Bivariant K-theory via correspondences

BIV ARIANT K-THEOR Y VIA CORRESPONDENCES HEA TH EMERSON AND RALF MEYER Abstrac t. W e use correspondences to deﬁne a purely topological equiv arian t biv a riant K -theory for spaces wi th a prop er groupoid action. Our notion of correspondence diﬀers slightly from that of Connes and Skandalis. Our con- struction uses no special features of equiv ar ian t K -theory . T o highligh t this, we construct biv ariant extensions f or arbitrary equiv ariant multiplicativ e co- homology theories. W e formulate necessary and suﬃcient conditions for certain dualit y isomor- phisms in the top ological biv ariant K -theory and ve rify these conditions in some cases, including s m ooth manifolds with a smooth cocompact action of a Lie group. One of these dualit y isomorphisms reduces biv ariant K -the ory to K -theory with support cond itions. Since similar dualit y isomorphisms exist in Kasparo v theory , the topological and analytic biv ariant K -theories agree if there is such a duality isomorphism. 1. Introduction Kasparov’s biv ariant K -theory is the main to ol in non-commutativ e top ology . Its deep analytic prop erties are resp onsible for many a pplications of C ∗ -algebra meth- o ds in top ology such as the Novik ov conjecture or the Gromov–La wson–Rosenber g conjecture. B ut some o f its applications – such as the co mputation of equiv ar iant Euler characteristics and Lefsch etz inv ariants in [8, 11] – should not require an y diﬃcult analysis and should therefore proﬁt fro m a purely top olog ical substitute for Kaspa rov’s theory . Our goa l here is to construct such a theory in terms of corresp ondences . Already in 19 80, Paul Baum and Ronald Douglas [4] prop osed a topo logical de- scription of the K -ho mology K K ∗ ( C 0 ( X ) , C ) of a space X , which was so on extended to the biv aria nt case by Alain Connes and Geor ges Skandalis [6]. Equiv ariant gen- eralisations with so mewhat limited sco pe were considered in [2, 19]. One might hop e that a to po logical biv a riant theor y deﬁned along these lines could b e shown to agree with Kaspa rov’s a nalytic theory KK G ∗  C 0 ( X ) , C 0 ( Y )  under some ﬁnite- ness ass umptions. But even in the case of non-equiv ar iant K -homology , a complete pro of app eared only rec en tly in [5]. The following problem cr eates new diﬃculties in the equiv ar iant case. P art of the data of a geometric cycle in the sense of Paul Ba um is a v ector bundle. But equiv ariant vector bundles a re sometimes in to o sho rt supply to generate equiv ari- ant K -theory . Let VK 0 G ( X ) be the Grothendieck gro up of the additiv e catego ry of G -equiv ariant complex vector bundles over a prop er G -spac e X . The functor ( X, A ) 7→ VK 0 G ( X, A ) for ﬁnite G -CW-pair s need not satisfy excision. T o get a re asonable cohomology theory , we need more general cycles as in Graeme Segal’s deﬁnition o f r epresentable K -theory in [21]. The G -equiv aria n t representable K -theory RK ∗ G ( X ) for loc ally compact group oids G a nd lo cally compact, pro per 2000 Mathematics Subject Classiﬁc ation. 19K35, 46L80. Heath Emerson was supported b y a National Science and Engineering Council of Canada (NSER C) Disco very gran t. Ralf Mey er wa s supported by the German Researc h F oundation (Deutsc he F orsc h ungsgemeinsc haft (DF G)) through the Institutional Strategy of the Universit y of Göttingen. 1 2 HEA TH EMERSON AND RALF MEYER G -spaces X is studied in [9]. Ther e are several equiv alent deﬁnitions, using a v ari- ant of Ka sparov theor y , K -theory for pro jectiv e limits of C ∗ -algebra s, or e quiv a riant families of F redholm op erators . F urthermore, [9] studies non-representable equiv ar i- ant K -theor y K ∗ G ( X ) and K -theor y with Y -c ompact support RK ∗ G ,Y ( X ) , where Y and X are G -spaces with a G -map X → Y . All three theories may b e describ ed b y F r edholm-op erator- v a lued maps – a reasonably satisfactor y homotopy theoretic picture. How ever, ev en after replacing VK 0 G ( X ) by RK 0 G ( X ) , equiv ar iant vector bundles still play an impo rtant role in v arious ar guments with corresp ondences . Fir st, the pro of that the top ological and ana lytic biv aria n t K -theories agree for smo oth mani- folds requires ce rtain equiv aria n t v e ctor bundles, whic h only exist under additional tech nical assumptions. S econdly , w e need some equiv ar iant G -vector bundles to comp ose cor resp ondences. If corresp ondences are deﬁned as in [7], then co mpos ing them r equires a transversality condition. In the equiv ar iant case , this can no longer be achi eved by a p erturbation a rgument. A basic exa mple is the pair of maps { 0 } → R ← { 0 } fro m a po in t to the pla ne se nding the p oint to the origin. This is equiv ariant with res pect to the action of Z / 2 by a r otation around the origin. These t wo maps cannot b e per turbed to be transverse to each o ther since the o rigin is the only ﬁxed-p oint. Ba um and Block [2] sug gest how to comp os e such co rresp ondences despite this. This trick uses v e ctor bundle mo diﬁcation a nd thu s an ample supply of vector bundles. Therefore, to get an elegant theory , we ha ve mo diﬁed t wo details in the deﬁnition of a corresp ondence. Our c hanges to the deﬁnition hav e the nice side eﬀect that our theory no longer uses a ny sp ecial features of K - theory and extends almost litera lly to any equiv aria nt m ultiplicative cohomolog y theory . W e work in this genera l setting for conceptual reaso ns and in order to prepa re for the construction of a n equiv a riant biv a riant Chern character. In the no n-equiv ariant case, Mar tin J akob has descr ibed the homology and biv aria n t cohomology theories asso cia ted to a cohomolog y theor y along similar lines in [14, 15]. A G -equiv aria n t c orr esp ondenc e fro m X to Y is a G -spac e M with G -equiv aria nt contin uous maps X b ← − M f − → Y and with s ome equiv ariant K -theory datum ξ on M . In [7], b is prop er, f is smoo th and K -oriented, and ξ is a vector bundle ov er M . W e do not require b to b e prop er. Instead, we let ξ ∈ RK ∗ G ,X ( M ) b e a G -equiv ar iant K -theory c lass with X - c omp act supp ort ; thu s b and ξ combine to an element of KK G ∗  C 0 ( X ) , C 0 ( M )  . Ro ughly sp eaking , instead of requiring the ﬁbres o f b to b e compact, we requir e that b restricts to a pro pe r map on the supp or t of ξ . F urthermore, we let f b e a K -oriented normal ly non-singular map in the sense of [12]. Roughly sp eaking , these are maps together with a fa ctorisation V ˆ f / / E π E   X ζ V O O f / / Y , where V is a G -vector bundle ov er X with zer o section ζ V : X → V , E is a G -vector bundle ov er Y with bundle pro jection π E : E → Y , and ˆ f : V → E is an op en embedding. The normally non-singular map is K -oriented if V a nd E are eq uiv a ri- antly K -oriented. F or technical reasons, we r equire E to be trivia l, that is, pulled back from the ob ject spa ce o f o ur group oid, and V to b e subtrivial, that is, a direct summand in a trivial G -vector bundle. The triviality o f E is needed to deﬁne the BIV ARIANT K-THEOR Y VIA CORRESPONDE NCES 3 comp osition of normally non-singula r maps. The subtriviality of V is needed to bring cor resp ondences in to a standard form; o therwise the bundle pro jection π V would not b e the trace of a normally no n-singular ma p. Thom iso morphisms and functorialit y suﬃce to construct purely top olo gical wrong-way maps for no rmally non-sing ular maps. In co n trast, the c onstruction of wrong-wa y elements for arbitrar y smo oth maps is ana lytical (see [12]). If we used smo o th maps instead of nor mally non-singula r maps, then co rresp ondences would not in g eneral describe equiv ar iant K -theory corr ectly , and since we prove the equiv alence o f the topolog ical and ana lytic biv ariant K - theory by a reduction to K -theory , this would hav e a disastrous eﬀect o n the gener al framew ork. On the other hand, for man y prop er group oids, a n y smo oth map ha s an essent ially unique normal factorisation, so that there is no diﬀerence b etw een smooth nor mally no n- singular maps and smo oth maps. But this requir es so me tec hnical co nditions on the group oid, which then have to appea r in all impo rtant theorems. W e use normally non-singular maps here to av oid such technical conditions. Before we discuss corresp ondences further, we must discuss the kind of equiv ar i- ance w e allow. Although we are mainly in teres ted in the case of group actions on spaces, we develop our whole theory in the setting where G is a numer ably pr op er group oid in the sense of [12]. Numerably pr op er group oids combine Abels’ n umer- ably prope r g roup actio ns ([1]) with Haar systems. If the group oid G is not pr op er – say an inﬁnite discr ete group – then we r eplace it b y the group oid G ⋉ E G for a univ ersal prop er G -s pace and pull back all G -spaces to G ⋉ E G -spa ces. This do es not change KK G ∗  C 0 ( X ) , C 0 ( Y )  if G acts prop erly (or amenably) on X (se e [10, 18]). Analysis plays no role in the construction of our biv ar iant coho mology theories. Hence we do no t need our spaces to b e lo c ally compact – para compact Hausdo rﬀ is go o d enough. F or actions of num erably prop er group oids on paracompact Haus- dorﬀ spaces, pull-backs o f equiv ar iant v ector bundles a long equiv ar iantly homotopic maps a re isomo rphic, equiv ar iant vector bundles ca rry inv ariant inner pro ducts, and extensions of equiv ariant vector bundles split. W e remov ed the prope rness condition on b in order to simplify the construction of the intersection pro duct. In the us ual approach, the inte rsection pro duct of cor- resp ondences only works under a transversality assumption, which can b e a ch ieved b y pe rturbing the maps inv olved. This p erturbation no longer works equiv ariantly . As mentioned a bove, Paul Baum a nd Jonathan Blo ck suggest in [2, 3] to use vector bundle mo diﬁcation to ov er come this, but while this works w ell in cer tain situations we found it hard to formalis e. With our non-prop er cor resp ondences, we can br ing any cor resp ondence in to a standard form for which transversalit y is automatic. This inv olves our equiv alence relation of Thom modiﬁca tion, which r eplaces the vector bundle mo diﬁcation of P a ul Baum. The diﬀerence is that we use the total space of the vector bundle instead of a sphere bundle. This is p ossible b ecaus e w e allow non-prop er corr esp ondences. W rite the o riented normally non-singular map f from M to Y in a corr esp ondence as a triple ( V , E , ˆ f ) for oriented G -vector bundles V and E ov er M and Y and an op en em bedding ˆ f : | V | ֒ → | E | . Thom mo diﬁcation alo ng the vector bundle V replaces the giv en corresp ondence by one that inv olves the total space of V in the middle, and where f b ecomes the normally non-singular map | V | ⊆ | E | ։ Y ; suc h normally non-singular maps a re also ca lled sp e cial normal ly non-singular submersions . Th us any co rresp ondence is equiv alent to a sp ecial one, that is, one with a specia l nor mally non-singular submersion f . Notice that the map | V | ։ M → X is almost never prop er. Since any map is transverse to a sp ecial normally non-singular submersion, it is easy to describ e the intersection product for s pecia l co rresp ondences and to c heck 4 HEA TH EMERSON AND RALF MEYER that it has the expec ted prop erties, including functorialit y of the c anonical map to Kasparov theory . Since corr esp ondences that a ppea r in pra ctice are usually not sp ecial, w e deﬁne a notion of tra nsversalit y for gener al corresp ondences and des crib e in tersection pro ducts more directly in the transverse case. The wrong -wa y functorialit y for normally non-s ingular maps in [12] provides a natural transfor mation (1.1) c kk ∗ G ( X, Y ) → KK G ∗  C 0 ( X ) , C 0 ( Y )  . The main result of this ar ticle is that (1.1) is an isomo rphism if G is a proper lo cally compact group oid with Haar system and X admits a nor mally non-s ingular ma p to Z × [0 , 1) o r to Z , where Z is the ob ject space of G . In the non-equiv a riant case, this assumption means that X × R n carries a structure of smo oth ma nifold with b oundar y for some n ∈ N . In the equiv ar iant cas e, the existence of such a nor mally non- singular map implies that t here is a G - vector bundle E ov er Z such that X × Z | E | is a bundle of smoo th manifolds with bo undary over Z , with a ﬁbrewise smo oth action of G . Con versely , such a smo oth structure yields a normally non-singular ma p X → Z under a tec hnical assumption ab out e quiv a riant v ector bundles. The additional technical assumption holds, for instance, if G = G ⋉ U for a discre te gr oup G and a ﬁnite-dimensional prop er G -space U with uniformly bo unded isotropy gr oups, or if G is a compact gro up and X is compact. The pr o of that (1.1) is an isomorphism for such spaces X is based on Poincaré dualit y . If X admits a no rmally non- singular ma p to Z × [0 , 1) , then w e desc ribe a G -space P that is dual to X in the sense that ther e are natural isomo rphisms c kk ∗ G ⋉ X ( X × Z U, X × Z Y ) ∼ = c kk ∗ G ( P × Z U, Y ) , (1.2) c kk ∗ G ( X × Z U, Y ) ∼ = c kk ∗ G ⋉ X ( X × Z U, P × Z Y ) (1.3) for any G -s paces Y and U , and similarly for the analytic theory KK instead of c kk . The corres po nding duality in Kasparov theory is studied in [10], where suﬃcien t and necess ary conditions for it are established. These cr iteria carr y ov er a lmost