p0O "f= f O p and each f":Gp-1(b) -! G(p0)-1(fb) is a section-preserving fiber G-homotopy equivalence. Define an S-sphere space to be a sphere space such that each G- fibration p: Gp-1(b) -! Gb has a given orientation b in (S ; S); we do not assu* *me any compatibility among these orientations at this point. A map of S-sphere spa* *ces is a map of sphere spaces such that each f":Gp-1(b) -! G(p0)-1(fb) preserves orientation. Thus, given (f"; f) and the orientations of the fibers of p0, the* *re are unique orientations of the fibers of p such that (f"; f) is a map of S-sphere s* *paces. An S-sphere space p: E -! B is said to be an S-fibration if it satisfies the S-covering homotopy property (S-CHP): Given an S-sphere space q :D -! A, a G-homotopy h: A x I -! B, and a G-map "h0:D -! E such that ("h0; h0) is a map of S-sphere spaces, there is a G-homotopy "hthat starts at "h0and covers h and is such that ("h; h) is a map of S-sphere spaces. Here q x I :D x I -! A x I has the evident structure of S-sphere space determined by that of q and thus, v* *ia ("h0; h0), by that of p. We have the following consistency observation. Lemma 23.1. An S-sphere space p: E -! B is an S-fibration if and only if p is a G-fibration with an orientation in (S ; S) that induces the given orientation* *s of the restrictions p: Gp-1(b) -! Gb. Proof.If p is a G-fibration with an orientation in (S ; S), then the ordinary G* *-CHP immediately implies the S-CHP; any covering homotopy is automatically a map of S-spaces because the orientations of orbits of points connected by paths in the* * fixed point spaces of B are compatible. Conversely, the S-CHP obviously implies the G- CHP, and it also implies that the orientations of orbits have the compatibility* * on paths required to specify an orientation of p. Definitions 23.2. Define a topological category FG (n; S) over OG as follows. I* *ts objects are pairs (x; ), where x is an object of FG (n) and is an orientation * *of x in (S ; S). Its morphisms are the maps of S-fibrations, with the function spa* *ce topology. Neglect of orientation gives a functor f : FG (n; S) -! FG (n) over O* *G . We define several functors on FG (n; S), and we abbreviate notation by writing F = FG (n; S). (i)Define a covariant functor E = EG (n; S) from F to G-spaces by sending an object (x; ) to the total space of x and sending a map in F to the underlying map of total spaces. (ii)Define a covariant functor B = BG (n; S) from F to G-spaces by sending an object (x; ) to the base space of x and sending a map in F to the underlying map of base spaces. Thus, with the notation of x20, B = Oss.We have an evident natural transformation p: E -! B. 60 S.R. COSTENOBLE, J.P. MAY, AND S. WANER (iii)Given an S-sphere space p: E -! B, define a contravariant functor PE from F to spaces by sending ff = (x; ) to the space of maps of S-fibrati* *ons x -! p; PE is specified on morphisms by precomposition. This is to be viewed as a principalization construction. We define PB by sending ff to the space of maps on base spaces of maps of S-fibrations x -! p, and we have the evident natural map PE -! PB. For each ff, PE(ff) -! PB(ff) is a nonequivariant fibration (as in [29, I.3.3]). (iv)For each object ff = (x; ) of F , define a covariant functor E [ff] from* * F to spaces by sending an object fi to the space F (ff; fi) of maps ff -! fi * *in F ; E [ff] is specified on morphisms by postcomposition. There is an analogo* *us functor B[ff] that sends fi to the space of maps on base spaces of morph* *isms in F (ff; fi), and there is an evident natural map p[ff]: E [ff] -! B[ff* *]. Theorem 23.3. The G-space BFG (n; S) = B(*; FG (n; S); BG (n; S)) classifies