Duality, correspondences and the Lefschetz map in equivariant KK-theory: a survey

DUALITY, CORRESPONDENCE S AND T HE LEFSCHETZ MAP IN EQUIV ARIANT KK-THEOR Y: A SU R VEY HEA TH EMERSON Abstract. W e surve y wo rk by the author and Ralf Meyer on equiv ari ant KK-theory . Dualit y pla ys a k ey role in our approac h. W e ha v e organized this surve y around the obj ectiv e of computing a certain homotopy-in v ari an t of a space equipped with a (g enerally proper ) action of a groupoi d. This inv ariant is called the Lefsch etz map. The Le fschet z map associates an e quiv arian t K- homology class to an equiv arian t Kasparo v s elf -morphism of a space X. W e wa nt to describ e i t explicitly i n the setting of bundles of s mo oth manifolds o v er the base space of a group oid, i n which groupoid elemen ts act b y diﬀeomor- phisms betw een ﬁbres. T o get the required description we de velop a topological model of equiv arian t KK-theory by wa y of a theory of corresp ondences, build- ing on ideas of Paul Baum, Alain Connes and Georges Skandalis in the 1980’s. This mo del agrees with the analytic mo del for bundles of smo oth manifolds under some technica l con ditions related to the existence of equiv arian t ve ctor bundles. Sub ject to these conditions we obtain the desired computation of the Lefsc hetz map in purely topological terms. Finally , we describe a ge neraliza- tion of the classical Lefschetz ﬁxed-p oint formula to apply to corresp ondences, instead of just maps. The papers [13], [11], [12], [16] p resent a study of the equiv a riant Kasparov groups KK G  C 0 ( X ) , C 0 ( Y )  where G is a lo cally c ompact Hausdo rﬀ group oid with Haar system and X and Y are G -spaces, usua lly with X a prop er G -spa ce. This pr o- gram builds on work of Ka sparov, Connes and Skandalis done mainly in the 1980’s. A t that po int , the main interest was the index theor em o f Atiy ah a nd Singe r and its gener alisations, and later , the Dirac dual-Dirac metho d and the Noviko v conjec- ture. F o r us, the goal is to develop E uler c haracteristics and Lefsc hetz form ulas in equiv a riant KK-theory . Via the Ba um-Connes isomorphism – when it applies – this contributes to noncommutativ e top ology and index theor y . Our progra m sta rted in [1 3] where we fo und the L efschetz m ap in connection with a K-theory pro blem. W e will g ive the deﬁnition of the Lefschetz map in the ﬁrs t section, but for now record that it has the form (0.1) Lef : KK G ⋉ X ∗  C 0 ( X × Z X ) , C 0 ( X )  → KK G ∗ ( C 0 ( X ) , C 0 ( Z )  , where w e alwa ys deno te b y Z the ba se space of the group oid. This map is deﬁned under certain somewhat technical circumsta nces, but, again, these nor mally in volv e prop er G - spaces X . The domain of the Lefschetz map is very closely related to t he simpler-lo oking group KK G ∗  C 0 ( X ) , C 0 ( X )  : the la tter group maps in a natural wa y to the domain in (0.1) and th is map is an isomorphism when the anc hor map X → Z is a prop er map. This means th at the Lefschetz map ca n b e used to assign an inv aria nt , which is a n equiv a riant K-ho mology cla ss, to an e quiv ar iant K asparov self-morphism o f X . W e call this class the L efschetz invariant of the map. It bears consideration even when G is the tr ivial gro upo id, and the rea der can do worse than 2000 Mathematics Subje ct Class iﬁc ation. 19K35, 46L80. Heath Emerson was supported by a National Science and Engineering Council of Canada (NSER C) Di sco v ery gran t. 1 2 HEA TH EM ERSON to co nsider this cas e to b egin with, although by doing so one misses the applica tions to noncomm utative top olog y . The deﬁnition of (0.1) uses the notion of an abstr act dual for X . Abstra ct duals for a given G - space X are not unique but the Lefschet z map do es no t dep end on the choice of a dual, only on the existence of one. Abstract duals do not a lwa ys exist either: a Cantor set X do e sn’t hav e an abstract dual ev en if G is trivial. But if X is a smo o th G -manifold, with G acting smo othly and prop erly on X , then X has an abs tract dual, a nd if G is a group, then any G -simplicial complex has a n abstract dual due to [13]. The L efsch etz map is functorial for G -maps X → X ′ in a w ay made explicit in §1.9 (see Theorem 1.26). In brief, it is a homotopy invariant of the G -space X . In particular, since Lef do esn’t depend on the dual us ed to to compute it, one can t ry to compute the Lefschet