literally to the top olog ical version of Kas parov theory . In particular, (1.3) for U = Z identiﬁes c kk ∗ G ( X, Y ) ∼ = c kk ∗ G ⋉ X ( X, P × Z Y ) , KK G ∗  C 0 ( X ) , C 0 ( Y )  ∼ = KK G ⋉ X ∗  C 0 ( X ) , C 0 ( P × Z Y )  ∼ = RK ∗ G ,X ( P × Z Y ) , where RK ∗ G ,X ( P × Z Y ) is the G -equiv ar iant K -theory o f P × Z Y with X -compact suppo rt. The map in (1.1) is an isomo rphism if X = Z beca use our biv ariant K -theory extends ordinary K - theory . Hence (1.1) is an isomo rphism whenever X has a duality isomorphism (1.3). Using the results in [12], this implies that (1.1) is in vertible provided X is a smo oth G - manifold with b oundary and some technical assumptions ab out G -vector bundles are satisﬁed. W e do not ex pect our topo logical biv ar iant K -theory to ha ve go o d pro pe rties in the same generality in whic h it may be deﬁned. Equiv a riant K asparov theory has go o d excision pro per ties (long exact sequences) for pro per actions in complete generality; in con trast, w e w o uld b e s urprised if the same were true for o ur top o- logical theo ry . W e have not studied its e xcision prope rties, but it seems likely that excision r equires some tec hnical assumptions. Corr esp ondingly , we do not exp ect our theory to ag ree with Kasparov theory in all cases. W e would lik e to e xtend our thanks to Paul Baum for a n umber of interesting conv ersa tions on the sub ject of topo logical KK -theory . BIV ARIANT K-THEOR Y VIA CORRESPONDE NCES 5 2. Correspondences W e ﬁrst deﬁne corresp ondences. Then we deﬁne equiv alence of corresp ondences, using the elementary rela tions of equiv alence o f normally non-singular map, b or- dism, and Thom mo diﬁcation. Equiv alence classes of F-or ien ted corresp ondences will be shown to form a group, whic h w e denote by ˆ f ∗ ( X, Y ) or ˆ f ∗ G ( X, Y ) in the G -equiv ariant case. The intersection pro duct deﬁning the compo sition of corresp ondences is only well- deﬁned under a transversality condition (see [7]). W e restrict attention to sp ecial corresp ondences at some p oint to r ule o ut this pr oblem. The Thom mo diﬁcation allows us to r eplace any corresp ondence by one who se normally non-sing ular map is a sp ecial normally no n-singular submersion, a nd this implies the transversality condition for all in tersection pro ducts. Thus w e turn ˆ f ∗ in to a Z -graded category . W e also deﬁne an exterior pro duct that turns it into a Z -gra ded symmetric monoidal category . Before we start with this, we br ieﬂy recall s ome prereq uisites for our theory fro m [9, 12]. Throughout this article, all top ologica l spa ces, including all topo logical group- oids, are assumed to be paracompa ct a nd Hausdorﬀ. W e shall use the notion of a numerably prop er gro upoid intro duced in [12]. Eq uiv a rian t v ector bundles for n umera bly prop er actions behave lik e non-equiv ariant v ector bundles: equiv ar i- ant sections or equiv ar iant vector bundle mo rphisms extend from closed inv ariant subspaces, vector bundle extensio ns split equiv ar iantly , and pull-bac ks along equiv- ariantly homotopic maps a re isomorphic. An action of a loc ally compact group oid with Haar system on a lo cally compact space is n umera bly prop er if and only if the action is prop er and the orbit space is para compact. As in [12], we write | V | for the total space, π V for the bundle pro jection, a nd ζ V for the zer o section of a v ector bundle V . W e reserve the arrows ֌ , ։ , and ֒ → for zer o sections, vector bundle pro jections, and op en embeddings, r esp ectively . A G -vector bundle is called t rivial if it is pulled back from the ob ject spa ce o f G , and subtrivial if it is a direct summand in a trivial G -vector bundle. Since our constructions use no specia l proper ties of K -theory , we mo stly w o rk with a gener al equiv ariant m ultiplicativ e coho mology theory F G as in [12]. Given F G and a G - space X , the F-cohomology F ∗ G ,X ( Y ) of Y with X - compact supp ort and the notion of an F-orient ed G - vector bundle are deﬁned in [12]. An F-orien ted v ector bundle V over Y has a Thom isomorphism F ∗ X ( Y ) ∼ = F ∗ X ( | V | ) . F urthermore, F ∗ G ,X is functorial for o pen embeddings. The cohomolo gy theory we are most interested in is (repr esentable) equiv aria n t K -theory . The resulting K - theory with X -compa ct supp ort a grees with the corre- sp onding theor y deﬁned in [9]. Actually , equiv ar iant (r epresent able) K -theory is deﬁned in [9] only for lo cally compact prop er G - spaces, so that w e should imp ose such restrictions whenever w e want to sp ecia lise to K -theory . Normally non-singular maps are in tro duced in [12]. Since some details of this deﬁnition will b ecome crucia l here, w e recall it: Deﬁnition 2.1. Let G b e a numerably pro pe r g roup oid w ith ob ject space Z and let X a nd Y b e G -spaces. Let F b e a G -equiv aria n t m ultiplicative cohomolog y theory . An F-oriented normal ly non-singular G -map from X to Y consists of • V , an F-oriented subtrivial G -vector bundle ov er X ; • E , an F-orie n ted G -vector bundle ov e r Z ; • ˆ f : | V | ֒ → | E Y | , an open em b edding (that is, ˆ f is a G -equiv ar iant map from | V | o n to a n op en subset of | E Y | = | E | × Z Y that is a homeo morphism with resp ect to the s ubspace topolo gy from | E Y | ). 6 HEA TH EMERSON AND RALF MEYER In addition, we assume that the dimensions of the ﬁbres of the G -vector bundles V and E are bo unded ab ov e by some n ∈ N . The tr ac e of a no rmally non-singular map is the G - map f : = π E Y ◦ ˆ f ◦ ζ V : X ֌ | V | ֒ → | E Y | ։ Y . Its de gr e e is dim V − dim E if this lo ca lly co nstant function o n X is constant (other- wise the degr ee is not deﬁned). Its stable normal bund le is [ V ] − [ E X ] , viewed as an element in the Grothendiec k g roup of the monoid of F-oriented subtrivial G - vector bundles on X . The normally non-sing ular G -map ( V , E , ˆ f ) is called a norma l ly non-singular emb e dding if E = 0 , so that π E Y = Id Y and f = ˆ f ◦ ζ V ; it is called a sp e cial normal ly non-singular submersion if V = 0 , so that ζ V = Id X and f = π E Y ◦ ˆ f . The assumption that E should be trivial is needed to deﬁne the comp osition o f normally non-singular maps – this requires extending the G -vector bundle E Y to larger spa ces, which only works in a ca nonical wa y for trivial G -vector bundles. As a consequence, a vector bundle pr o jection π V : | V | → Y is the trace of a sp ecial normally non-s ingular submersion if and only if V is t rivial. If V is subtrivial, then we may at least lift π V to a n ormally non- singular map (see [12]). Since some manipulations with co rresp ondences req uire π V to have such a no rmally non- singular lifti ng, we need V to be subtrivial in Deﬁnition 2 .1. This assumption is already made in [12], but it only b ecomes relev ant her e. An F-or ien ted nor mally non-singula r map f from X to Y genera tes a wrong-wa y map f ! : F ∗ Z ( X ) → F ∗ Z ( Y ) , see [1 2]. The notion of equiv alence for normally non- singular maps is base d on a natural notion of iso topy and on a lifting alo ng tr ivial G -vector bundles. W e refer to [12] for the deﬁnition of the co mpos ition and exterior pro duct of normally non-singular maps. The top ological wrong-wa y functoria lit y f 7→ f ! is w ell-deﬁned o n eq uiv a lence classes and is compatible with comp osition and exterior pro ducts. Now let X and Y b e smo oth G -manifolds (see [12]). Then w e may also c onsider smo oth normally non-singular maps from X to Y . F or s uch maps , w e require a smo oth structure on the G -vector bundle V (this is automatic for E ) a nd assume that ˆ f is a ﬁbrewise diﬀeomorphism. Smo o th e quiv a lence for smo oth normally non- singular maps is based on smo oth isotopies and lifting. Under suitable tec hnical h yp otheses, an y s moo th map X → Y lift s to a smoo th nor mally no n-singular map, which is unique up to smo oth equiv alence . F or instance, this works if G is a compact group and X is compact, or if G = G ⋉ Z for a discrete g roup G and a prop er G -CW- complex Z with ﬁnite c ov ering dimension and with uniformly b ounded size of the isotropy gro ups (see [12]). There a re also examples of compa ct group oids for whic h all this fails, that is, there ma y b e s mo oth maps that do not lift to smo oth normally non-singular maps . These counterexamples oblige us to use norma lly non-singular maps. 2.1. The de ﬁnition of correspo ndence. Deﬁnition 2.2. A ( G -e quivariant, F -oriente d ) c orr esp ondenc e from X to Y is a quadruple ( M , b, f , ξ ) , where • M is a G -space ( M for middle); • b : M → X is a G -map ( b for backw ards); • f : M → Y is an F-oriented normally non-singular G -map ( f for forwards); • ξ b elongs to F ∗ X ( M ) ; here we use b to view M as a space over X . The de gr e e of a corres po ndence is the sum of the degrees of f and ξ (it need not be deﬁned). BIV ARIANT K-THEOR Y VIA CORRESPONDE NCES 7 A corr espo ndence ( M , b , f , ξ ) is called pr op er if b : M → X is prop er. Then any closed subset of M – including M itself – is X -compact, so that F ∗ X ( M ) ∼ = F ∗ ( M ) . Our deﬁnition deviates from previous o nes (see [4, 7, 19]) in tw o aspec ts: w e do not r equire b to b e prop er, and we let f b e a nor mally non-singular map instead of a smoo th map. W e have explained in the int ro duction wh y these c hanges are helpful. Example 2.3 . A prop er G -map b : Y → X yields a corres po ndence b ∗ : = ( Y , b, Id Y , 1) from X to Y , where Id Y denotes the identit y no rmally non- singular map on Y and 1 ∈ F ∗ ( Y ) ∼ = F ∗ X ( Y ) is the unit elemen t. Example 2.4 . An F- oriented nor mally non-singular G -map f : X → Y yields a corresp ondence f ! : = ( X , Id X , f , 1) from X to Y . Example 2.5 . Any cla ss ξ in F ∗ ( X ) yields a cor resp ondence ( X, Id X , Id X , ξ ) fro m X to itself. Deﬁnition 2 . 6. The su m of tw o corr esp ondences is their disjoint union: ( M 1 , b 1 , f 1 , ξ 1 ) + ( M 2 , b 2 , f 2 , ξ 2 ) : = ( M 1 ⊔ M 2 , b 1 ⊔ b 2 , f 1 ⊔ f 2 , ξ 1 ⊔ ξ 2 ) . This uses [1 2, Lemma 4.30] and is w ell-deﬁned, asso ciative, and commutativ e up to isomorphism. The empty cor resp ondence with M = ∅ acts a s zero. An y corresp ondence decomp os es uniquely a s a sum of corresp ondences o f deg ree j for j ∈ Z . T o s ee this, write f = ( V , E , ˆ f ) , and decomp ose M into the disjoin t subsets where V , f ∗ ( E ) , and ξ have certain degrees. The dimension assumption ensures that only ﬁnitely ma n y non-empty pieces ar ise. 2.2. Equiv alence of corresp ondences. Now w e deﬁne when tw o c orresp ondences are equiv alent. F or this, we in tr o duce several e lemen tary relations, which to gether generate equiv alence. Besides isomor phism, we need equiv alence of the nor mally non-singular maps, bor dism, and Thom mo diﬁcation; the latter replaces the notions of vector bundle mo diﬁcation used in [2, 4]. The o nly re ason not to call it by that name is to av oid c onfusion with the tw o diﬀere n t notions that alre ady g o by it. It is clear when t wo co rresp ondences ar e isomorphic . In the following, w e ta c- itly w ork with isomor phism classes of corresp ondences all the time. Equivalenc e of normal ly non-singular maps simply means that we consider the co rresp ondences ( M , b, f 0 , ξ ) and ( M , b, f 1 , ξ ) equiv a lent if f 0 and f 1 are equiv alent F-oriented nor- mally non-singular maps. Deﬁnition 2 . 7. A b or dism of c orr esp ondenc es from X to Y consists of • W , a G -s pace; • b , a G -map fro m W to X ; • f : = ( V , E , ˆ f ) , an F-oriented normally no n-singular G -map from W to Y × [0 , 1] – that is, V is a subtrivial G - vector bundle ov er W , E is a G -vector bun- dle ov er Z , and ˆ f is an o pen embedding from | V | into | E Y × [0 , 1] | ∼ = | E Y | × [0 , 1] – with the additional prop ery that there are subsets ∂ 0 W , ∂ 1 W ⊆ W such that ˆ f − 1  Y × { j }  = π − 1 V ( ∂ j W ) ⊆ | V | for j = 0 , 1 ; • ξ ∈ F ∗ X ( W ) . Example 2.12 explains the r elationship to the more tra ditional notion of bor dism. A bor dism Ψ = ( W, b, f , ξ ) from X to Y restr icts to corresp ondences ∂ j Ψ = ( ∂ j W , b | ∂ j W , f | ∂ j W , ξ | ∂ j W ) 8 HEA TH EMERSON AND RALF MEYER from X to Y , where f | ∂ j W denotes the no rmally non-singular map ( V | ∂ j W , E , ˆ f j ) from ∂ j W to Y with ˆ f ( v ) = ( ˆ f j ( v ) , j ) for v ∈ π − 1 V ( ∂ j W ) , j = 0 , 1 . W e call these corr espo ndences ∂ 0 Ψ and ∂ 1 Ψ b or dant and write ∂ 0 Ψ ∼ b ∂ 1 Ψ . Finally , w e incorp orate Thom iso morphisms: Deﬁnition 2.8. Let Ψ : = ( M , b , f , ξ ) b e a cor resp ondence from X to Y and let V be a subtrivial F-or ien ted G - vector bundle o ver M . Let π V : | V | ։ M b e the bundle pro jection, viewed a s an F-orient ed normally non-singular G -map. The Thom mo diﬁc ation Ψ V of Ψ with re spec t to V is the co rresp ondence  | V | , b ◦ π V , f ◦ π V , τ V ( ξ )  ; here f ◦ π V denotes the compo sition of F-oriented normally non-singular maps and τ V denotes the Tho m isomorphism F ∗ X ( M ) → F ∗ X ( | V | ) for V , shifting degrees b y + dim( V ) and g iven by comp osing pull-back with multiplication by an assumed Thom class , or orientation class in F dim V X ( | V | ) (see [12, Deﬁnition 5.1]). Deﬁnition 2.9. Equivalenc e of corre spo ndences is the equiv ale nce re lation on the set of c orresp ondences from X to Y ge nerated by equiv alence of normally non- singular ma ps, bo rdism, a nd Thom mo diﬁcation. Let ˆ f ∗ ( X, Y ) b e the set of equiv- alence classes of co rresp ondences fr om X to Y . Equiv alence preserves the degree and the addition of corresp ondences, so that F ∗ ( X, Y ) b ecomes a gr aded monoid. W e will sho w below that rev ersing the F-orient ation on f provides additiv e in- verses, so that ˆ f ∗ ( X, Y ) is a gr aded Abelia n group. 