S-fibrations. That is, the set of equivalence classes of S-fibrations over a G* *-CW complex X is in natural bijective correspondence with the set [X; BFG (n; S)]G . The G-map Bf : BFG (n; S) -! BFG (n) represents the forgetful functor from equivalence classes of S-fibrations to equivalence classes of spherical G-fibra* *tions. Proof.We sketch the argument, referring the reader to [21, x9] and [29, x2] for more detailed accounts of analogous proofs. The latter source gives the proof * *of Theorem 21.2, and we are simply elaborating the argument to take account of the orientations. We abbreviate F = FG (n; S), etc. For a sphere space p: E -! X, there is a natural way to construct a spherical G-fibration p: E -! X and a natural map j :p -! p of sphere spaces; j is a fiber G-homotopy equivalence if p is a G-fibration, in which case a fiber homotopy inverse :p -! p is essentiall* *y a path lifting function. See [21, x3] and [29, 1.2]. If p is an S-sphere space, t* *hen p is an S-sphere space and j is a map of S-sphere spaces. The map p is said to be* * a G-quasifibration if each pH is a quasifibration, and the map j :E -! E is then a weak G-equivalence (see [23] for a modernized treatment of quasifibrations). * *We say that p is an S-quasifibration if it is a G-quasifibration with an orientati* *on in (S ; S). (While the pullback of a quasifibration need not be a quasifibration, * *the restriction of a sphere space over an orbit is a G-fibration, and it therefore * *makes sense to talk about orientations of sphere spaces.) For any contravariant functor Y :F -! U , the canonical map p = B(id; id; p): B(Y ; F ; E ) -! B(Y ; F ; B) is a G-quasifibration. Moreover, it has a canonical orientation in (S ; S). In * *fact, since G acts only on the E and B coordinates, each orbit in the base space is a copy of the base space of a particular object (x; ) of F , and the inverse imag* *e of that orbit is a copy of the total space of x. Remembering , we see that we have canonical orientations of the restrictions of p to orbits. Because the morphis* *ms of F are orientation preserving, these orientations of orbits satisfy the requi* *site compatibility to specify an orientation of p in (S ; S). We claim that the S-fibration p: B(S ; F ; E ) -! B(S ; F ; B) is universal. Pulling p back along G-maps, we obtain a natural transformation from the functor [X; B(S ; F ; B)]G to the functor that assigns to X the set of equivalence classes of S-fibrations over X, and our claim is that is a natural EQUIVARIANT ORIENTATION THEORY 61 bijection. We construct an inverse natural transformation . Let p: E -! X be an S-fibration. We have a pair of maps X oo"__B(PE; F ; B) __q_//_B(*; F ; B) = BF : Here q is induced by the unique natural map PE -! * and ffl is induced by the composite PE F B -! PB F B -! X, where the second arrow is given by evaluation of maps of base spaces. We claim that ffl is a weak G-equivalenc* *e. Granting this, our assumption that X is a G-CW complex ensures that there is a map g :X -! B(PE; F ; B), unique up to G-homotopy, such that ffl O g ' id. We define (p) to be the homotopy class of f = q O g. We first verify our claim about ffl, then verify that is the identity transf* *or- mation, and finally verify that is a natural automorphism; it follows formally that must be the identity. For each object ff of F and any Y , the canonical map ffl: B(Y ; F ; E [ff]) -! Y (ff) induced by the evident evaluation map Y F E [ff] -! Y (ff) is a homotopy equiv- alence by the argument of [22, Prop. 9.9]. We apply this with Y = PE for an S-fibration p: E -! X to obtain a map of fibrations B(PE; F ; E [ff])"__//_PE(ff) | | | | fflffl| " fflffl| B(PE; F ; B[ff])____//PB(ff): The map " of total spaces is an equivalence, fibers map by equivalences, and th* *us the map " of base spaces is a weak equivalence. If ff = (x; ) and x has base or* *bit G=H, then PB(ff) is a subspace of BH ; similarly, B(PE; F ; B[ff]) is a subspac* *e of B(PE; F ; B)H . Since the maps ffl are weak equivalences for all ff, a little a* *nalysis of fixed point spaces shows that fflH :B(PE; F ; B)H -! XH is a weak equivalence for all H; see [29, 2.3.2]. This verifies our claim that ffl is a weak G-equiva* *lence. Now consider (p), p: E -! X. We must check that if f = q O g is the classifying G-map that we have constructed, then the pullback of the universal S-fibration along f is equivalent to p. The following schematic diagram gives t* *he idea. "q E _____//Ecoo______""_______B(PE;cF;GE7)_________________//B(*;7F;8E8) | | GGG ooo | qqq | | | GH1GG "goooo p| qqq | | | GG ooo | qqqf" | | | -1 o | K -1q | | | g (p) ______________________//_f (p) |p | | | | | | | | | | | | | | | | | | fflffl|_fflffl| || " fflffl| q || fflffl| ______X Hoo_______________B(PE; F; B) _________________//_B(*; F; B) X HHHHHH | | p77p HHHHH | | ppp HHHHH|H | pppf=qOgp HHfflffl| fflffl|ppp X ____________________________X 62 S.R. COSTENOBLE, J.P. MAY, AND S. WANER Here K is given by the universal property of the pullback f-1 (p) and H1 is obtained at the end of a homotopy which starts at "fflO "gand covers any homoto* *py from ffl O g to the identity; H exists since E -! X satisfies the S-CHP. Finally, to consider (f), f :X -! B(S ; F ; B), we consider the diagram _(f)____________________________________________* *______________________________________________________________@ _________________________________________________________* *______________________________________________________________@ _______________________________________________________________* *______________________________________________________________@ ________________________________________________**___________________* *______________________________________________________________@ X oo____"_____B(Pf B(*; F ; E ); F ; B)___q_____//BF f || |B(Pf";id;id)| |||| fflffl| fflffl| || BF oo____"______B(PB(*; F ; E ); F ; B)____q_____//_BF : We have already noted that the arrows ffl are weak G-equivalences. The maps q a* *re G-quasifibrations, and we claim that the fibers of the bottom map q are (nonequ* *iv- ariantly) weakly contractible. It follows that q is also a weak G-equivalence; * *there is no problem of equivariance since G acts only on B. By the G-Whitehead the- orem, the two weak G-equivalences of the bottom row induce automorphisms of represented functors on G-CW complexes. We conclude from the diagram that the composite is an automorphism of the functor [X; BF ]G , as required. To see that the fibers of the bottom map q are weakly contractible, observe f* *irst that PE(ff) is weakly equivalent to PE(ff) for any S-quasifibration p: E -! X. Observe next that there is an identification (PB(Y ; F ; E ))(ff) ~=B(Y ; F ; E [ff]) for any object ff of F and any Y ; we have observed that the right side is equi* *valent to Y (ff). Taking Y to be the trivial functor *, we see that (PB(*; F ; E ))(f* *f) is contractible. This implies the conclusion. 