z inv ariant of a g iven morphism using tw o diﬀerent dua ls and thereby get an identit y in eq uiv ar iant K -homology (see §2). Suc h examples (w orked out in [13] and [15]), seemed to us interesting enough to supp or t making a systematic study of the Lefschetz map. Howev er, to g et started on this question one obviously has to ﬁrst describ e the morphisms f themselves in so me kind o f satisfactory w a y . T o this end, we have extended the theory o f corresp ondences initiated by Ba um, Connes, Skandalis and others, to the equiv a riant situation, in the pap er [16]. This e xtension presents some new features, and we will devote a part o f this survey to explaining them. O f cours e the theory of corres po ndences is useful and imp or tant in its own right. But it is designe d for intersection theory beca use of the w a y corr espo ndences are compo sed using coincidence spaces and transversalit y . The theory of cor resp ondences re quires the groupo id G to b e pr op er . The Baum- Connes conjecture allows us to reduce to this case sub ject to a weak er as sumption that we expla in b elow. Let G b e pr op er a nd X and Y b e G -spa ces. A G -e quivari ant c o rr esp ondenc e fr om X to Y is a qua druple ( M , b, f , ξ ) where M is a G -spa ce, b : M → X is a G -map (not necessa rily prop er), ξ is an equiv arian t K-theory class with co mpact vertical supp ort along the ﬁbre s of b , and f is a K-orie n ted n ormal ly non-singular map from M to Y (see Deﬁnition 3.15). F or exa mple, if G is a co mpact group, Y is a point and M is compact, then an normally non-singular map M → Y is the sp eciﬁcation o f an orthogo nal repres en tation of G on s ome R n , an equiv arian t vector bundle V o v er M , and an open equiv a riant em bedding ˆ f : V → R n . T o construct a n example of such a triple, assume that M has been giv en the structure of a smo oth ma nifold, and that G a cts smo o thly . In this case we may app eal to a theorem of Mostow to em be d M in a ﬁnite-dimensional linear representation of G , then take V to b e the normal bundle to the em bedding. There is a topo logically deﬁned equiv a lence relation on corresp ondences that makes the se t o f equiv alence c lasses of G -equiv arian t corr esp ondences from X to Y the morphism set c kk ∗ G ( X, Y ) in a Z / 2-g raded ca tegory c kk G which maps naturally to KK G . F or exa mple, let G b e compact gro up and let bo th X a nd Y b e the one-p oint space. Let M b e a smo oth, co mpact, equiv ariant ly K-oriented, even-dimensional G -manifold, ξ ∈ K 0 G ( M ) b e an eq uiv ar iant K- theory class for M represented by an equiv arian t vector bundle V on M . By embedding M in a ﬁnite-dimensiona l representation of G as in the previous para graph, we can endow the map from M to Y : = ⋆ with the structure of a smo o th, K-o riented, nor mal ma p, a nd we o btain a G -equiv a riant co rresp ondence ( M , ⋆, ⋆, ξ ) from a p oint to itself. This yields a class in c kk 0 G ( ⋆, ⋆ ). Applying the na tural map c kk G 0 ( ⋆, ⋆ ) → KK G 0 ( C , C ) ∼ = Rep( G ) maps this corr esp ondence to the G -e quivaria nt t op olo gic al index of D V in the sense of [1], where D V is the Dirac oper ator on M t wisted b y the equiv ariant v ector bundle V . By the Atiy ah-Singer Index theoreom, this agree s with the G - equiv a riant analytic DUALITY AND CORRESPONDENCES 3 index of D V in Rep( G ), obtained by consider ing the diﬀ erence of ﬁnite-dimensional G -representations o n the k ernel and cok ernel o f D V . The combination of t he A ti yah-Singer index theorem and t he t heory of equiv ari- ant corr esp ondences r epresents a powerful to ol, b ecause while the index theorem allows us to translate analytic pr oblems into top ologica l ones, the theory of corr e- sp ondences allows us to manipulate this top olog ical data in in teresting ways. F or an example of this pro ce ss in connectio n with the re presentation theory of complex semisimple Lie groups, see [17]. In terms of the Lefschetz map, the fa ct that corr esp ondences ca n be comp osed in an essentially t op ologica l fashion has the consequence that the Lefshetz inv aria n ts of self-corr esp ondences of a smo o th G -manifold X , or, or mor e pr ecisely , of their images in KK G , ca n b e computed in terms of consideratio ns of transversality . W e explain the outcome of th is computation in Sectio n 4 : the gist is that the Lefsc hetz in v a riant of a smo oth e quiv ar iant self-corr