2.3. Examples of b ordism s. W e establish that b ordism is a n equiv alence relation and that it contains homo top y for the maps b : M → X and isotopy for the normally non-singular maps f : M → Y . W e also construct so me important examples of bo rdisms. Prop osition 2.10 . The r elation ∼ b is an e quivalenc e r elation on c orr esp ondenc es fr om X to Y . Pr o of. Let Ψ = ( M , b, ( V , E , ˆ f ) , ξ ) b e a co rresp ondence from X to Y . Deﬁne W : = M × [0 , 1] , b ′ : = b ◦ p , V ′ : = p ∗ ( V ) , ˆ f ′ ( v , t ) =  ˆ f ( v ) , t  , ξ ′ : = p ∗ ( ξ ) , where p : W → M is the co o rdinate pro jection. Then ( W , b ′ , ( V ′ , E , ˆ f ′ ) , ξ ′ ) is a bo rdism be t ween Ψ and itself, so that ∼ b is reﬂexive. If Ψ = ( W, b, ( V , E , ˆ f ) , ξ ) is a bor dism fro m X to Y , s o is ( W , b, ( V , E , σ ◦ ˆ f ) , ξ ) , where σ : | E Y | × [0 , 1] → | E Y | × [0 , 1] maps ( e, t ) to ( e, 1 − t ) . This exc hanges the roles of ∂ 0 Ψ and ∂ 1 Ψ , proving that ∼ b is symmetric. Let Ψ 1 = ( W 1 , b 1 , ( V 1 , E 1 , ˆ f 1 ) , ξ 1 ) and Ψ 2 = ( W 2 , b 2 , ( V 2 , E 2 , ˆ f 2 ) , ξ 2 ) b e bo rdisms such that the co rresp ondences ∂ 1 Ψ 1 and ∂ 0 Ψ 2 are isomorphic. H ence E 1 ∼ = E 2 – we may even assume E 1 = E 2 – and there is a homeo morphism ∂ 1 W 1 ∼ = ∂ 0 W 2 compatible with the other structure. It a llows us to glue together W 1 and W 2 to a G -space W 12 : = W 1 ∪ ∂ 1 W 1 ∼ = ∂ 0 W 2 W 2 and b 1 and b 2 to a G -map b 12 : W 12 → X . The G - vector bundles V 1 and V 2 combine to a G -vector bundle V 12 on W 12 , whic h inherits an F-orientation by [12, Lemma 5.6]. The classes ξ 1 and ξ 2 combine to a class ξ 12 ∈ F ∗ X ( W 12 ) by the Ma yer–Vietoris sequence for F ∗ X . Rescale ˆ f 1 and ˆ f 2 to o pen embeddings from | V 1 | and | V 2 | to | E Y | × [0 , 1 / 2 ] and | E Y | × [ 1 / 2 , 1] tha t map π − 1 V 1 ( ∂ 0 W 1 ) to | E Y | × { 0 } , π − 1 V 1 ( ∂ 1 W 1 ) and π − 1 V 2 ( ∂ 0 W 2 ) to | E Y | × { 1 / 2 } , a nd π − 1 V 2 ( ∂ 1 W 2 ) to | E Y | × { 1 } . These combine to an op en em b edding ˆ f 12 from | V 12 | BIV ARIANT K-THEOR Y VIA CORRESPONDE NCES 9 in to | E Y | × [0 , 1] . This yields a b or dism ( W 12 , b 12 , ( V 12 , E 12 , ˆ f 12 ) , ξ 12 ) from ∂ 0 Ψ 1 to ∂ 1 Ψ 2 . Th us ∼ b is transitive.  Lemma 2.11. L et ( M , b 0 , f 0 , ξ ) b e a c orr esp ondenc e fr om X t o Y . L et b 0 b e ho- motopic to b 1 and let ˆ f 0 b e isotopic to ˆ f 1 . Then the c orr esp ondenc es ( M , b 0 , f 0 , ξ ) and ( M , b 1 , f 1 , ξ ) ar e b or dant. Pr o of. The b or dism is co nstructed as in the pro o f that bordism is reﬂexive; but this time, b ′ is replaced b y a ho motopy b et ween b 0 and b 1 , and ˆ f ′ b y an isotopy betw een ˆ f 0 and ˆ f 1 .  Example 2.12 . Let X and Y b e smo oth manifolds and let W b e a smo o th ma nifold with b oundary ∂ W , decomp os ed in to t wo disjoint subsets: ∂ W = ∂ 0 W ⊔ ∂ 1 W . Let ξ ∈ F ∗ X ( W ) , let b : W → X b e a smo oth map, and let f : W → Y be a smoo th map that is F-oriented in the sense that f ∗ ( T Y ) ⊕ N W is F-o riented o r, equiv alently , [ T X ] − f ∗ [ T Y ] is stably F-oriented. W e wan t to constr uct a bor dism from this data. W e deﬁne the stable nor mal bundle N W as the restriction of N DW to W , where D W : = W ∪ ∂ W W is the double of W – a smo oth manifold. Recall that N DW is the normal bundle o f a smoo th embedding h : D W → R n for some n ∈ N . W e lift f to a normally non-s ingular map Φ = ( V , R n , ˆ f ) from W to Y × [0 , 1 ] as follows. Let k : W → [0 , 1 ] be a smooth map with ∂ j W = k − 1 ( j ) for j = 0 , 1 and with non-v anishing ﬁrst deriv ative on ∂ W . Then ( f , h | W , k ) : W → Y × R n × [0 , 1] iden tiﬁes W with a neat submanifold of Y × R n × [0 , 1] (see [13, page 30]). The T ubular Neigh b our ho o d Theo rem for smoo th manifolds with b oundary shows that ( f , h | W , k ) extends to a diﬀeomorphism ˆ f from its normal bundle V ∼ = f ∗ ( T Y ) ⊕ N W ⊕ R on to an o pe n subset of Y × R n × [0 , 1 ] . W e get an F-oriented norma lly non-singular map Φ : = ( V , ˆ f , R n ) from W to Y × [0 , 1 ] . Putting everything tog ether, we get a b ordism o f corresp ondences ( W , b , Φ , ξ ) with ∂ 0 W a nd ∂ 1 W a s s peciﬁed. F urthermore, the trace of Φ lifts f to a map W → Y × [0 , 1] , so that Φ | ∂ j W is a norma lly non-singular map with trace f | ∂ j W : ∂ j W → Y . Example 2.13 . Let ( M , b, f , ξ ) be a corr esp ondence from X to Y . Let M ′ ⊆ M b e an open G -inv aria n t subset and as sume that there is ξ ′ ∈ F ∗ X ( M ′ ) that is mapp ed to ξ by the canonica l map F ∗ X ( M ′ ) → F ∗ X ( M ) . W e claim that the corr espo ndences ( M , b, f , ξ ) and ( M ′ , b | M ′ , f | M ′ , ξ ′ ) are borda nt . Here f | M ′ denotes the comp ositio n of f with the op en embedding M ′ ֒ → M , viewed as a norma lly non- singular ma p; if f = ( V , E , ˆ f ) , then f | M ′ = ( V | M ′ , E , ˆ f | M ′ ) . The underlying space of the bor dism is the G -inv ariant o pe n subset W : = M ′ × { 0 } ∪ M × (0 , 1] ⊆ M × [0 , 1 ] with the subspace top ology , induced G -action, and the obvious maps to X and Y (see the pro of o f Prop osition 2.10). W e may pull back ξ ′ to a cla ss in F ∗ X ( M ′ × [0 , 1 ]) , which then extends to a class in F ∗ X ( W ) whos e restrictions to M ′ × { 0 } and M × { 1 } are ξ ′ and ξ , re spec tiv ely . This yields the requir ed b ordism betw een ( M , b, f , ξ ) and ( M ′ , b | M ′ , f | M ′ , ξ ′ ) . It is unclear from our deﬁnition of b or dism which subsets ∂ 0 W and ∂ 1 W of W are p ossible. The following deﬁnition provides a criter ion for this: Deﬁnition 2.14. Let W b e a G -s pace. A closed G - in v ariant subset ∂ W is called a b oundary of W if the embedding ∂ W × { 0 } ∼ = ∂ W → W ex tends to a G -equiv ariant op en embedding c : ∂ W × [0 , 1) ֒ → W ; the map c is called a c ol lar for ∂ W . 10 HEA TH EMERSON AND RALF MEYER If ∂ W ⊆ W is a b oundary , then we let W ◦ : = W \ ∂ W b e the interior o f W . W e iden tify ∂ W × [0 , 1) with a subset o f W using the c ollar. The following lemma uses the auxiliar y o rient ation-preser ving diﬀeomorphism: ϕ : R ∼ = − → (0 , 1 ) , t 7→ 1 2 + t 2 √ 1 + t 2 . Notice that ϕ ( − t ) = 1 − ϕ ( t ) . Lemma 2.15. L et ∂ 0 W ⊔ ∂ 1 W ⊆ W b e a b oundary. Then ther e is an op en emb e d- ding h : W × R ֒ → W ◦ × [0 , 1] with the fol lowing pr op erties: • h ( w, t ) =  w, ϕ ( t )  for w / ∈ ∂ W × [0 , 1 / 2 ) ; • h ( w, t ) ∈ W ◦ × (0 , 1) for w ∈ W \ ∂ W ; • h  ( w, 0 ) , t  =   w, ϕ ( − t ) / 2  , 0  for w ∈ ∂ 0 W ; • h  ( w, 0 ) , t  =   w, ϕ ( t ) / 2  , 1  for w ∈ ∂ 1 W . Pr o of. Let A : = W \ ∂ W × [0 , 1 / 2 ] . W e put h ( w, t ) : =  w, ϕ ( t )  if w ∈ A to fulﬁl the ﬁrst condition; this maps A × R homeomorphically on to A × (0 , 1) . On ∂ W × [0 , 1 / 2 ] × R , w e connect the prescrib ed v alues on ∂ W × { 0 , 1 / 2 } × R by an aﬃne homotopy; that is, if w ∈ ∂ W , s ∈ [0 , 1 / 2 ) , and t ∈ R , then h  ( w, s ) , t  : =      w, s − ( s − 1 / 2 ) ϕ ( − t )  , 2 sϕ ( t )  if w ∈ ∂ 0 W ,   w, s − ( s − 1 / 2 ) ϕ ( t )  , 1 − 2 sϕ ( − t )  if w ∈ ∂ 1 W . A routine computation shows that h maps ∂ W × (0 , 1 / 2 ] × R homeomo rphically onto a relatively op en subset of itself. Hence h is an op en embedding on W × R .  Example 2.16 . Let ( M , b, f , ξ ) be a corresp ondence from X to Y and let V b e a subtrivial F-or ient ed G -vector bundle ov er M equipped with some G - in v a riant inner pro duct. Let S V ⊆ D V ⊆ | V | be the unit sphere and unit disk bundles and let π D : D V → M a nd π S : S V → M be the canonical pro jections. The pro jection π V : | V | ։ M is a n F-or ient ed normally non-singular ma p by [12, Example 4.25]. The embedding S V → | V | is a normally no n-singular em b edding with constant nor mal bundle R for a suitable tubular neig h bo urho o d, say , (2.17) S V × R ֒ → | V | , ( v , t ) 7→ v ·  2 − 2 ϕ ( t )  , with the auxiliary function ϕ ab ov e. Hence π S is an F-orien ted normally non- singular map. W e get a corresp ondence  S V , b ◦ π S , f ◦ π S , π ∗ S ( ξ )  from X to Y . W e claim that this cor resp ondence is bor dant to the empt y cor resp ondence. W e wan t to construct a b ordis m ( W , b ′ , f ′ , ξ ′ ) with W = D V , b ′ : = b ◦ π D , ξ ′ : = π ∗ D ( b ) , ∂ 0 W = ∅ , ∂ 1 W = S V . The F-oriented normally non-singular map f ◦ π V : | V | → Y pulls back to an F-oriented nor mally non-sing ular map ( f π V ) × [0 , 1 ] : | V | × [0 , 1] → Y × [0 , 1] . W e let f ′ be the comp osition of ( f π V ) × [0 , 1] with the F-o riented normally non-singular embedding ( D V × R , h ) from D V to | V | × [0 , 1] , where h : D V × R → ( D V ) ◦ × [0 , 1] ⊆ | V | × [0 , 1 ] is the o pen embedding constructed in Lemma 2.15. Her e we use the co l- lar S V × [0 , 1) ֒ → D V , ( v , t ) 7→ v · (1 − t ) . The op en em b edding h | S V : S V × R ֒ → | V | is is otopic to the tubular neighbourho o d fo r S V in (2.17). Hence the bo undary of ( W , b ′ , f ′ , ξ ′ ) is equiv alent to  S V , b ◦ π S , f ◦ π S , π ∗ S ( ξ )  . Example 2.18 . Let Ψ = ( M , b , f , ξ ) b e a cor resp ondence from X to Y . Let − f de- note f with the opp osite F-or ient ation a nd let − Ψ : = ( M , b , − f , ξ ) . Up to bor dism, this is in verse to Ψ , that is, the disjoint union Ψ ⊔ − Ψ is b ordant to the empty cor - resp ondence. As a cons equence, bo rdism classes of G -equiv aria nt corresp ondences from X to Y form an Ab elian g roup. BIV ARIANT K-THEOR Y VIA CORRESPONDE NCES 11 The bordism Ψ ⊔ − Ψ ∼ b ∅ is, in fact, a specia l case of Example 2.16 wher e V : = M × R is the constant v ec tor bundle o f ra nk 1 ; hence the disk bundle D V is simply M × [0 , 1] and the unit s phere bundle is M ⊔ M . The sign co mes from the orientation-reversal o n one bo undary comp onent in L emma 2.15. The las t t wo examples allow us to re late the Thom mo diﬁcatio n in Deﬁnition 2 .8 to the vector bundle modiﬁca tions used in [19] and [2, 4]. Let Ψ : = ( M , b , f , ξ ) b e a corresp ondence fro m X to Y and let V be a subtrivial F-oriented G -vector bundle over M . Since the bundle pro jection π V : | V | ։ M is not prop er, the Thom modiﬁcation ma kes no sense in the setting of [2, 4, 19]. Let ¯ V b e the unit sphere bundle in V ⊕ R . This co n tains | V | as an op en subset, whose complement is ho meomorphic to M via the ∞ -section. Excision for F yields a canonical map F ∗ X ( | V | ) ∼ = F ∗ X ( ¯ V , M ) → F ∗ X ( ¯ V ) . Let ¯ τ V : F ∗ X ( M ) → F ∗ X ( ¯ V ) be its comp osition with the Thom isomorphism. The pro jection π V : | V | ։ M extends to a n F-oriented normally non-singula r map ¯ π V : ¯ V → M . W e get a correspo ndence  ¯ V , b ◦ ¯ π V , f ◦ ¯ π V , ¯ τ V ( ξ )  . This is pre- cisely the vector bundle mo diﬁcation used by Jeﬀ Rav en in [19]. Example 2.13 shows that  V , b ◦ π V , f ◦ π V , τ V ( ξ )  ∼ b  ¯ V , b ◦ ¯ π V , f ◦ ¯ π V , ¯ τ V ( ξ )  . Thu s the Thom mo diﬁca tion is bor dant to Rav en’s v ector bundle modiﬁcatio n of Ψ . The notion of vector bundle mo diﬁcation in [2, 4] is slight ly diﬀerent from Rav en’s. The clutc hing construction in [4] do es not inv olve the full Thom class, it only uses its non-trivial half. Recall that the Thom class τ V ∈ RK ∗ G ,M ( ¯ V , M ) in K -theory restricts to the Bott generator in K n ( S n , ⋆ ) in each ﬁbre. Since the dimension v an- ishes on this r elative K -g roup, the Thom cla ss is a diﬀerence o f tw o vector bundles. One is the clutching construction of [4], the other is pulled back fro m M . Leav- ing out this second half yields a b orda n t corr espo ndence b ecause of Example 2.16, which yields a b or dism  ¯ V , b ◦ ¯ π V , f ◦ ¯ π V , ¯ π ∗ V ( δ )  ∼ b ∅ for any δ ∈ F ∗ X ( M ) . This is wh y the tw o notions of vector bundle mo diﬁcation used by Ba um and Rav en are almos t equiv alent. The only diﬀerence is that – unlike Baum’s – Rav en’s vector bundle mo diﬁcation cont ains the direc t s um–disjoin t unio n relation when combined with b ordism (see [19, Prop osition 4.3.2]). Lemma 2 .19. L et Ψ 1 = ( M , b , f , ξ 1 ) and Ψ 2 = ( M , b, f , ξ 2 ) b e two c orr esp ondenc es fr om X to Y with the same data ( M , b, f ) and let Ψ + : = ( M , b , f , ξ 1 + ξ 2 ) . The c orr esp ondenc es Ψ 1 ⊔ Ψ 2 and Ψ + ar e e quivalent. Pr o of. W e are going to construct a bo rdism b etw een the Tho m mo diﬁcations of Ψ 1 ⊔ Ψ 2 and Ψ + along the constant 1 -dimensional G -vector bundle R . Let W : = [0 , 1 ] × R \ { (0 , 0) } and ∂ j W : = W ∩ { j } × R . Th us ∂ 0 W ∼ = R ⊔ R and ∂ 1 W ∼ = R . The b ordism we seek is of the for m ( W × M , b ◦ π 2 , f ′ , ξ ) for a certain ξ ∈ F ∗ X ( W × M ) . Here π 2 : W × M → M is the ca nonical pro jection, and f ′ is the exterior pro duct the op en em b edding W ֒ → [0 , 1 ] × R with f . Excision shows that restriction to ∂ 0 W induces an iso morphism F ∗ X ( W × M ) ∼ = F ∗ X  ( R ⊔ R ) × M  ∼ = F ∗ +1 X ( M ) ⊕ F ∗ +1 X ( M ) . Hence there is a unique ξ ∈ F ∗ X ( W × M ) whose restriction to ∂ 0 W is ξ 1 ⊔ ξ 2 and whose restriction to ∂ 1 W is ξ 1 + ξ 2 . This provides the desired b ordism betw een the Thom mo diﬁcations of Ψ 1 ⊔ Ψ 2 and Ψ + along R .  