24. Moore loops and the classification of representations In this slightly digressive final section, we prove an analogue of Theorem 21* *.2 for representations of fundamental groupoids. Contrary to our previous conven- tions, unless otherwise specified we let G be any topological group, not necess* *arily compact Lie, here. Fix a groupoid R over OG . We may as well assume that R is skeletal. Recall the definition of a representation from Definition 7.1 and res* *trict attention to representations R : G X -! R. We say that two such representations R and R0are isomorphic if there is an isomorphism OE : R -! R0over OG . In terms of the definition of a map of representations in Definition 7.1, we are requiri* *ng the functor on the domain category G X of R and R0to be the identity. The following result generalizes [2, 3.8]. Rigorously, we should only claim it as a conjectur* *e since, except in the discrete case, we have not filled in the details of one step of t* *he proof (see Lemma 24.8 below). Theorem 24.1. The G-space BR classifies representations of G X in R. That is, for G-CW complexes X, [X; BR]G is in natural bijective correspondence with the set of isomorphism classes of representations R : G X -! R. When G is compact Lie, we have a G-map B! : BVG (n) -! BVG (n), and simi- larly for topological G-bundles and spherical G-fibrations. The following expec* *ted comparison is checked by comparing the proofs of Theorems 21.2 and 24.1. EQUIVARIANT ORIENTATION THEORY 63 Corollary 24.2. The G-map B! : BVG (n) -! BVG (n) represents the natural transformation that sends a G-n-plane bundle p : E -! B to the representation p* : G B -! VG (n). The G-maps B! : BT opG (n) -! BTopG (n) and B! : BFG (n) -! BFG (n) represent the analogous natural transformations on topological G-bundles and sp* *her- ical G-fibrations. We shall use the Moore loop category of X to prove Theorem 24.1. This category is related to G X as VG (n) is related to VG (n) and is of independent interest. Definition 24.3. Let X be a G-space. The Moore loop category G X is the category whose objects are the G-maps x: G=H - ! X and whose morphisms x -! y, y : G=K -! X, are the triples (; r; ff), where ff: G=H -! G=K is a G-map, r 0 is a real number, and : G=H x [0; r] -! X is a path of length r from x to y O ff in XH . Composition is induced by concatenation of paths and addition of real numbers; paths of length zero give identity morphisms. Regard paths as defined on [0; 1] by letting them be constant on [r; 1] and topologize* * the set of maps x -! y as a subspace of the space of maps [0; 1] -! XH . Thus the space of self-maps of an object x over the identity map of G=H is the Moore loop space (XH ; x). Let ss :G X -! OG be the functor given by ss(x) = G=H and ss(!; ff) = ff. Again, the functor ss is continuous. We have used paths of varying length to obtain a category, but this makes the construction of a functor ! : G X -! G X awkward, especially in view of the use of paths of length zero. One way to proc* *eed is to first extend paths of length r to paths on [-1; r] that are constant on [* *-1; 0], next to use the evident linear isomorphisms [-1; r] -! [0; 1] to rescale paths * *to paths defined on [0; 1], and finally to pass to equivalence classes of paths. * *This gives a continuous functor !, and it induces a G-map B! : BG X -! BG X. Remark 24.4. A quite different topologization of G X is studied in [24]. It is * *used to construct a G-map f : X -! Y , where Y is a kind of "K(G X; 1)", namely a G-space Y such that f* : G X -! G Y is an equivalence of categories and each component of each Y H is a K(ss; 1). The proof of Theorem 24.1 will be a direct application of two basic results a* *bout Moore loop categories. The first generalizes the classical nonequivariant weak * *equiv- alence X ' BX for connected spaces X [21, 14.3]. Proposition 24.5. For G-spaces X, there is a natural weak G-equivalence between X and BG X. When X is a G-CW complex, there is a weak G-equivalence i : X -! BG X that is natural up to G-homotopy. If G is a compact Lie group, i is a G-homotopy