12 HEA TH EMERSON AND RALF MEYER 2.4. Sp ecial corresp o ndences. W e use Thom mo diﬁcations to bring corresp on- dences int o a standard form. This greatly simpliﬁes the deﬁnition of ˆ f ∗ ( X, Y ) and is needed for the co mpos ition pro duct. Deﬁnition 2.20. A cor resp ondence ( M , b, f , ξ ) or a bor dism ( W, b, f , ξ ) is ca lled sp e cial if f is a sp ecial nor mally no n-singular submersion. Example 2.21 . The co rresp ondence b ∗ for a prop er G -map b : M → X des crib ed in Example 2.3 is sp ecial. Recall that a sp ecial norma lly no n-singular submersio n from X to Y is a norma lly non-singular map of the form ( X , ˆ f , E ) , where ˆ f identiﬁes X with a n op en subset of | E Y | . In a sp ecial corres po ndence, we ma y replace M b y this subset of | E Y | , so that ˆ f b ecomes the identit y map. Hence a sp ecia l corre spo ndence from X to Y is equiv alent to a qua druple ( E , M , b, ξ ) , where • E is an F-oriented G -vector bundle ov er Z ; • M is an op en subset of | E Y | ; • b is a G -ma p from M to X ; • ξ ∈ F ∗ X ( M ) ; a sp ecial b ordism fro m X to Y is equiv alent to a quadruple ( E , W, b, ξ ) , where • E is an F-oriented G -vector bundle ov er Z ; • W is an op en subset of | E Y | × [0 , 1] ; • b is a G -ma p from W to X ; • ξ ∈ F ∗ X ( W ) . Observe that ∂ t W : = W ∩  | E Y | × { t }  for t = 0 , 1 – viewed as op en subsets of | E Y | – automatically have the prop erties r equired in Deﬁnition 2.7. This is why a sp ecial bo rdism fr om X to Y is nothing but a sp ecial corresp ondence from X to Y × [0 , 1] (this is not true for general b ordisms). The b oundaries of a specia l bo rdism are the sp ecial corresp o ndences ( E , ∂ t W , b | ∂ t W , ξ ) for t = 0 , 1 . Thom mo diﬁcations of sp ecial corresp ondences need not b e specia l an y more, unless we mo dify b y a tr ivial G -vector bundle (compa re [12, Example 4.2 5]). Deﬁnition 2.22. Let Ψ = ( E , M , b, ξ ) b e a sp ecial c orresp ondence from X to Y and let V b e an F-oriented G -vector bundle over Z . The Thom mo diﬁc ation of Ψ b y V is the sp ecial corresp ondence Ψ V : =  E ⊕ V , M × Z | V | , b ◦ π V M , τ V ( ξ )  from X to Y , w here τ V : F ∗ X ( M ) → F ∗ X ( | V M | ) = F ∗ X ( M × Z | V | ) is the Thom isomorphism for the induced F -orientation on V M . Theorem 2.23. Any c orr esp ondenc e ( M , b, f , ξ ) is e quivalent to a sp e cial c orr e- sp ondenc e. Two sp e cial c orr esp ondenc es ar e e quivalent if and only if they have sp e cial ly b or dant Thom mo diﬁc ations by G -ve ctor bund les over Z . Pr o of. Let Ψ = ( M , b , f , ξ ) b e a corresp ondence fro m X to Y and let f = ( V , E , ˆ f ) , where V is a subtrivial F-oriented G -vector bundle ov er M , E is an F-o riented G -vector bundle o ver Z , and ˆ f is an o pen em b edding from | V | into | E Y | . The Thom mo diﬁcation of Ψ a long V is a corresp ondence that inv olves the comp osite normally non-singular map | V | ։ M f − → Y , which is equiv alent to the spec ial normally non- singular submersion ( ˆ f , E ) (see [12, Exa mple 4.24]). This yields a sp ecial corresp o ndence equiv alent to Ψ . W e may do the same to a b ordism Ψ = ( W , b, f , ξ ) ; write f = ( V , E , ˆ f ) , then the Thom modiﬁcation of W a long V is a sp ecial b or dism, whos e bo undaries a re the Thom modiﬁcatio ns of the b oundar ies ∂ 0 Ψ and ∂ 1 Ψ of Ψ along the re strictions of V to ∂ 0 W a nd ∂ 1 W , respec tiv ely . BIV ARIANT K-THEOR Y VIA CORRESPONDE NCES 13 F or sp ecial cor resp ondences Ψ 1 and Ψ 2 , w e wr ite Ψ 1 ∼ sb Ψ 2 if there is a sp ecial bo rdism b etw een Ψ 1 and Ψ 2 , and Ψ 1 ∼ s Ψ 2 if there are G - vector bundles V 1 and V 2 ov er Z with Ψ V 1 ∼ sb Ψ V 2 . An argument as in the pro o f of Prop osition 2.10 sho ws that ∼ sb is an equiv alence relation. W e claim that ∼ s is an equiv alence relation a s well. This follows as in the pr o of of [12, Lemma 4 .16] using the following obs erv ation: if V 1 and V 2 are t wo G -vector bundles ov er Z , then the Tho m modiﬁca tion along V 1 follow ed b y the Thom mo diﬁcation along V 2 yields the Thom modiﬁcatio n along V 1 ⊕ V 2 . It is clear that tw o sp ecial corres po ndences are equiv alent if Ψ 1 ∼ s Ψ 2 beca use Thom mo diﬁcation by G -vector bundles o ver Z and spe cial b or dism ar e contained in the relations that generate the eq uiv a lence o f corres po ndences. T o show that the tw o relations are equal, we m ust check the following. Let Ψ 1 and Ψ 2 be corresp ondences and let Ψ ′ 1 and Ψ ′ 2 be the a sso ciated specia l corresp ondences as above. If Ψ 1 and Ψ 2 are r elated by an equiv alence of normally non-sing ular maps, a bordis m, or a Thom mo diﬁcation, then Ψ ′ 1 ∼ s Ψ ′ 2 . W e may further split up equiv a lence of nor mally non-singular maps into iso topy and lifting of norma lly non-singular maps a nd only hav e to consider these tw o specia l cases. Let Ψ 1 = ( M , b , f , ξ ) with f = ( V , E , ˆ f ) . W e hav e alrea dy observed a bove that bo rdism is co n tained in ∼ s . This a lso cov ers isotopy of normally non-singular maps, which is a special case of b ordism b y Lemma 2.11. Now suppo se that Ψ 2 = ( M , b , f E 2 , ξ ) , wher e f E 2 = ( V ⊕ E M 2 , ˆ f E 2 , E ⊕ E 2 ) is the lifting of f along a G - vector bundle E 2 ov er Z . Then Ψ ′ 2 is the Thom mo diﬁcation of Ψ ′ 1 along E 2 , so that Ψ ′ 1 ∼ s Ψ ′ 2 . Finally , let Ψ 2 be the Thom mo diﬁcation of Ψ 1 along some subtrivial G -vector bundle V 2 ov er M , that is, Ψ 2 =  V 2 , b ◦ π V 2 , f ◦ π V 2 , τ V 2 ( ξ )  . Let V ⊥ 2 and E 2 be G -vector bundles o ver M a nd Z with V 2 ⊕ V ⊥ 2 ∼ = E M 2 , let ι : | V 2 | ⊕ V ⊥ 2 → E M 2 be the isomorphism. Then f ◦ π V 2 =  π ∗ V 2 ( V ⊥ 2 ⊕ V ) , ( Id E 2 × Z ˆ f ) ◦ ( ι × M Id | V | ) , E 2 ⊕ E  ; here we use that ι × M Id | V | iden tiﬁes the total spac e of π ∗ V 2 ( V ⊥ 2 ⊕ V ) with | π ∗ V 2 ( V ⊥ 2 ⊕ V ) | ∼ = | V 2 | × M | V ⊥ 2 | × M | V | ∼ = | E M 2 | × M | V | ∼ = E 2 × Z | V | . Now it is routine to c heck that Ψ ′ 2 is isomorphic to the Tho m mo diﬁcation of Ψ ′ 1 along E 2 .  Theorem 2.2 4. Ther e is a natur al isomorphi sm ˆ f ∗ ( Z, Y ) ∼ = F ∗ Z ( Y ) for al l G -sp ac es Y . Pr o of. Theorem 2.23 shows that ˆ f ∗ ( Z, Y ) is the set of ∼ s -equiv alence classes of sp ecial cor resp ondences from Z to Y . Let ( E , M , b, ξ ) b e a sp ecial co rresp ondence from Z to Y as a bove. The map b must be the a nch or map of M b y G -eq uiv a riance and therefore extends to  : | E Y | → Z . Thus E xample 2.1 3 provides a special bo rdism ( E , M , b, ξ ) ∼ sb ( E , | E Y | , , ¯ ξ ) , where ¯ ξ ∈ F ∗ Z ( | E Y | ) is the imag e of ξ under the canonical map F ∗ Z ( M ) → F ∗ Z ( | E Y | ) . Hence we may restrict a tten tion to specia l corresp ondences with M = | E Y | . The same argument for specia l bor disms b etw een such corresp ondences shows that any sp ecial b ordism ex tends to a c onstant o ne. Thu s sp ecial co rresp ondences with M = | E Y | are equiv alent if a nd only if they b ecome equa l after a Thom mo diﬁcation. Now ( E , | E Y | , , ¯ ξ ) is the Thom mo diﬁcation of  0 , Y , , τ − 1 E Y ( ¯ ξ )  along E Y . Hence any class in ˆ f ∗ ( Z, Y ) is represented b y (0 , Y , , η ) for a unique η ∈ F ∗ Z ( Y ) . Thus F ∗ Z ( Y ) ∼ = ˆ f ∗ ( Z, Y ) .  14 HEA TH EMERSON AND RALF MEYER This result may seem rather trivial, but it is the place where many of our techni- cal mo diﬁcations of the notion of a corr esp ondence are used. Wit hout the nor mal factorisation o r without assuming the subtrivialit y of the vector bundle V in a normally non-singular map, w e could not simplify cycles for ˆ f ∗ ( Z, Y ) as above. F urthermore, Theorem 2.24 is the one case where we use the deﬁnition of ˆ f ∗ ( Z, Y ) to co mpute the theory . O ur pro of that biv ar iant top ologica l and analytic K -theory are equal will use duality to r educe the general case to this sp ecia l case. The dual- it y a rgument only uses forma l pro per ties o f the corr esp ondence catego ry and some rather sp ecial corresp ondences whic h are needed to g enerate the dualit y isomor- phisms (the latter requir e an additional, geometric hypothesis). 2.5. Comp os ition of corresp ondences. W e ﬁrst deﬁne the c ompo sition only for special corresp ondences . Let Ψ 1 : = ( E 1 , M 1 , b 1 , ξ 1 ) a nd Ψ 2 : = ( E 2 , M 2 , b 2 , ξ 2 ) be special cor resp ondences from X to Y and from Y to U , respectively . Their comp osition pro duct Ψ 1 # Ψ 2 is a sp ecial cor resp ondence ( E , M , b, ξ ) from X to U . W e let E : = E 1 ⊕ E 2 and for m M : = M 1 × Y M 2 using the maps M 1 ⊆ | E Y 1 | ։ Y and b 2 : M 2 → Y . W e iden tify M with a n ope n subset of | E U | as follows: M = M 1 × Y M 2 ∼ =  ( u, e 1 , e 2 ) ∈ U × Z E 1 × Z E 2 = | E U |   ( u, e 2 ) ∈ M 2 and ( b 2 ( u, e 2 ) , e 1 ) ∈ M 1  W e deﬁne b : M → X by b ( m 1 , m 2 ) : = b 1 ( m 1 ) and let ξ : = ξ 1 ⊗ Y ξ 2 ∈ F 0 X ( M ) b e the exterior pro duct o f ξ 1 ∈ F 0 X ( M 1 ) and ξ 2 ∈ F 0 Y ( M 2 ) ; more precisely , ξ 1 ⊗ Y ξ 2 ∈ F 0 X ( M ) denotes the restriction of the exter ior pro duct ξ 1 × Z ξ 2 in F 0 X × Z Y ( M 1 × Z M 2 ) to M 1 × Y M 2 ; we may change the supp or t condition bec ause X × Z Y -compact subsets of M 1 × Y M 2 are X -compa ct. This yields a sp ecial cor resp ondence ( E , M , b, ξ ) , whic h w e denote by Ψ 1 # Ψ 2 or Ψ 1 # Y Ψ 2 and call the c omp osition pr o duct of Ψ 1 and Ψ 2 . Our pro duct construction applies equally well to sp ecial bor disms, so that pro d- ucts of sp ecially bor dant sp ecial corresp ondences remain sp ecially b orda n t. The de- gree is a dditiv e for pro ducts, and our pro duct co mm utes with Thom mo diﬁcations b y G -vector bundles over Z on the ﬁr st or second factor. As a result, Theorem 2.23 shows that we we get a gr ading-preser ving, bi-additive map # Y = # : ˆ f ∗ ( X, Y ) × ˆ f ∗ ( Y , U ) → ˆ f ∗ ( X, U ) . Lemma 2. 25. The pr o duct map # is asso ciative and tu rn s ˆ f ∗ into a Z -gr ade d additive c ate gory. Pr o of. The asso ciativity of # is ro utine to chec k. W e get a category beca use we also hav e identit y corresp ondences. The morphism spaces are Z -gra ded Abelian groups. F or an additive category , we also need pro ducts and a zero ob ject. It is easy to see that the empty G -space is a zero ob ject and that Y 1 ⊔ Y 2 is b oth a copro duct and a pro duct of Y 1 and Y 2 in the catego ry ˆ f ∗ .  The exterior pr o duct o f tw o c orresp ondences is deﬁned by applying × Z to a ll ingredients. Exterio r pro ducts of sp ecial corr espo ndences remain sp ecial. Theorem 2.2 6. With the c omp osition pr o duct and exterior pr o duct deﬁne d ab ove, ˆ f ∗ b e c omes a Z -gr ade d symmetric monoida l addi t ive c ate gory; the unit obje ct is Z . Pr o of. This is a n analogue of [12, Pr op osition 4 .26], which is just as trivial t o prov e. It is ro utine to chec k that the ex terior pro duct is functorial for intersection pro ducts of specia l corresp ondences. It is asso ciative and commu tative and has unit ob ject Z , up to certain natura l homeomorphisms; these ar e na tural with resp ect to ordinary maps a nd normally non-singular maps and hence natural with r esp ect to corresp ondences . Thus ˆ f ∗ is a symmetric mono idal category (see [20]).  BIV ARIANT K-THEOR Y VIA CORRESPONDE NCES 15 Recall that a prop er G -map b : Y → X y ields a c orresp ondence b ∗ from X to Y b y Example 2.3. Lemma 2. 2 7. T he map b 7→ b ∗ is a c ontr avariant, symmetric monoidal funct or fr om the c ate gory of G - sp ac es with pr op er G -maps to t he c ate gory ˆ f 0 of maps in ˆ f ∗ of de gr e e 0 . Pr o of. F unctoriality means that b ∗ 1 # b ∗ 2 = ( b 1 ◦ b 2 ) ∗ for prop er G -maps b 1 : Y → X and b 2 : U → Y . Being symmetric monoidal means that if b 1 : Y 1 → X 1 and b 2 : Y 2 → X 2 are pr op er G -maps, then b ∗ 1 × Z b ∗ 2 = ( b 1 × Z b 2 ) ∗ . Both s tatemen ts a re obvious b ecause all corresp ondences in volved are sp ecial.  More generally , consider pairs ( b , ξ ) where b : Y → X is a G - map and ξ ∈ F ∗ X ( Y ) . This b ecomes a special corresp ondence with f = Id; denote this corresp ondence by ( b, ξ ) ∗ . The comp osition o f such corresp ondences inv olves co mpo sing the maps b and taking an exterior pro duct of the cohomology classes . Mo re precisely , let b 1 : Y → X , b 2 : U → Y , ξ 1 ∈ F ∗ X ( Y ) , ξ 2 ∈ F ∗ Y ( U ) , then ( b 1 , ξ 1 ) ∗ # ( b 2 , ξ 2 ) ∗ = ( b 1 ◦ b 2 , ξ 1 ⊗ Y ξ 2 ) ∗ , where ξ 1 ⊗ Y ξ 2 ∈ F ∗ X ( U ) denotes the r estriction of the exterior pro duct ξ 1 × Z ξ 2 in F ∗ X × Z Y ( Y × Z U ) to the gr aph o f b 2 : U → Y . A normally no n-singular G -map f : X → Y y ields a cor resp ondence f ! from X to Y by E xample 2.4. W e cla im that this is a grading-preser ving, symmetric mon- oidal functor from the ca tegory of normally non-singular maps to the category of c orresp ondences, that is, it is compatible with products and exter ior pr o ducts. Compatibilit y with exterior pro ducts and degr ees is trivial, and c ompatibilit y with pro ducts is trivial for special normally non-singular submersions. T o prov e func- toriality for all no rmally non-sing ular maps, we need to know when pro ducts of non-sp ecial co rresp ondences ar e given by an in ters ection pro duct recipe. This re- quires a notion of tr ansversalit y . Let Ψ 1 : = ( M 1 , b 1 , f 1 , ξ 1 ) and Ψ 2 : = ( M 2 , b 2 , f 2 , ξ 2 ) b e cor resp ondences from X to Y and from Y to U . W rite f 1 = ( V 1 , E 1 , ˆ f 1 ) and f 2 = ( V 2 , E 2 , ˆ f 2 ) . Let M : = M 1 × Y M 2 = { ( m 1 , m 2 ) ∈ M 1 × M 2 | f 1 ( m 1 ) = b 2 ( m 2 ) } , b : M → E , ( m 1 , m 2 ) 7→ b 1 ( m 1 ) , ξ : = ξ 1 ⊗ Y ξ 2 in F 0 X ( M ) ; here f 1 also denotes the t race of the normally non-singular map f 1 . T o get a corresp ondence fro m X to Y , we need a nor mally no n-singular map f = ( E , ˆ f , V ) from M to U ; its tr ace should b e the pro duct of the co ordinate pr o jection M → M 2 with the trace of f 2 . W e put E : = E 1 ⊕ E 2 , V : = pr ∗ 1 ( V 1 ) ⊕ pr ∗ 2 ( V 2 ) with the induced F-orientations. In genera l, there need not be an op en em bedding | V | ֒ → | E U | : this is where we need transversalit y . Deﬁnition 2.28. Let M 1 , M 2 , and Y b e G -spa ces, let f = ( V 1 , E 1 , ˆ f 1 ) b e a normally non-singular G -map fro m M 1 to Y , and let b 2 : M 2 → Y b e a G -map. View | V 1 | as a s pace o ver Y v ia | V 1 | ⊆ | E Y 1 | ։ Y . L et V M 1 be the pull-back o f V 1 to M : = M 1 × Y M 2 along the canonical pro jection pr 1 : M → M 1 . W e call f 1 tr ansverse to b 2 if the map ζ V 1 × Y Id M 2 : M : = M 1 × Y M 2 → | V 1 | × Y M 2 extends to an op en embedding from | V M 1 | in to | V 1 | × Y M 2 . T wo corr esp ondences Ψ 1 and Ψ 2 as ab ov e are tr ansverse if f 1 is transverse to b 2 . 16 HEA TH EMERSON AND RALF MEYER More precisely , a transverse pa ir of maps is a triple consisting of a map b 2 , a normally non-singular map f 1 = ( V 1 , E 1 , ˆ f 1 ) , and an op en em b edding from | V M 1 | in to | V 1 | × Y M 2 . A transverse pair o f corres po ndences is a similar triple. The total space of V M 1 is | V M 1 | =  ( v 1 , m 2 ) ∈ | V 1 | × M 2   f 1 ◦ π V 1 ( v 1 ) = b 2 ( m 2 )  . This may diﬀer dr astically from | V 1 | × Y M 2 =  ( v 1 , m 2 ) ∈ | V 1 | × M 2   π E Y 1 ◦ ˆ f 1 ( v 1 ) = b 2 ( m 2 )  . R emark 2.29 . The transversality condition Deﬁnition 2.28 asserts that the embed- ding M 1 × Y M 2 → | V 1 | × Y M 2 be a normal ly non-singular embedding and that its normal bundle b e pr ∗ 1 ( V 1 ) . Before we c heck that trans versalit y ensures that the int ersection pro duct of t wo corresp ondences exists and represents their pro duct, we compar e it to the usual notion of transversality for smo oth maps. Example 2.30 . Let M 1 , M 2 , and Y be smo oth ma nifolds, let b : M 2 → Y be a smo oth map, and let f =  V , R n , ˆ f  be the lifting of a smo oth map ϕ : M 1 → Y using a smo oth embedding h : M 1 → R n . Th us V is the normal bundle of the embedding ( ϕ, h ) ◦ ζ V : M 1 → Y × R n . Assume that the maps b a nd ϕ are tra nsverse in the usual sense that D m 1 ϕ ( T m 1 M 1 ) + D m 2 b ( T m 2 M 2 ) = T y Y for all m 1 ∈ M 1 , m 2 ∈ M 2 with y : = ϕ ( m 1 ) = b ( m 2 ) . Then M 1 × Y M 2 is a smo oth submanifold o f M 1 × M 2 and hence a smo oth manifold. Since the map | V | ⊆ | E Y | ։ Y is a submersion, | V | × Y M 2 is a smo oth manifold as w ell. The map ζ V × Y Id M 2 is a smo oth embedding beca use it is the r estriction of the smo oth embedding ζ V × Id M 2 . W e claim that its normal bundle is the pull-back of V to M 1 × Y M 2 . This follows from the vector bundle isomo rphisms T M = T ( M 1 × Y M 2 ) ∼ = pr ∗ 1 ( T M 1 ) ⊕ ( pr 1 ◦ f 1 ) ∗ ( T Y ) pr ∗ 2 ( T M 2 ) , T ( | V | × Y M 2 ) ∼ = pr ∗ 1 ( T | V | ) ⊕ T Y pr ∗ 2 ( T M 2 ) ζ ∗ V ( T | V | ) ∼ = T M 1 ⊕ V , which com bine to sho w that the c okernel of the vector bundle map T ( M ) → T ( | V | × Y M 2 ) is pr ∗ 1 ( V ) . Thus ζ V × Y Id M 2 is a smo oth nor mally non-singular em- bedding with the r equired normal bundle (compa re Remark 2 .29) so that b and ϕ are transverse in the sense of Deﬁnition 2.2 8. Now w e return to the problem of computing the pro duct of tw o non-sp ecial corresp ondences Ψ 1 and Ψ 2 . W e follo w our previo us notation and, in particular, deﬁne M , b : M → X , ξ ∈ F ∗ X ( M ) , and the G -vector bundles V and E as ab ov e. The total space of V is | V 1 | × Y | V 2 | , whic h a grees with the tota l space of the pull-back of V 2 to | V 1 | × Y M 2 . Note that to fo rm | V 1 | × Y | V 2 | it makes no diﬀerence which map from | V 1 | to Y we use: f 1 ◦ π V 1 or π E Y ◦ ˆ f ; [12, Prop os ition 2.22] shows that only the homoto p y cla ss of the ma p | V 1 | → Y matters. If our corresp ondences are transverse, there are norma lly non-singular embed- dings M → | V 1 | × Y M 2 → | V 1 | × Y | V 2 | with normal bundles π ∗ 1 V 1 (denoted V M 1 in Deﬁnition 2 .28) and π ∗ 2 V 2 , resp ectively; the ﬁrst norma lly non-s ingular map is the transversality assumption, the second one is obvious b ecause V 2 is a G -vector bundle ov er M 2 . Compositio n yields a normally non-singular em bedding M → | V 1 | × Y | V 2 | with no rmal bundle V , that is, an op en embedding from | V | into | V 1 | × Y | V 2 | . BIV ARIANT K-THEOR Y VIA CORRESPONDE NCES 17 When we ﬁr st repla ce our tw o corr esp ondences by s pecia l ones and then take their in tersection pro duct, we r eplace M 1 b y | V 1 | and M 2 b y | V 2 | and construct a sp ecial nor mally non-singular s ubmersion | V 1 | × Y | V 2 | ֒ → | E U | ։ U ; we comp ose the op en embedding | V 1 | × Y | V 2 | ֒ → | E U | from this prev ious construction with the op en embedding | V | ֒ → | V 1 | × Y | V 2 | fro m transversalit y to g et an o pen em b edding ˆ f : | V | ֒ → | E U | . This pro duces the desir ed F- oriented normally non-singular ma p ( V , ˆ f , E ) fr om M to U and hence a co rresp ondence from X to U , called the int erse ction pr o duct o f Ψ 1 and Ψ 2 and deno ted Ψ 1 # Ψ 2 or Ψ 1 # Y Ψ 2 . This is only deﬁned if Ψ 1 and Ψ 2 are tra nsverse and, at ﬁrst sight , dep ends on the c ho ice o f the op en embedding in the deﬁnition of transversality . Theorem 2.31. If the c orr esp ondenc es Ψ 1 and Ψ 2 ar e tr ansverse, then their in- terse ction pr o duct is e quivalent to t he c omp osition pr o duct of the e quivalent sp e cial c orr esp ondenc es. Pr o of. The Thom mo diﬁcation of Ψ 1 # Ψ 2 along V is a sp ecial corresp ondence ( E , | V | , b, ξ ) fr om X to Y , where we use ˆ f to view | V | as an open subset of | E Y | and let b b e the compos ition of the bundle pro jection | V | ։ M with the map M → M 1 → X . When we ﬁr st Thom mo dify Ψ 1 and Ψ 2 along V 1 and V 2 to make them sp ecial and then co mpos e, we get a sp ecial corresp ondence ( E , V ′ , b ′ , ξ ′ ) with V ′ = | V 1 | × Y | V 2 | ; w e hav e seen that this cont ains | V | as an op en subset. The map b ′ extends b on | V | , and ξ ′ extends ξ . Hence the t wo corresp ondences via | V | and V ′ are sp ecially b ordant b y E xample 2.13.  Example 2.32 . A pair of cor resp ondences Ψ 1 and Ψ 2 is transverse if Ψ 1 is sp ecial, regardles s o f Ψ 2 , for V 1 is the 0 -vector bundle in this cas e, making the condition in Deﬁnition 2.28 trivially satisﬁed. Example 2.33 . If b 2 : M 2 ։ Y is a v ector bundle pro jection, then Ψ 1 and Ψ 2 are transverse. T o see this, note that M 1 × Y M 2 and | V 1 | × Y M 2 are the total spaces of the pull-backs of the v ector bu ndle M 2 to M 1 and to | V 1 | under the maps f : M 1 → Y a nd π E Y ◦ ˆ f : V 1 → Y . Now the maps π E Y ◦ ˆ f and f ◦ π V 1 are homo topic by a homo topy which is co nstant o n the zero se ction. Hence the corresp onding pull-bac k s of M 2 are isomorphic via an isomorphism which is the iden tit y o n the zero section, and, in particula r, | V 1 | × Y M 2 is homeomorphic to | V M 1 | via a homeomorphism which is the iden tity on the zer o section (see [12, P rop osition 2.22]). Ther efore we are left with a vector bundle W : = f ∗ 1 ( M 2 ) on M 1 . It is obvious that the zero-s ection embedding W → π ∗ V 1 ( W ) is normally non-sing ular with normal bundle π ∗ V 1 ( W ) : = ( f ◦ π V 1 ) ∗ ( M 2 ) as requir ed. Example 2.34 . Let ( M , b , f , ξ ) be a corresp ondence from X to Y . Then ( b, ξ ) ∗ is a corresp ondence from X to M and f ! is a co rresp ondence from M to Y . These t wo are transverse b y Example 2.33, and their co mpo sition pro duct ( b, ξ ) ∗ # f ! is the given cor resp ondence ( M , b, f , ξ ) . Finally , we use E xample 2.33 to show that the compos ition of corresp ondences generalises the comp ositio n of normally non-singular maps: Corollary 2. 3 5. L et f 1 : X → Y and f 2 : Y → U b e F -oriente d normal ly non- singular maps. Then f 1 ! # f 2 ! = ( f 2 ◦ f 1 )! . Pr o of. Let Ψ j be the canonical representativ es for f j ! for j = 1 , 2 . W e conclude that M = X , b = Id, ξ = 1 , and the transv er sality condition is automatic b y Example 2.33. Hence the pro duct is represented by the intersection pro duct Ψ 1 # Ψ 2 b y Theorem 2.31 . This is of the form f ! for a normally non-singular map 18 HEA TH EMERSON AND RALF MEYER f : X → U . Insp ection shows that f ag rees with the pro duct of no rmally non- singular maps f 2 ◦ f 1 .  As in Kasparov theory , w e may combine exterio r pr o ducts and comp osition pro d- ucts to an op eratio n (2.36) # U : ˆ f i ( X 1 , Y 1 × Z U ) × ˆ f j ( U × Z X 2 , Y 2 ) → ˆ f i + j ( X 1 × Z X 2 , Y 1 × Z Y 2 ) , ( α, β ) 7→ ( α × Z Id X 2 ) # ( Id Y 1 × Z β ) , which is a gain as so ciative and graded comm utative in a suitable sense. This op er- ation will b e used heavily in §3 to cons truct dualit y iso morphisms. Recall that ˆ f ∗ ( Z, Y ) ∼ = F ∗ Z ( Y ) is the F -cohomolog y of Y with Z -compact s uppor t (Theorem 2.24). Since ˆ f ∗ is a category , corre spo ndences act o n F ∗ Z ( Y ) ; this extends the wrong-way maps for F-oriented normally non-s ingular maps in [12, Theor em 5.16]. 2.6. Comp os ition of smo oth corresp ondences using transv ersality . W e re- cov er the tra nsversalit y form ula for the comp os ition of tw o smo oth co rresp ondences in genera l p osition of Connes and Skandalis [6]. Theorem 2.37. L et M 1 , M 2 , and Y b e smo oth G -manifolds; let Φ = ( V , E , ˆ f ) b e a smo oth normal ly non-singular G -map fr om M 1 to Y with tr ac e f and let b : M 2 → Y b e a smo oth G -m ap. These two maps ar e tr ansverse if D m 1 f ( T m 1 M 1 ) + D m 2 b ( T m 2 M 2 ) = T y Y for al l m 1 ∈ M 1 , m 2 ∈ M 2 with y : = f ( m 1 ) = b ( m 2 ) . Pr o of. The argument is liter ally the same as in the non-equiv ar iant case, see Exam- ple 2.30. The T ubular Neighbour ho o d w e need exists by [12, Theorem 3.18].  Corollary 2. 38. L et Φ 1 = ( M 1 , b 1 , f 1 , ξ 1 ) and Φ 2 = ( M 2 , b 2 , f 2 , ξ 2 ) b e smo oth c orr esp ondenc es fr om X t o Y and fr om Y to U , r esp e ctively. A ssume that b oth M 1 and M 2 admit smo oth n ormal ly non-singu lar maps to Z , so that we lose nothing if we view f 1 and f 2 as F -oriente d smo oth maps. A ssu me also that f 1 and b 2 ar e tr ansverse as in The or em 2.37 . Then M 1 × Y M 2 is a smo oth G - manifold with a smo oth normal ly n on-singular map to Z as wel l. The interse ct ion pr o duct of the two c orr esp ondenc es ab ove is Φ 1 # Y Φ 2 =  M 1 × Y M 2 , b 1 ◦ π 1 , f 2 ◦ π 2 , π ∗ 1 ( ξ 1 ) ⊗ π ∗ 2 ( ξ 2 )  , wher e π j : M 1 × Y M 2 → M j for j = 1 , 2 ar e the c anonic al pr oje ctions. Pr o of. If M 1 and M 2 admit smoo th normally non-sing ular maps to Z , then so do es M 1 × Y M 2 beca use it embeds in M 1 × Z M 2 , which em beds in | E 1 ⊕ E 2 | = | E 1 | × Z | E 2 | if M j embeds in | E j | for j = 1 , 2 . Under this ass umption, we can replace all smo oth normally non-singular maps by mer e smo o th maps, so tha t it suﬃces to describ e the traces and the F -orientations of the nor mally non-singular maps we a re dealing with. Hence the assertion follows from the construction o f the in ters ection pro duct for trans verse corre spo ndences in §2.5. W e leav e it to the r eader to wr ite down the F-orientation that the ma p f 2 ◦ π 2 inherits.  3. Duality isomorphisms There is a canonica l notion of duality in ˆ f ∗ beca use it is a symmetric mono idal category: t wo G -spa ces X and P are dual in ˆ f ∗ if there is a na tural isomorphism ˆ f ∗ ( X × Z U 1 , U 2 ) ∼ = ˆ f ∗ ( U 1 , P × Z U 2 ) for a ll G - spaces U 1 and U 2 . This is equiv alent to the symmetric dua lit y iso morphism ˆ f ∗ ( P × Z U 1 , U 2 ) ∼ = ˆ f ∗ ( U 1 , X × Z U 2 ) . BIV ARIANT K-THEOR Y VIA CORRESPONDE NCES 19 But this notion does not cov e r familiar duality isomorphisms for non- compact smo oth manifolds. Let X b e a s mo oth manifold and let T X b e its tangent space. Then there are natur al isomorphisms RK ∗ ( X ) ∼ = K ∗ ( T X ) , K ∗ ( X ) ∼ = K ∗ X ( T X ) betw een the representable K -theory of X and the K - homology of T X , and betw ee n the K -homolo gy o f X and the K -theory of T X with X -compact supp ort. These tw o dualit y iso morphisms are gener alised in [10], following Gennadi Kasparov [16]. The abstract conditions in [10] that are equiv alent to the existence of such duality is o- morphisms only use forma l prop erties o f equiv aria nt Kasparov theo ry and therefore carry ov er to the geometric setting we consider here. W e s ketc h this g eneralisation in this section. The tw o duality isomorphisms ha ve the follo wing imp ortant applications. The ﬁrst duality is use d in [10] to deﬁne equiv aria n t Euler characteristics a nd equiv ar iant Lefschet z inv ariants in equiv ar iant K -homology; the g eometric coun terpart of this dualit y descr ibed below will b e used in a forthcoming article to compute Euler characteristics and Lefschetz in v a riants of cor resp ondences in geometric terms. The second duality a llows, in particular, to reduce biv a riant K -gr oups to K -theory with suppo rt conditions. This will be used b elow to show that the top ological and analytic versions o f biv ar iant K -theory agr ee if there is a duality isomorphism. Our notion o f dualit y do e s no t cont ain Spanier–Whitehead dualit y as a sp ecial case. The main issue is that we req uire the d ual of X to be a space ov er X . This seems unav oidable for the second duality iso morphism and rules out taking a complement of X in some ambien t space as in Spanier –Whitehead Dualit y . 