equivalence. Proof.Nonequivariantly, the Moore path space P X = P (X; x) has a right action of the Moore loop space X = (X; x) and, when X is path connected, there is a natural weak equivalence X oo"__B(P X; X; *)__q__//B(*; X; *) = BX: Here " is induced by the endpoint evaluation map p : P X -! X and q is induced by the trivial map P X -! *. The equivariant generalization is precisely parall* *el. 64 S.R. COSTENOBLE, J.P. MAY, AND S. WANER There is a contravariant Moore path functor PG X : G X -! U . For an object x : G=H -! X of G X, PG X(x) is the Moore path space P (XH ; x), points of which may be viewed as G-maps G=H x[0; 1] -! X. For a map (; r; ff) : x -! y, y : G=K -! X, and a path (; s) 2 P (XK ; y), PG X(; r; ff)(; s) = (( O ff) . ; r + s) 2 P (XH ; x): We construct a natural weak G-equivalence X oo"__B(PG X; G X) __q_//_B(*; G X) = BG X: The G-maps PG X(x) x G=H -! X given by evaluation of paths at cosets gH and end-point evaluation give rise to a G-map PG X GX O ss -! X that induces the G-map ". The natural map from PG X to the trivial functor * induces the G-map q. Since the map PG X -! * is a spacewise equivalence, q is a weak G-equivalenc* *e. We must show that " is a weak G-equivalence. This means that the H-fixed map "H : B(PG X; G X)H -! XH is a weak equivalence for any G-space X and any H G. Recall that B(PG X; G X)H = B(PG X; G X; OG(G=H; ss(-))) and consider the following diagram: ` ` (XH ; x)______________ (XH ; x) [x] [x] | | | | ` fflffl| ` fflffl| B(PG X; G X; G X(x; -))___'__// P (XH ; x) [x] [x] | | | | fflffl| fflH fflffl| B(PG X; G X; OG(G=H; ss(-)))_______//XH : The disjoint unions are taken over the isomorphism classes of objects in the fi* *ber (G X)G=H over G=H, or equivalently over the components of XH . It is stan- dard that the right column is a fibration, and the left column is a quasifibrat* *ion by the argument of [21, 7.6]. By another standard argument [21, 7.5], the space B(PG X; G X; G X(x; -)) is contractible, and so is P (XH ; x). This implies that "H is a weak equivalence. When X is a G-CW complex, the composite of " and a G-CW approximation fl : B(PG X; G X) -! B(PG X; G X) is a G-homotopy equivalence. Choosing an inverse and composing with q O fl, we obtain the required weak G-equivalence i : X -! BG X. If G is a compact Lie group, results of Waner [28] imply that BG X has the homotopy type of a G-CW complex, so that i is a G-homotopy equivalence by the G-Whitehead theorem. The second basic result about the Moore loop category generalizes the discrete special case of the classical nonequivariant weak equivalence G ' BG for topo- logical groups G [21, 8.7]. Proposition 24.6. Let ss : R -! OG be a groupoid over OG . Then there is an equivalence : G BR -! R of groupoids over OG , natural up to isomorphism over OG . EQUIVARIANT ORIENTATION THEORY 65 Proof.We mimic the nonequivariant argument in [21, 8.7]. We define a natural equivalence : R ! BR over OG and let be any chosen inverse to . Note that we are working here with our original fundamental groupoid G BR based on paths of length one. The G-space of zero simplices of BR is the disjoint union over objects x 2 R of the orbit G-spaces G=H, where ss(x) = G=H. We let (x) be the point 1x eH in the orbit corresponding to x. Writing 1 for the unique point in the standard 0-simplex, we can write (x) = | * [ ]1x; 1|. Write (t; 1 * *- t), 0 t 1, for the points in the standard 1-simplex. If ! :x -! y is a morphism of R with ss(!) = ff : G=H -! G=K, let (!) be the homotopy class of the path t 7! | * [!]1x; (t; 1 - t)| from (x) to (y) O ff. This is a morphism in G BR. We see that is a functor by using the way in which 2-simplices in BR are