3.1. Preparations. Fir st we need some notation. T o emphasise the group oid un- der considera tion, we now wr ite F ∗ G ( U ) and ˆ f ∗ G ( X, Y ) instead of F ∗ ( U ) and ˆ f ∗ ( X, Y ) . Let U be a G -space. Recall that G ⋉ U -spaces ar e G - spaces with a G -map to U . Hence a cohomolog y theor y F ∗ G for G -spaces restricts to one for G ⋉ U -spaces. W e denote the latter by F ∗ G ⋉ U and get a corres po nding biv ar iant theory ˆ f ∗ G ⋉ U ( X, Y ) for G ⋉ U -spaces X and Y . The specia l co rresp ondences ( E , M , b , ξ ) that en ter in its deﬁnition diﬀer from the ones for ˆ f ∗ G ( X, Y ) in the following ways: • E is an F-oriented G -vector bundle on U , not on Z ; • the map b : M → X is a G -ma p ov er U . The ﬁrst mo diﬁcation is o f little impor tance: if we mak e the mild assumption that any G -vector bundle o ver U is subtrivial, then we may use Thom modiﬁcation to reduce to G ⋉ U -equiv ariant cor resp ondences whose G - vector bundle ov er U is trivial. But the second co ndition has a signiﬁcant eﬀect. Theorem 2.24 genera lises to an isomorphism (3.1) ˆ f ∗ G ⋉ X ( X, Y ) ∼ = F ∗ G ,X ( Y ) for an y G ⋉ X -space Y , where the righ t hand side denotes the X -compactly s up- po rted version of F ∗ G . This is quite diﬀerent than ˆ f ∗ G ( X, Y ) . The functoriality prop erties o f normally non-singula r maps ca rry ov er to c orre- sp ondences. First, a map ϕ : U 1 → U 2 induces a symmetric monoida l functor ϕ ∗ : ˆ f ∗ G ⋉ U 2 ( X, Y ) → ˆ f ∗ G ⋉ U 1 ( ϕ ∗ X , ϕ ∗ Y ) . W e often write U 1 × U 2 α instead o f ϕ ∗ α for α ∈ ˆ f ∗ G ⋉ U 2 ( X, Y ) . Secondly , if all G ⋉ U 2 -vector bundles ov er U 1 are subtrivial (that is, direct summands of G -vector bundles pulled back from U 2 ), then there is a fo rgetful functor ˆ f ∗ G ⋉ U 1 ( X, Y ) → ˆ f ∗ G ⋉ U 2 ( X, Y ) 20 HEA TH EMERSON AND RALF MEYER in the opp osite direction, where we view G ⋉ U 1 -spaces as G ⋉ U 2 -spaces b y comp osing the anchor map to U 1 with ϕ . W e usually denote the image of g ∈ ˆ f ∗ G ⋉ U ( X, Y ) under the forgetful functor by g ∈ ˆ f ∗ G ( X, Y ) . R emark 3.2 . When we comp ose morphisms, we so metimes dro p pull-back functors and forgetful functors from our no tation. F or instance, if Θ ∈ ˆ f ∗ G ⋉ X ( X, X × Z P ) and D ∈ ˆ f ∗ G ( P, Z ) for t wo G -spaces X and P , then Θ # P D ∈ ˆ f ∗ G ⋉ X ( X, X ) ∼ = F ∗ ( X ) denotes the pr o duct of Θ a nd  ∗ ( D ) ∈ ˆ f ∗ G ⋉ X ( X × Z P, X ) where  : X → Z is the anchor map and we iden tify X × Z Z ∼ = X . Deﬁnition 3.3 . Let X b e a G -spac e a nd let Y 1 and Y 2 be tw o G -spaces ov er X . Then w e may view Y 1 × Z Y 2 as a G -space ov er X in tw o diﬀerent wa y s, using the ﬁrst or second co or dinate pro jection follow ed by the anc ho r map Y j → X . T o distinguish these tw o G ⋉ X -spaces, we underline the facto r whose X -structure is used. Thus the groups ˆ f ∗ G ⋉ X ( X, X × Z P ) and ˆ f ∗ G ⋉ X ( X, X × Z P ) for a G ⋉ X -s pace P are diﬀerent. 3.2. The t wo dualit y isomorphis ms. Throughout this section, X is a G -space , P is a G ⋉ X -space, a nd D ∈ ˆ f − n G ( P, Z ) for some n ∈ Z ; U is a G ⋉ X -space, and Y is a G - space. W e assume througho ut that all G -vector bundles ov er X are subtrivial. W e are going to deﬁne tw o dualit y maps in volving this data and then analys e when they are inv ertible, following [10]. The ﬁrst duality map for ( X, P, D ) with co eﬃcients U and Y is the map (3.4) PD ∗ : ˆ f i G ⋉ X ( U, X × Z Y ) → ˆ f i − n G ( P × X U, Y ) , g 7→ ( − 1) in ( P × X g ) # P D . The se c ond duality map for ( X, P , D ) with co eﬃcients U and Y is the map (3.5) PD ∗ 2 : ˆ f i G ⋉ X ( U, P × Z Y ) → ˆ f i − n G ( U, Y ) , f 7→ ( − 1) in f # P D . In b oth cases , the o verlines denote the forge tful functor ˆ f ∗ G ⋉ X → ˆ f ∗ G . The second dualit y map is particularly in tere sting for U = X : then it maps F i G ,X ( P × Z Y ) ∼ = ˆ f i G ⋉ X ( X, P × Z Y ) to ˆ f i − n G ( X, Y ) by Theo rem 2.24. Necessary and suﬃcien t co nditions for a nalogous duality maps in K asparov the- ory to be isomor phisms a re a nalysed in [10]. These carry over litera lly to our s etting beca use they only use formal pr op erties of Ka sparov theory . Theorem 3.6. Fix X , P , D and U . The ﬁrst duality map is an isomorphism for al l G -sp ac es Y if and only if ther e is Θ U ∈ ˆ f n G ⋉ X  U, X × Z ( P × X U )  with the fol lowing pr op erties: (i) ( P × X Θ U ) # P D = ( − 1) n Id P × X U in ˆ f 0 G ( P × X U, P × X U ) ; (ii) ( − 1) in Θ U # P × X U ( P × X g ) # P D = g for al l g ∈ ˆ f i G ⋉ X ( U, X × Z Y ) and al l G -sp ac es Y . F urthermor e, the inverse of PD ∗ is of the form (3.7) PD : ˆ f i − n G ( P × X U, Y ) → ˆ f i G ⋉ X ( U, X × Z Y ) , f 7→ Θ U # P × X U f , and Θ U is determine d uniquely. Supp ose that Θ ∈ ˆ f n G × X ( X, X × Z P ) satisﬁes Θ # P D = Id X in ˆ f n G × X ( X, X ) . Then the fol lowing c onditions (iii) and (iv) imply (i) and (ii) : (iii) the fol lowing diagr am in ˆ f ∗ G c ommutes: P × X U P × X Θ U / / ( P × X U ) × X Θ   P × Z ( P × X U ) 5 5 ∼ = ( − 1) n ﬂip u u l l l l l l l l l l l l l l ( P × X U ) × Z P. BIV ARIANT K-THEOR Y VIA CORRESPONDE NCES 21 (iv) Θ U # P × X U ( P × X g ) = Θ # X g in ˆ f i + n G ⋉ X ( U, X × Z P × Z Y ) for al l g ∈ ˆ f i G ⋉ X ( U, X × Z Y ) and al l G -sp ac es Y . Pr o of. Condition (i) means that Θ U ∈ ˆ f n G ⋉ X ( U, X × Z P × X U ) satisﬁes PD ∗ (Θ U ) = Id P × X U . Hence (i) is necessary for PD ∗ to b e inv ertible a nd determines Θ U uniquely . The asso cia tivit y o f # and the graded co mm utativity of exterior pro ducts yield PD ∗ ◦ PD ( f ) = ( − 1) in ( P × X Θ U ) # P × X U ( f # P D ) = ( − 1) n ( P × X Θ U ) # P D # P × X U f for f ∈ ˆ f i − n G ( P × X U, Y ) . Hence PD ∗ ◦ PD is t he iden tity map if and only if Condition (i) holds. Then the inv erse of PD ∗ can only b e P D. By deﬁnition, PD ◦ PD ∗ ( g ) = ( − 1 ) in Θ U # P × X U ( P × X g ) # P D for all g ∈ ˆ f i G ⋉ X ( U, X × Z Y ) . Hence (ii) is equiv a len t to PD ◦ PD ∗ = Id. As a result, the maps PD and PD ∗ deﬁned as in (3.4) and (3.7) are in verse to each other if and only if Conditions (i) and (ii) hold. Now assume that there is a class Θ as ab ove. Condition (iii) implies (i); a nd (iv) implies (ii) b ecause of the graded comm utativit y of exterior pro ducts: ( − 1) in Θ U # P × X U ( P × X g ) # P D = ( − 1) in Θ # X g # P D = g # X Θ # P D = g for all g .  A cla ss Θ a s ab ov e exists and is equal to Θ X if the ﬁrst dualit y map is an isomorphism for U = X . Hence the existence of Θ is a harmless as sumption for our purp oses. Theorem 3. 8. Fix X , P , D and U . The se c ond duality map PD ∗ 2 is an isomor- phism for al l G - sp ac es Y if and only if ther e is e Θ U ∈ ˆ f n G ⋉ X ( U, P × Z U ) with the fol lowing pr op erties: (i) e Θ U # P D = ( − 1) n Id U in ˆ f 0 G ( U, U ) ; (ii) ( − 1) in e Θ U # U g # P D = g in ˆ f i G ⋉ X ( U, P × Z Y ) for al l G -sp ac es Y and al l g ∈ ˆ f i G ⋉ X ( U, P × Z Y ) . F urthermor e, (i) determines e Θ U uniquely, and the inverse of PD ∗ 2 is of the form (3.9) PD 2 : ˆ f i − n G ( U, Y ) → ˆ f i G ⋉ X ( U, P × Z Y ) , f 7→ e Θ U # U f . Supp ose that Θ ∈ ˆ f n G × X ( X, X × Z P ) satisﬁes Θ # P D = Id X in ˆ f n G × X ( X, X ) . Then the fol lowing c onditions (iii) and (iv) imply (i) and (ii) : (iii) the fol lowing diagr am in ˆ f ∗ G c ommutes: U e Θ U / / U × X Θ   P × Z U 8 8 ∼ = ( − 1) n ﬂip x x r r r r r r r r r r U × Z P. (iv) e Θ U # U g = Θ # X g in ˆ f i + n G ⋉ X ( U, P × Z P × Z Y ) for al l g ∈ ˆ f i G ⋉ X ( U, P × Z Y ) and al l G -sp ac es Y . Pr o of. Condition (i) means that P D ∗ 2 ( e Θ U ) = ( − 1) n Id U in ˆ f 0 G ( U, U ) . Hence there is a unique e Θ U satisfying (i) if PD ∗ 2 is inv ertible. Deﬁne a map PD 2 as in (3.9). Th e deﬁning prop erty of e Θ U and the gr aded comm uta tivit y o f exterior pro ducts yield PD ∗ 2 ◦ PD 2 ( f ) = ( − 1) in e Θ U # U f # P D = ( − 1) n e Θ U # P D # U f = f 22 HEA TH EMERSON AND RALF MEYER for all f ∈ ˆ f i − n G ( U, Y ) . Hence the inv erse of PD ∗ 2 can only b e P D 2 . W e compute PD 2 ◦ PD ∗ 2 ( g ) = ( − 1 ) in e Θ U # U g # P D for all g ∈ ˆ f i G ⋉ X ( U, P × Z Y ) , so that (ii) is eq uiv a len t to PD 2 ◦ PD ∗ 2 = Id. Now a ssume that there is Θ ∈ ˆ f n G ⋉ X ( X, X × Z P ) with Θ # P D = Id X . Then Condition (iii) implies (i) and (iv) implies (ii), using the g raded co mm utativit y of exterior pro ducts.  Deﬁnition 3.10 . Let n ∈ Z and let X be a G -space. A symmetric dual for X is a quadruple ( P, D , Θ , e Θ) , where • P is a G ⋉ X -space, • D ∈ ˆ f − n G ( P, Z ) , • Θ ∈ ˆ f n G ⋉ X ( X, X × Z P ) ∼ = F n G ,X ( X × Z P ) (se e Deﬁnition 3.3), and • e Θ ∈ ˆ f n G ⋉ X ( X, X × Z P ) ∼ = F n G ,X ( X × Z P ) satisfy the following conditions: (i) Θ # P D = I d X in the ring ˆ f 0 G ⋉ X ( X, X ) ∼ = F 0 G ( X ) ; (ii) ( P × X Θ) # P × Z P ﬂip = ( − 1) n ( P × X Θ) in ˆ f n G ( P, P × Z P ) , where ﬂip de- notes the p ermutation ( x, y ) 7→ ( y , x ) on P × Z P ; (iii) e Θ = ( − 1) n Θ in ˆ f n G ( X, X × Z P ) ; (iv) Θ # P ( P × X g ) = Θ # X g in ˆ f i + n G ⋉ X  X , X × Z P × Z Y ) ∼ = F i + n G ,X ( X × Z P × Z Y ) for all g ∈ ˆ f i G ⋉ X ( X, X × Z Y ) ∼ = F i G ,X ( X × Z Y ) and all G -spaces Y ; (v) e Θ # X g = Θ # X g in ˆ f i + n G ⋉ X  X , P × Z ( P × Z Y )  ∼ = F i + n G ,X ( P × Z ( P × Z Y )  for all g ∈ ˆ f i G ⋉ X ( X, P × Z Y ) ∼ = F i G ,X ( P × Z Y ) and a ll G - spaces Y . W e hav e used Theorem 2.24 rep eatedly to simplify ˆ f ∗ G ⋉ X ( X, ) to F ∗ G ,X ( ) . Most of the data a nd conditions ab ov e take place in F ∗ G ,X ( ) . Theorem 3.11. If the sp ac e X has a symmetric dual and if every G -e quivariant ve ctor bund le over X is subtrivial, then t he maps in (3.4) , (3.7) , (3.5) , and (3.9) for U = X yield isomorphisms F i G ,X ( X × Z Y ) ∼ = ˆ f i G ⋉ X ( X, X × Z Y ) ∼ = ˆ f i − n G ( P, Y ) , F i G ,X ( P × Z Y ) ∼ = ˆ f i G ⋉ X ( X, P × Z Y ) ∼ = ˆ f i − n G ( X, Y ) for al l G -sp ac es Y . Pr o of. The conditions for a symmetric dual in Deﬁnition 3.10 are Θ # P D = Id X and the Conditions (iii) and (iv) in Theo rems 3.6 and 3 .8 with Θ X = Θ and e Θ X = e Θ . Hence the isomo rphisms fo llow from Theor ems 2 .24, 3.6, and 3.8.  R emark 3.12 . Theorem 3.11 has a con verse: the conditions in Deﬁnition 3.10 are necessary for the duality maps to be inv e rse to each other. If X has a symmetric dual, then Conditions (iii) and (iv) in Theor em 3.6 are also necessar y for the ﬁrst dualit y iso morphism, and Conditions (iii) and (iv) in Theorem 3 .8 are necessary for the second duality iso morphism. Analogous statements ab out dualit y isomor phisms in K asparov theory are estab- lished in [10], and the pro ofs carry ov er almost literally . R emark 3.13 . The v ar iants of the duality iso morphisms with diﬀerent suppo rt con- ditions co nsidered in [10, Theorems 4.50 and 6.11] also work in our g eometric theory , of course. But w e will no t use these v ar iants here. W e re mark, how ever, that the constructions of sy mmetric duals b elow are s uﬃcien tly lo cal to g ive duality isomo r- phisms with diﬀerent supp ort conditions as well. BIV ARIANT K-THEOR Y VIA CORRESPONDE NCES 23 3.3. Dualit y for certain G -spaces. As b efore, G is a numerably pr op er group oid with o b ject spac e Z . W e are going to establish dualit y isomorphisms for spaces with certain prop erties. The following deﬁnition lists our require men ts: Deﬁnition 3.14. A G -space X is c alled normal ly non-singular if there is a normally non-singular G -map from X to Z × [0 , ∞ ) and if all G -vector bundles on X are subtrivial. Recall that a no rmally non-singula r G -map fro m X to Z × [0 , ∞ ) is a triple Φ : = ( V , E , ˆ f ) , where V is a subtrivial G -vector bundle ov e r X , E is a G -vector bundle ov er Z , and ˆ f is an op en embedding from | V | in to | E | × [0 , ∞ ) . If ther e is a normally no n-singular G - map from X to Z , then there is one to Z × [0 , ∞ ) a s well beca use the map Z → Z × [0 , ∞ ) , z 7→ ( z , t ) , is the trace o f a normally non-sing ular map for a ll t > 0 . Under some technical assumptions ab out equiv ariant vector bundles, normally non-singular maps X → Z cor resp ond to smo oth structures on X × Z E for so me G -vector bundle E ov er Z , a nd norma lly non- singular maps X → Z × [0 , ∞ ) corresp ond to a structure o f smo oth G -manifold with bo undary on X × Z E for so me G -vector bundle E over Z . The tec hnical assumptions here are rela ted to the ﬁnite or bit t yp e assumption in the Mostow Embedding Theorem. F or instance, if G is a compa ct gr oup, then a smo oth G -manifold with bo undary X admits a normal map to Z × [0 , ∞ ) if and o nly if it has ﬁnite or bit t ype . But this as sumption does not yet ensur e that all G -vector bundles ov er X are subtrivial. F or e xample, let X b e the integers a nd G be the circle with the trivial action on X . Using the identiﬁcation X ∼ = b G w e get an obvious G -eq uiv a riant complex line bundle on X whic h is not subtrivia l b ecause it contains inﬁnitely many inequiv a lent irreducible repr esentations of G . F or more information on no n-singular spaces, see [12]. Let X b e a no-sing ular G -space and let Φ : = ( V , ˜ E , ˆ f ) be a normally non-s ingular G -map from X to Z × [0 , ∞ ) . W e assume ˜ E to b e F -oriente d and with a wel l-deﬁne d dimension, and we let d : = dim ˜ E + 1 . W e imp ose no restrictions on the G - vect or bundle V ; thus the no rmally non-singular map Φ need not be F-oriented. R emark 3 .15 . Assume that a ny G -vector bundle over Z is a direct summand in an F-oriented one; this is automatic if F is cohomo logy , equiv a riant K -theory , or equiv ariant KO -theory . Then a lifting of our o riginal normally non-singular map replaces ˜ E by a n F-orient ed G -vector bundle. Hence our a ssumption that ˜ E b e F-oriented is no los s of genera lit y . Similarly , if the ﬁbre dimensions of E are merely b ounded ab ov e by some N ∈ N , then lifting along the lo cally constant G -vector bundle with ﬁbre R N − dim E z at z ensures that dim ˜ E z = N for all z ∈ Z , without aﬀecting the F-orientation. Hence our assumptions on ˜ E can alw ays b e achiev ed by lifting Φ . W e use ˆ f to iden tify | V | with an op en subset of | ˜ E | × [0 , ∞ ) a nd th us drop ˆ f from our no tation fro m now on. Let ∂ V : = | V | ∩ | ˜ E | × { 0 } . This is an open subset of | ˜ E | × { 0 } ∼ = | ˜ E | . Let E : = ˜ E ⊕ R and P : =  ∂ V × ( −∞ , 0]  ∪ | V | . Then P is an op en subset of | E | . Since | E | is F-o riented and d -dimensional, the sp ecial normally non-sing ular submers ion P ֒ → | E | ։ Z provides D ∈ ˆ f − d G ( P, Z ) . There is a canonical defor mation r etraction from P onto | V | ⊆ P : h : P × [0 , 1] → P , h  ( v , s ) , t  : = ( v , s · t ) for v ∈ ∂ V , s ∈ ( −∞ , 0] , t ∈ [0 , 1] , and h ( v , t ) = v fo r v ∈ | V | , t ∈ [0 , 1] . W e view P as a space ov e r X using the map π V ◦ h 0 : P → | V | ։ X . 