attached to the 1-skeleton. To show that is an equivalence of groupoids over OG , it suffices to show th* *at (24.7) : R(x; y) -! G BR((x); (y)) is a homeomorphism for each pair of objects x and y of R. We believe that the following lemma holds in general, but our proof is only complete when G is disc* *rete. Lemma 24.8. ss(R(x; y)) = ss(G BR((x); (y))) in OG (ss(x); ss(y)). Proof.Since ss O = ss, ss(R(x; y)) ss(G BR((x); (y))). Let ss(x) = G=H and fix ff : ss(x) -! ss(y). We must show that if there is a path in (BR)H that con* *nects (x) to (y) O ff, then there is a map ! : x -! y in R such that ss(!) = ff. Usin* *g a cellular approximation argument, if there is such a path, then it can be deform* *ed to a path in the simplicial 1-skeleton of (BR)H . If G is discrete, this skelet* *on is a graph, and paths in it are equivalent to reduced finite edge paths. The edges a* *re of the form () or ()-1 for morphisms of R. Using source lifting and divisibility, we can deduce the result by induction on the number of edges. We believe that t* *he covering property of Definition 5.1(i) can be used to adapt the argument to gen* *eral topological groups G, but we have not worked out the details. In view of the lemma and Proposition 5.4, to prove that the map of (24.7) is a homeomorphism in general, it suffices to prove that, for each object x of R, * *the restriction of to a map AutR (x) -! AutGBR ((x)) of discrete groups is a bijection and thus an isomorphism. Consider the diagram Aut (x)____i____//_B(*; R; ssR(x; -)) | | | | fflffl| fflffl| B(*; R; R(x; -))____//_P B(*; R; ssR(x; -)) | | | | fflffl| fflffl| B(*; R; ssR(x; -))____B(*; R; ssR(x; -)): The (ordinary) loop and path spaces on the right use paths based at (x), and the group of components of the displayed loop space is AutGBR ((x)). The maps i and are defined by i(!)(t) = | * [!]1!(0); (t; 1 - t)| and (| * [!n; : :;:!1]!0; u|)(t) = | * [!n; : :;:!1; !0]1!0(0); (tu; 1 - t)* *|; 66 S.R. COSTENOBLE, J.P. MAY, AND S. WANER where u 2 n. The right column of the diagram is a fibration, and the left column is a quasifibration, as in [21, 7.6]. Both total spaces are contractible, so i * *is a weak equivalence. This implies that = ss0(i): AutR (x) ! AutGB ((x)) is a bijectio* *n, as required. Proof of Theorem 24.1.We define inverse isomorphisms and between the two functors of X in the statement. A G-map f : X -! BR induces the representation (f) specified as the composite G X _f*_//_G BR____//_R: A representation R : G X -! R induces the G-map (R) specified as the com- posite X __i_//_BG X_BR_//_BR: By the definitions and the naturality of , (R) = O BR* O i* = R O O i*: This representation is isomorphic to R since O i* : G X -! G X is isomorphic over OG to the identity functor. Therefore is the identity functor. Similarly, by the naturality of i, (f) = B O Bf* O i = B O i O f: Since is an isomorphism over OG , B is a weak G-equivalence, as is i. Therefore B O i : BR -! BR is a weak G-equivalence and the composite is an auto- morphism. It follows formally that this automorphism must be the identity. References [1]S. R. Costenoble and S. Waner. Equivariant orientations and G-bordism theor* *y. Pacific J. Math. 140 (1989), 63-84. [2]S. R. Costenoble and S. Waner. Fixed set systems of equivariant infinite lo* *op spaces. Trans Amer. Math. Soc. 326(1991), 485-505. [3]S. R. Costenoble and S. Waner. The equivariant Thom isomorphism theorem. Pa* *cific J. Math. 152 (1992), 21-39. [4]S. R. Costenoble and S. Waner. Equivariant Poincare duality. Michigan Math.* * J. 39 (1992), 325-351. [5]S. R. Costenoble and S. Waner. The equivariant Spivak normal bundle and equ* *ivariant surgery. Michigan Math. J. 39 (1992), 415-424. [6]S. R. Costenoble and S. Waner. Equivariant simple Poincare duality. Michiga* *n Math. J. 40 (1993), 577-604. [7]S. R. Costenoble, S. Waner, and Y. Wu. The equivariant Euler Characteristic* *. J. Pure and Applied Algebra 70 (1991), 227-249. [8]P. Deligne and D. Mumford. The irreducibility of the space of curves of giv* *en genus. Institut des Hautes Etudes Scientifiques publications mathematiques 36(1969), 75-109. [9]T. tom Dieck. Transformation groups and representation theory. Springer Lec* *ture Notes in Math. Vol. 766. 1979. [10]T. tom Dieck. Transformation groups. Studies in Math. 8, Walter de Gruyter.* *1987. [11]K.H. Dovermann and R. Schultz. Equivariant surgery theories and their perio* *dicity proper- ties. Springer Lecture Notes in Mathematics Vol 1443. 1990. [12]A. Elmendorf. System of fixed point sets. Trans. Amer. Math. Soc. 277 (1983* *), 275-284. [13]J.P. May, et al. Equivariant homotopy and cohomology theory. NSF-CBMS Regio* *nal Confer- ence Series in Mathematics No. 91. Amer. Math. Soc. 1996. [14]Z. Fiedorowicz, H. Hauschild, and J.P. May. Equivariant algebraic K-theory.* * Springer Lecture Notes in Mathematics Vol. 967. 1983, 23-80. EQUIVARIANT ORIENTATION THEORY 67 [15]A. Grothendieck. R^evetementsetale et groupe fondemental. Lecture Notes in * *Math. 224, Springer-Verlag. 1971. [16]R. Lashof and J.P. May. Generalized equivariant bundles. Bull. Soc. Math. B* *elgique 38 (1986), 265-271. [17]R. Lashof and M. Rothenberg, G-smoothing theory, Proc. Symp. Pure Math. 32 * *Part I, Amer. Math. Soc. (1978), 211-266. [18]L.G. Lewis, J.P. May, and M. Steinberger (with contributions by J.E. McClur* *e). Equivariant stable homotopy theory. Springer Lecture Notes in Mathematics Vol. 1213. 198* *6. [19]S. Mac Lane. Categories for the working mathematician. Second edition. Spri* *nger-Verlag. 1998. [20]M. Mandell and J.P. May. Equivariant orthogonal spectra. Memoirs Amer. Math* *. Soc. To appear. [21]J.P. May. Classifying spaces and fibrations. Memoirs Amer. Math. Soc. 155 (* *1972). [22]J.P. May. (with contributions by F. Quinn, N. Ray, and J. Tornehave). E1 ri* *ng spaces and E1 ring spectra. Lecture Notes in Mathematics Vol. 577. Springer-Verlag 1977. [23]J.P. May Weak equivalences and quasifibrations, Lecture Notes in Mathematic* *s Vol. 1425, Springer-Verlag 1990, 91-101. [24]J.P. May. G-spaces and fundamental groupoids. Appendix to "An equivariant N* *ovikov con- jecture" by J. Rosenberg and S. Weinberger. Journal of K-theory 4 (1990), 50* *-53. [25]J.P. May. Equivariant orientations and Thom isomorphisms. In Tel Aviv Topol* *ogy Confer- ence: Rothenberg Festschrift. Contemporary Mathematics Vol 231, 1999, 227-24* *3. [26]R.J. Piacenza. Homotopy theory of diagrams and CW-complexes over a category* *. Canadian J. Math. 43(1991), 814-824. [27]P. Traczyk, On the G-homotopy equivalence of spheres of representations, Ma* *th. Z. 161 (1978), 257-261. [28]S. Waner. Equivariant homotopy theory and Milnor's theorem. Trans. Amer. Ma* *th. Soc. 258 (1980), 351-368. [29]S. Waner. Equivariant classifying spaces and fibrations. Trans. Amer. Math.* * Soc. 258 (1980), 385-405. [30]S. Waner. Classification of oriented equivariant spherical fibrations. Tran* *s. Amer. Math. Soc. 271 (1982), 313-323. [31]S. Waner. Equivariant orientation theory. Preprint, Hofstra University (198* *6). Department of Mathematics, 103 Hofstra University, Hempstead, NY 11549 E-mail address: Steven.R.Costenoble@Hofstra.edu Department of Mathematics, University of Chicago, Chicago, IL 60637 E-mail address: may@uchicago.edu Department of Mathematics, 103 Hofstra University, Hempstead, NY 11549 E-mail address: matszw@hofstra.edu