24 HEA TH EMERSON AND RALF MEYER This construction simpliﬁes if we use a normally non-singular G -map ( V , E , ˆ f ) from X to Z . Then P : = | V | , view ed as a space ov er X via π V : | V | ։ X ; the pair ( E , ˆ f ) is a sp ecial normally non-s ingular submers ion fro m P to Z a nd provides D ∈ ˆ f − d G ( P, Z ) . Specia lising further, if G is a compact gro up, so that Z = ⋆ , then a normally non-singular map X → Z is equiv alent to a tubular neighbourho o d ˆ f : | V | ֒ → R n for an embedding X → R n with normal bundle V . W e ha ve now descr ibed the ingredien ts P a nd D of the symmetric dual, which ﬁx the duality isomorphisms PD ∗ and P D ∗ 2 b y (3.4) and (3.5) and thus determine the other ingredients Θ and e Θ . W e now des crib e these. Since P is an op en subset of | E | , X × Z P is an op en subset of X × Z | E | . The latter is the total space of the trivial G - vector bundle E X ov er X . The zero section of V followed by the em b edding | V | : = P ֒ → | E | provides a G -equiv ariant section of E X . Lemma 3.16. A ny se ct ion of E X with values in X × Z P ⊂ X × Z E = E X is the tr ac e of a G ⋉ X -e quivariant normal ly non-singular emb e dding fr om X to X × Z P . Pr o of. This is a n ea sy sp ecial case o f the equiv ariant T ubular Neighbourho o d The- orem, which we establish by hand. W e wan t to deﬁne a G ⋉ X -equiv ar iant o pen embedding ι : | E X | = X × Z | E | → X × Z P by ι ( x, e ) : =  x, ζ V ( x ) + e · R ( ζ V ( x ) , e )  with R ( p, e ) =  ( p ) k e k + 1 ; here ζ V : X → | V | ⊆ | E | is the zero section of V a nd + a nd · denote the addition and scalar multiplication in the vector bundle E ; k e k is the norm from a G -inv a riant inner pro duct on E , which exists by [12] bec ause G is numerably prop er; and  : P → (0 , 1] is a G - in v a riant function c hosen s uc h that p + e ∈ P for ( p, e ) ∈ P × Z | E | with k e k <  ( p ) – this ensur es that ι ( x, e ) ∈ X × Z P for all ( x, e ) ∈ X × Z | E | . F or each p ∈ P , there are an o pen neigh b ourho o d U p and ǫ p > 0 s uc h that p ′ + e ∈ P for ( p ′ , e ) ∈ P × Z | E | with p ′ ∈ U p and k e k < ǫ p beca use P is o pen in | E X | . Since P is pa racompact by o ur s tanding a ssumption on top ologica l spaces, we may use a par tition o f unity to ﬁnd a cont in uous function  : P → (0 , ∞ ) with  ( p ) ≤ ǫ p for all p ∈ P . W e can replace  by a G -inv ar iant function, using that G is n umerably prop er.  Since the G -vector bundle E is F-orient ed, s o is the normally non-singular em- bedding ( E X , ι ) fro m X to X × Z P . W e let Θ : = ( E X , ι )! ∈ ˆ f d G ⋉ X ( X, X × Z P ) ∼ = F d G ,X ( X × Z P ) . Now we mo dify this construction to get e Θ . View | E X | as a spa ce ov er X via π ′ : | E X | → X , ( x, e ) 7→ π V ◦ h 0  ζ V ( x ) + e · R ( ζ V ( x ) , e )  . Let D ( E X ) b e the unit disk bundle of E X with res pect to the chosen metric. The restrictions of b oth π E X and π ′ to D ( E X ) ar e prop er maps to X , that is, D ( E X ) is X -compact when we view | E X | as a space ov er X us ing one of these maps. Ev en more, D ( E X ) × [0 , 1] is X -co mpact with r esp ect to ¯ π : | E X | × [0 , 1] → X , ( x, e, t ) 7→ π ′ ( x, t · e ); this is a homotopy betw een π E X and π ′ . The F-o rient ation τ E of E X may be re presented by a cohomology cla ss supp orted in D ( E X ) . Hence we get ¯ τ E ∈ F ∗ G ,X  | E X | × [0 , 1] , ¯ π  and, b y restriction, τ E ∈ F ∗ G ,X  | E X | , π ′  ; here the maps on the right sp ecify how to view the s paces on the left as spaces over X . BIV ARIANT K-THEOR Y VIA CORRESPONDE NCES 25 Now we can d escrib e e Θ ∈ ˆ f d G ⋉ X ( X, X × Z P ) : it is the class of the special corresp ondence (0 , | E X | , π ′ , τ E ) from X to X × Z P . Here we view | E X | as an op en subset of X × Z P using the map ι constructed ab ov e and as a space ov er X using π ′ . As a class in F d G ,X ( X × Z P ) , we hav e e Θ = ι !( τ E ) . Theorem 3.17 . L et X b e a normal ly non-singular G -sp ac e. Assume that any G -ve ctor bun d le over Z is c ont aine d in an F -oriente d one. The data ( P, D , Θ , e Θ) pr ovides a symmetric dual for X . H enc e t her e ar e duality isomorphisms F i G ,X ( X × Z Y ) ∼ = ˆ f i − n G ( P, Y ) , F i G ,X ( P × Z Y ) ∼ = ˆ f i − n G ( X, Y ) for al l G -sp ac es Y . Pr o of. W e mu st chec k Conditions (i) – (v) in Deﬁnition 3.1 0. The dualit y isomor- phisms then follow from Theorem 3.11. The comp osition of ι with the o pen embedding P ֒ → | E | is isotopic to the identit y map on | E X | . Thu s the computation in [12 , Example 4.25] yields Co ndition (i) : Θ # P D = I d X in ˆ f 0 G ⋉ X ( X, X ) . Condition (ii) asserts that P × X Θ ∈ ˆ f d G ( P, P × Z P ) is ro tation inv ariant up to the sign ( − 1) d . This amounts to an asse rtion ab out the map P × X ι . First we construct an isotopy from P × X ι to a slightly more symmetric map. By deﬁnition, P × X ι : P × X X × Z | E | ∼ = P × Z | E | ֒ → P × Z P maps ( p, e ) 7→  p, 0 · h 0 ( p ) + e · R (0 · h 0 ( p ) , e )  , where h 0 : P → | V | is the canonical retraction a nd · is the sca lar multiplication in V , that is, 0 · v = ζ V ◦ π V ( v ) . The op en embeddings ( p, e ) 7→  p, t · h t ( p ) + e · R ( t · h t ( p ) , e )  for t ∈ [0 , 1] provide an isotopy from P × X ι to the map ι P : P × Z | E | ֒ → P × Z P, ( p, e ) 7→  p, p + e · R ( p, e )  . The matrices A t : =  1 + t − t t 1 − t  , A − 1 t : =  1 − t t − t 1 + t  are inv ers e to ea ch other. Applying A t for t ∈ [0 , 1 ] to ι P provides an is otopy ( p, e ) 7→ A t ·  p p + e · R ( p, e )  =  p − e · R ( p, e ) t p + e · R ( p, e ) · (1 − t )  of ope n em b eddings | E P | ֒ → | E | × Z | E | betw e en ι P and ﬂip ◦ ι P ◦ Ψ , wher e Ψ( p, e ) : = ( p, − e ) . These maps take v a lues in P × Z P by construction of R ( p, e ) . The isotopies P × Z ι ∼ ι P ∼ ﬂip ◦ ι P ◦ Ψ ∼ ﬂip ◦ ( P × Z ι ) ◦ Ψ constructed a bove provide a n op en embedding (3.18) κ : | E P | × [0 , 1] ֒ → P × Z P × [0 , 1] with κ 0 = P × X ι and κ 1 = ﬂip ◦ ( P × X ι ) ◦ Ψ . The linear is omorphism Ψ has the cla ss ( − 1 ) dim E = ( − 1) d in ˆ f 0 G ⋉ X ( X × Z E , X × Z E ) . The isotop y of open embeddings κ together with the F-orientation τ E ∈ F d G ,P ( | E P | ) provides a sp ecial bo rdism b etw een P × X Θ and ( − 1) d ( P × X Θ) # ﬂip. This ﬁnishes the pro o f of Condition (ii). Condition (iii) asser ts that the for getful functor to ˆ f G maps Θ and ( − 1) d e Θ to the sa me element in ˆ f d G ( X, X × Z P ) . The homotop y ¯ π b etw een the pro jection 26 HEA TH EMERSON AND RALF MEYER ¯ π 1 = π ′ : | E X | → X and the standard pro jection ¯ π 0 = π E X and ¯ τ E provide a sp ecial b ordism b etw een ( − 1) d e Θ : = ( | E X | , 0 , π ′ , τ E ) and ( | E X | , 0 , π E X , τ E ) . The latter is the sp ecia l corresp ondence asso ciated to Θ . This pr ov es Condition (iii). Condition (iv) a sserts Θ # P ( P × X g ) = Θ # X g for g ∈ ˆ f i G ⋉ X ( X, X × Z Y ) for an y G -space Y . W e may pull bac k g to a class in ˆ f i G ⋉ ( X × Z P ) ( X × Z P, X × Z P × Z Y ) along the coo rdinate pro jection X × Z P → X a nd along the pro jection X × Z P → X , ( x, p ) 7→ 0 · h 0 ( p ) . The pro ducts Θ # X g and Θ # P ( P × X g ) a re the comp osition pro ducts of Θ with these tw o pull-backs o f g . These compo sites agree beca use the tw o maps X × Z P ⇒ X restrict to homotopic maps on the r ange o f the embedding ι (use ¯ π ). The pull-back of g along this homotopy provides a b ordism of c orresp ondences that co nnects the tw o pro ducts in question. This ﬁnishes the pro of of Condition (iv). Condition (v) asser ts e Θ # X ξ = Θ # X ξ for ξ ∈ ˆ f i G ⋉ X ( X, P × Z Y ) ∼ = F i G ,X ( P × Z Y ) for any G -s pace Y . W e may rewr ite these pro ducts as comp osition pro ducts: (3.19) e Θ # X ξ = e Θ # X × Z P ( ξ × Z P ) , Θ # X ξ = Θ # X × Z P ( ξ × Z P ) with ξ × Z P ∈ F i G ,X × Z P ( P × Z Y × Z P ) ∼ = ˆ f i G ⋉ ( X × Z P ) ( X × Z P, P × Z Y × Z P ) . The pro ducts in (3.19) lie in ˆ f i + n G ⋉ X ( X, P × Z Y × Z P ) and ˆ f i + n G ⋉ X ( X, P × Z Y × Z P ) , resp ectively , so that w e must, more precisely , show that e Θ # ( ξ × Z P ) = Θ # ( ξ × Z P ) # ﬂip . The isomorphism in Theo rem 2.24 re places ξ × Z P ∈ F i G ,X × Z P ( P × Z Y × Z P ) b y the sp ecial corr esp ondence (0 , P × Z Y × Z P, b , ξ × Z P ) w ith b : P × Z Y × Z P → X × Z P, ( p 1 , y , p 2 ) 7→ ( π V h 0 ( p 1 ) , p 2 ) . Recall that Θ a nd e Θ are represented b y the G ⋉ X -eq uiv a riant sp ecial corresp on- dences ( | E X | , 0 , π E X , τ E ) a nd ( | E X | , 0 , π ′ , ( − 1) d τ E ) , where we use ι to view | E X | as an op en subset of X × Z P . Having represented o ur biv ariant cohomo logy classes b y specia l cor resp ondences, we may use the deﬁnition to compute the co mpos ition pro ducts in (3.19). In b oth cases, the G -s pace in the middle is M : = | E | × Z P × Z Y , viewed a s a n o pen subset of P × Z Y × Z P via λ : M : = | E | × Z P × Z Y ֒ → P × Z Y × Z P, ( e, p, y ) 7→  p, y , 0 · h 0 ( p ) + e · R (0 · h 0 ( p ) , e )  , and the F-class on M is τ E · λ ∗ ( ξ ) . But the maps to X are diﬀerent: for Θ # X ξ , we use P × Z Y × Z P and thus view M a s a space over X via ( e, p, y ) 7→ π V h 0 ( p ) ; for e Θ # X ξ , we use P × Z Y × Z P and thus view M as a space over X via ( e, p, y ) 7→ π V h 0  0 · h 0 ( p ) + e · R (0 · h 0 ( p ) , e )  . Thu s, we must compo se one of the co pies of λ with the ﬂip isomo rphism ﬂip : P × Z Y × Z P ∼ = − → P × Z Y × Z P , ( p 1 , y , p 2 ) 7→ ( p 2 , y , p 1 ) befo re we can compare them. Now we use the isotopy o f op en embeddings κ in (3.18). It connects λ = κ 0 × Z Y and ﬂip ◦ λ ◦ Ψ = κ 1 × Z Y , where Ψ maps e 7→ − e on E . The F-c ohomology class τ E · κ ∗ ( ξ ) on | E | × Z P × Z Y × [0 , 1] has X - compact support with respect to the map P × Z Y × Z P → X . This pro duces a s pecia l b ordism b etw een the products Θ # X ξ and e Θ # X ξ . Notice that the sig n ( − 1) d in the deﬁnition of e Θ cancels the sign pro duced by the automorphism Ψ ab ov e.  BIV ARIANT K-THEOR Y VIA CORRESPONDE NCES 27 R emark 3 .20 . With s ome a dditional eﬀor t, it ca n b e shown more generally that the maps PD ∗ and PD ∗ 2 provide iso morphisms ˆ f i G ⋉ X ( U, P × Z Y ) ∼ = ˆ f i − n G ( U, Y ) , ˆ f i G ⋉ X ( U, X × Z Y ) ∼ = ˆ f i − n G ( P × X U, Y ) , provided X is a non-singular G -space with b oundary , Y is any G -space, and U is a lo c al ly trivial G ⋉ X -s pace. Lo cal triviality means that there is a neighbourho o d W of the diag onal in X × Z X such that th e pull-backs of U to W along the t wo co ordinate pro jections W → X b ecome G -equiv ariantly ho meomorphic. The s ame condition is used in [10] to construct dualit y isomorphisms in equiv ariant Kaspar ov theory . 4. Comp arison to Kasp aro v theor y Now we restrict atten tion to the case where o ur c ohomology theory is equiv a ri- ant K -theory or, more precis ely , the repre sent able equiv ariant K -theory for loca lly compact G -spaces for a prop er lo cally compact gr oup oid G , see [9]. W e denote the topo logical biv ariant K -theory deﬁned above by c kk ∗ G ( X, Y ) : = ˆ f ∗ G ( X, Y ) for F = RK . W e wan t to compare it to the equiv ar iant Kaspa rov theory deﬁned in [17]. In order for b oth theories to be deﬁned, we requir e X and Y to be second count able, lo cally compact Hausdorﬀ spaces and G to be a pr op er, second countable, lo cally co mpact, Hausdorﬀ group oid with Haar system. Then G is numerably pro pe r by [12, Lemma 2.16]. R emark 4.1 . It follows from [9] that if G is a prop er group oid, X is a c o compact G -space, and Y is a G -spa ce with enough G -vector bundles, then RK ∗ G ,X ( Y ) can be decribed in ter ms of triples ( V + , V − , ϕ ) , wher e V ± are G -vector bundles on Y and ϕ : V + → V − is an equiv a riant vector bundle ma p that is an isomo rphism oﬀ an X -compact G - in v ariant closed subset. If Y is also a smoo th G -manifold, then a s imple a rgument with cross ed pro ducts implies that the vector bundles and ϕ can b e taken to b e smoo th. Comb ining these observ a tions with T heorem 4.8 gives a description of KK G ∗ ( C 0 ( X ) , C 0 ( Y )) in terms of smo oth corr esp ondences whose K -theory data ar e enco ded by smo oth G -equiv ariant vector bundles on Y which are smo othly isomorphic oﬀ an X -co mpact set. This is mor e in line with the tra ditional deﬁnitions. Theorem 4.2 . L et G b e a pr op er, se c ond c ountable, lo c al ly c omp act gr oup oid with Haar system and let X and Y b e se c ond c oun table, lo c al ly c omp act G -sp ac es. Ther e is a natu r al tr ansformation c kk ∗ G ( X, Y ) → KK G ∗  C 0 ( X ) , C 0 ( Y )  that pr eserves gr ad- ings, c omp osition pr o ducts, and exterior pr o ducts. It is an isomorphism if X has a symmetric dual in c kk ∗ . Pr o of. Let Ψ : = ( E , M , b , ξ ) be a sp ecial co rresp ondence from X to Y . The equi- v a riant K -theory RK ∗ G ,X ( M ) of M with X -co mpact supp ort is identiﬁed in [9 ] with KK G ⋉ X ∗  C 0 ( X ) , C 0 ( M )  ; here we view M as a space ov er X using b . In par ticular, ξ beco mes a clas s in KK G ⋉ X ∗  C 0 ( X ) , C 0 ( M )  , which maps to KK G ∗  C 0 ( X ) , C 0 ( M )  b y a forg etful functor. Since M is an op en subset of | E Y | , we may identify C 0 ( M ) with an ideal in C 0 ( | E Y | ) . The K -orientation for the G -vector bundle E Y ov er Y in- duces a KK G ⋉ Y -equiv alence b et ween C 0 ( | E Y | ) and C 0 ( Y ) . Putting all ingredients together, we ge t a class in KK G 0  C 0 ( X ) , C 0 ( Y )  , which we denote by KK(Ψ ) . 28 HEA TH EMERSON AND RALF MEYER The inv e rtible elemen t in KK G ⋉ Y  C 0 ( | E Y | ) , C 0 ( Y )  used abov e induces the Thom isomo rphism for the K -oriented G -vector bundle E Y . Since the composi- tion of tw o Thom isomorphisms is ag ain a Thom iso morphism for the direct sum vector bundle, it follows that KK(Ψ) = KK(Ψ E ′ ) if Ψ E ′ is the Thom mo diﬁcation of Ψ along a K -orien ted G - vector bundle E ′ ov er Z . Reca ll also that a sp ecial bo rdism of corres po ndences fro m X to Y is nothing but a spe cial corresp ondence from X to Y × [0 , 1] . Hence a b or dism pro duces a homotopy o f Kasparov cycles, so that KK(Ψ) is inv ar iant under sp ecial b ordisms. Thus Ψ 7→ K K(Ψ) is a well-deﬁned map c kk ∗ G ( X, Y ) → KK G ∗  C 0 ( X ) , C 0 ( Y )  . It is clear that this construction pr eserves gra dings. Compatibility with exter ior pro ducts is easy to chec k as well. W e c heck compatibility with co mpos ition pr o ducts. Recall that a sp ecial corr esp ondence Ψ = ( E , M , b, ξ ) is the pr o duct of one of the form ( b, ξ ) ∗ and f ! for a special normally non-singula r submersion f : M ⊆ | E Y | ։ Y . By construction, KK(Ψ) is the pro duct o f KK  ( b, ξ ) ∗  and KK( f !) . It is ea sy to see that Ψ 7→ KK(Ψ) is multiplicativ e on corr esp ondences of the sp ecial form ( b, ξ ) ∗ . Multiplicativit y for normally no n-singular maps – in particular, for sp ecial normally non-singular submersions – fo llows as in [12]. It remains to chec k m ultiplicativity for pro ducts of the form Φ! # ( b, ξ ) ∗ for a sp ecial normally non-s ingular submersion Φ = ( E , X ) with an op en subset X ⊆ | E Y | , a G -map b : U → Y , and ξ ∈ RK ∗ G ,Y ( U ) . The pro duct in c kk ∗ G ( X, U ) is the sp ecial corresp o ndence  E , X × Y U, p 1 , p ∗ 2 ( ξ )  , where p 1 : X × Y U → X and p 2 : X × Y U → U are the co ordina te pro jections. Now the assertion follows from the natura lit y of the Tho m isomorphism with respect to KK  ( b, ξ ) ∗  . More ex plicitly , let π E Y : | E Y | ։ Y b e the pro jection. Then the diagram C 0 ( X ) ⊆ / / Id X × Y ( b,ξ ) ∗   C 0 ( | E Y | ) Id | E | × Z ( b,ξ ) ∗   π E Y ! / / C 0 ( Y ) ( b,ξ ) ∗   C 0 ( X × Y U ) ⊆ / / C 0 ( | E U | ) p U ! / / C 0 ( U ) commut es in K K G . Hence KK  Φ! # Y ( b, ξ ) ∗  = KK(Φ!) ⊗ C 0 ( Y ) KK  ( b, ξ ) ∗  . This ﬁnishes the pro of that Ψ 7→ K K(Ψ) is a functor. Now assume that ( P , D , Θ , e Θ) is a K -orie n ted s ymmetric dual for X , so that Theorem 3.11 provides a n isomorphism c kk ∗ G ( X, Y ) ∼ = c kk ∗ G ⋉ X ( X, P × Z Y ) ∼ = RK ∗ G ,X ( P × Z Y ) . The imag es of D , Θ a nd e Θ in Kasparov theory satisfy analogues of the conditions in Deﬁnition 3.1 0 b ecause the tr ansformation from c kk to KK is compatible with comp osition and exterior pro ducts. F or the last tw o co nditions (iv) and (v), Theo- rem 2 .24 and its analog ue in K asparov theory show that there ar e the same auxiliary data g to co nsider in bo th theories. The same co mputations as in §3.2 yield KK G ∗  C 0 ( X ) , C 0 ( Y )  ∼ = KK G ⋉ X ∗  C 0 ( X ) , C 0 ( P × Z Y )  ∼ = RK ∗ G ,X ( P × Z Y ) . The la st isomorphism is contained in [9] (in fact, it is a deﬁnition in [9]; the results in [9] show that this deﬁnition agrees with the one used here). Hence we get the desired isomorphism.  W e leav e it to the reader to deﬁne KK(Ψ) for non-specia l corresp ondences di- rectly and to c heck that KK(Ψ) = KK(Ψ ′ ) if Ψ ′ is the sp ecial corresp ondence asso ciated to a cor resp ondence Ψ . BIV ARIANT K-THEOR Y VIA CORRESPONDE NCES 29 Corollary 4.3. The natur al tr ansformation c kk ∗ G ( X, Y ) → KK G ∗  C 0 ( X ) , C 0 ( Y )  is invertible if X is normal ly non-singular. Pr o of. Combine Theorems 3 .17 and 4.2.  R emark 4 .4 . It is not diﬃcult to c heck that in fact a symmetric dual in c kk G arising from a ﬁbrewise stable smo oth structure on X ma ps to a Kasparov dual in KK G . This follows from an examination of the pro of of [10, Theorem 7.11], which carr ies through with no changes fr om the smo o th c ase. 4.1. Smo oth corresp ondences. A smo oth G - manifold is a G -space with a ﬁbre- wise smo oth s tructure along the ﬁbr es o f the anchor map X → Z determined by an atlas for X consisting of open sets iso morphic to pro ducts U × R n where U ⊂ Z is an op en set. W e also require that the isomorphism intert wines the anchor map and the ﬁrst co ordina te pro jection, and that the change-of-v ariables ma ps are smo oth in the vertical dire ction. Theorem 4.5 ([12 ]) . L et X and Y b e smo oth G -manifolds, assume that ther e is a smo oth normal ly non-singular map fr om X to the obje ct sp ac e Z of G and that either T Y is subtrivial or that al l G -ve ctor bund les on X ar e subtrivial. Then any smo oth G - m ap fr om X to Y is the t r ac e of a smo oth n ormal ly n on-singular G -map, and two sm o oth normal ly non- singular maps fr om X to Y ar e smo othly e quivalent if and only if their tr ac es ar e smo othly homotopic. R emark 4.6 . Theo rem 4.5 fails for non-smo oth normally non-s ingular maps: for a smo oth manifold X , there may be several norma lly non- singular maps X → X × X whose trace is the diag onal em bedding. T aking into account orientations, one can chec k that smo oth eq uiv a lence classes of F-oriented smo oth normally non-singula r G -maps fr om X to Y cor resp ond bi- jectiv ely to pairs ( f , τ ) where f is the smo o th homo topy cla ss of a smo oth G - map from X to Y and τ is a stable F-orientation on [ T X ] − f ∗ [ T Y ] . Deﬁnition 4. 7 . Let X and Y be smoo th G - manifolds. A smo oth c orr esp ondenc e from X to Y is a corresp ondence ( M , b, f , ξ ) from X to Y where M is a smo oth G -manifold, b is a ﬁbrewise smo oth G -map, and f is a smo oth F-oriented norma lly non-singular G -ma p. Smo oth b or disms , sp e cial smo oth c orr esp ondenc es , and sp e cial smo oth b or disms a re deﬁned similarly . W e let smo oth e quivalenc e b e the equiv alence relation on smo oth cor resp ondences g enerated b y smo oth equiv alence of nor mally non-singular ma ps, smoo th b ordism, and Thom mo diﬁcation b y smoo th G -vector bundles. The same ar guments as in the non-smo oth cas e show that every s mo oth corr e- sp ondence is equiv alent to a sp ecial smo oth corresp ondence and that t wo sp ecial smo oth corresp ondences are smoothly equiv alent if a nd o nly if they hav e Thom mo diﬁcations by trivial G -vector bundles that a re related b y a sp ecial smo o th bor - dism (see Theorem 2.2 3). Let ( E , M , b, ξ ) b e a sp ecial co rresp ondence fro m X to Y . Any G -vector bundle ov er Z ca rries a unique smo oth structure for whic h it is a smo o th G -vector bundle, and this r estricts to a unique smo oth G -manifold structure on M . Hence a sp ecial smo oth corr esp ondence do es not carry additional structure, it merely has the addi- tional prope rty that the G -map b : M → X is ﬁbre wise smo oth. The same applies to smo oth b ordisms. Since in ters ection pr o ducts a nd exterior products of specia l smoo th corresp on- dences are again sp ecial smo oth corresp ondences, the smo oth cor resp ondences form a symmetric monoidal ca tegory as well. Theor em 2 .24 and the duality res ults in §3.2 work for the smoo th version of ˆ f ∗ G as well. 30 HEA TH EMERSON AND RALF MEYER Theorem 4 .8. L et X b e a smo oth normal ly non-singular G - m anifold. Then X has a smo oth symmetric dual. F urthermor e, the smo oth and n on- smo oth versions of ˆ f ∗ ( X, Y ) agr e e in this c ase, for any smo oth G -manifold Y . Pr o of. The constructio ns in §3.3 pro duce smo oth corresp ondences if we plug in a smo oth normally non-singula r map from X to Z , and the pro of of Theor em 3.17 still works in the smo oth v ersion of ˆ f ∗ . This duality isomorphism allows to identify the tw o versions of ˆ f ∗ as in the pro of of Theorem 4.2.  R emark 4.9 . If there is a smo oth normally non-singular map from M to Z , then as we hav e stated ab ov e, smo o th equiv alence classes of smo oth normally non-singular maps M → Y co rresp ond bĳectively to smo oth homotopy classes of s mo oth maps from M to Y , and since smo oth homotopy is a s pecia l case of smo oth bo rdism, we we may drop “smo oth equiv alence of normally no n-singular maps” from the deﬁnition of smo oth equiv alence in Deﬁnition 4.7. The problem with this observ ation is that we hav e little co n trol ab out M . It seems that we can o nly a pply it if al l smo oth G -manifolds M admit a smo oth normally non-singula r G - map to Z . This holds, for instance, if G = G ⋉ Z for a dis- crete group G and a ﬁnite-dimensional G -CW-complex Z with uniformly b ounded isotropy groups. If G is, say , a compact group, then we must restrict atten tion to smo oth co rresp ondences ( M , b, f , ξ ) where M is a smo oth G -manifold of ﬁnite orbit t ype . Summing up, the diﬀerence betw een smo oth ma ps and smo oth normally non- singular maps is usual ly insigniﬁca n t – but only under some m ild tec hnical as- sumption. O ur theory dep ends on nor mally non-singular maps because only this allows to re place a gener al co rresp ondence by a s pecia l one, but it do es not dep end on smo o thness. This is why w e develop ed our main theory without smo othness assumptions in the main par t of this ar ticle. 5. Outlook and co n cl uding remarks W e have e xtended an equiv ar iant co homology theor y to a biv ariant theory . In particular, this pr ovides a pur ely topo logical counterpart o f equiv a riant Ka sparov theory for prop er group oids. W e hav e used dualit y isomorphisms to iden tify the topo logical and ana lytic biv ariant K -theories , and established s uc h duality isomo r- phisms for smo oth G -ma nifolds with b oundary (under some tec hnical assumptions ab out equiv a riant vector bundles). It is known that any ﬁnite-dimensiona l CW-co mplex is homo top y equiv alent to a smoo th manifold with b oundar y and hence admits a symmetric dual. W e do not know whether a similar res ult ho lds equiv ariantly , say , for simplicial complexes with an action o f a ﬁnite group. Anyw ay , it is desirable for computations to construct symmetric duals for simplicia l complexes or ev en CW-complexes in ˆ f ∗ G directly , without mo delling them b y smo oth manifolds with b oundary . In biv a riant K asparov theory , s uch a symmetric dual for a simplicial complex is co nstructed in [8], but it in volves mildly non-commutativ e C ∗ -algebra s. It is an open pr oblem to replace this by a purely commut ative construction. An issue that w e hav e neglected here is excision. Since Kasparov theory satisﬁes very strong excision results for prop er actions, our top ological theory will satisfy excision whenever it ag rees with Kasparov theory . But we s hould not exp ect go o d excision results in complete gener ality: this is one of the p oints where a la c k o f enough G -vector bundles o ver Z should cause problems. The ﬁr st dualit y isomorphism ma y b e used to deﬁne equiv ariant Euler c harac- teristics and equiv ariant Lefsc hetz inv ar iants (see [8, 10]). Since we hav e translated BIV ARIANT K-THEOR Y VIA CORRESPONDE NCES 31 it to a purely topo logical biv ar iant K -theor y , we ca n now co mpute these inv ariants geometrically . W e will carr y this out for several examples in a forthcoming article. Although we are mainly in tere sted in K -theory a nd KO -theory her e, we ha ve allow ed mo re gener al equiv aria n t cohomolo gy theories in all our constructions. W e exp ect this to hav e several applications. First, since our construction of biv ar iant theories is functoria l with resp ect to natural transformations of cohomology theories, it should b e useful to constr uct biv a riant Cher n characters from biv ariant equiv a riant K -theor y to suitable biv a riant Bredon cohomo logy gr oups (at least for discrete gr oups). Secondly , w e hop e to deﬁne biv a riant v er sions of twisted K -theory within our framework. One approa c h to this views twisted K -theory as a PU ( H ) -equiv aria n t cohomolog y theor y , wher e PU ( H ) denotes the pro jective unitary group of a sepa- rable Hilb ert space H . A space with a t wist datum can b e describ ed a s a principal PU ( H ) -bundle. 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MR 0 271930 32 HEA TH EMERSON AND RALF MEYER E-mail addr ess : hemerson@ math.uvi c.ca Dep ar tmen t of Ma thema tics and St a tistics, University of Victoria, PO BO X 304 5 STN CSC, Victoria, B.C., Canada V8W 3P4 E-mail addr ess : rameyer@u ni-math. gwdg.de Ma thema tisches Institut and Courant Resear ch Centre “Higher Order Structures”, Georg-A ugust Universit ä t Göttingen, Bunsenstraße 3–5 , 37073 Göttingen, German y

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