esp ondence Ψ of a smooth G -manifold X in genera l p os ition, ca n b e describ ed in terms of a G -space c alled the c oincidenc e sp ac e F ′ Ψ of the corresp o ndence. The coincidence space inherits fro m the smo oth structure and K-orientation o n Ψ the structure of a smo o th and equiv arian tly K- oriented G -ma nifold which maps to X and repr esents an equiv a riant cor resp ondence from X to Z , thus a c ycle for c kk ∗ G ( X, Z ) and then a class in K K G ∗  C 0 ( X ) , C 0 ( Z )  . It represents the Lefschetz inv aria nt of the morphism repr esent ed by Ψ. See Theorem 4.12 for the exact statement. The ‘general po sition’ caveat is non-trivial: in the equiv a riant setting, it ma y not b e poss ible to p erturb a pair of eq uiv ar iant maps to make them transverse. The framew ork of equiv ar iant cor resp ondences allo ws us to treat this diﬃcult y using Bott P erio dicity , but w e do not dis cuss this m uc h here. F or example, if G is a compact gr oup, X a smo oth and compact ma nifold with a s mo oth actio n of G , then the Lefschetz inv ar iant of a s mo oth equiv a riant map f : X → X in general p ositio n is the ﬁxed-p oint set o f the ma p, which is a ﬁnite set of p oints p ermuted by G , orient ed b y an equiv a riant line bundle o ver this ﬁnite set. This bundle dep ends on o rientation data from the original map f in a manner which reduces to the classical choice of s igns at ea ch ﬁxed-p oint when G is trivia l. Thu s, the top olo gical mo del of the Lefschetz map provided by the theory o f cor - resp ondences yields an interpretation of Lef in terms of a a ﬁxed-po in t theo ry for corresp ondences . One natura lly asks when the map c kk ∗ G ( X, Y ) → KK G ∗  C 0 ( X ) , C 0 ( Y )  is a n is o- morphism. W e explain our results on this in §4.1; once again, they rely on dualit y in a crucial wa y . When they apply , the top olog ical and analytic Lefschetz maps are equiv ale n t. As ment ioned ab ov e, the Baum-Co nnes conjecture can b e used to reduce the non-prop er situation to the proper one under some weak er a ssumptions on the G action on X , namely that it b e top o lo gic al ly amenable . Thi s is expla ined in §3.1. P utting everything together g ives a co mputation o f the Lefschetz inv aria nt for quite a wide spectrum of smo oth G - spaces X . What is dualit y? It is cen tral to o ur whole f ramework, and a ccordingly we b egin the article with a discussion of it. It is well-known from the w ork of Kaspar ov and Connes and Ska ndalis (se e [20] a nd [8]) that if if X is a smo oth manifold, then there is a natural family o f isomorphisms KK( C 0 (T X ) , C ) ∼ = RK ∗ ( X ) : = KK X ( C 0 ( X ) , C 0 ( X )) where the gr oup oid equiv a riant KK group on the r ight is equiv ariant r epr esentible K-theory , o r K-theory with lo c al ly ﬁnite supp ort , denoted RK ∗ ( X ) b y Ka sparov. There is a gener alisation of this dualit y to the equiv ariant s ituation if G is a group oid acting smo othly and pro per ly on a bund le X → Z of smo o th manifolds ov er the base Z of G , and furthermore, the roles o f X a nd T X ca n b e in a sense reversed, so that 4 HEA TH EM ERSON one can establish a pair of na tural (in a tec hnical sense) familes of iso morphisms (0.2) KK G ∗ ( C 0 (T X ) ⊗ A, B ) ∼ = KK G ⋉ X ∗ ( C 0 ( X ) ⊗ A, C 0 ( X ) ⊗ B ) and (0.3) KK G ∗ ( C 0 ( X ) ⊗ A, B ) ∼ = KK G ⋉ X ∗ ( C 0 ( X ) , C 0 (T X ) ⊗ B ) for all G -C*-alg ebras A and B . (The tensor pro ducts are all in the catego ry of G -C*- algebras .) These results are proved in [11]. It is the ﬁrst kind of duality (0.2) which is relev an t for the Lefschetz map, and the second (0.3) that is used to prove that the map c kk G → KK G is an isomorphism in certain cases. The basic idea is that since the dualit y iso morphisms (0.3) are themselves induced by e quiv ar iant corres po ndences, dualit y can be used simul taneously in both the analytic and topolo gical categories to to reduce the question to a problem about monova riant KK-theory , that is, equiv a riant K-theory with supp ort conditions. What is needed to ma ke this work is then a top olo gical mo del of dualit y . The main new issue that app ear s is that our e quiv ar iant corr esp ondences r equire a g o o d supply of equiv arian t vector bundles and this forces conditions on the g roup oid G . These cons iderations hav e in fact alrea dy appear ed in the literature in co nnection with the (prop er) group oids G ⋉ E G in work by W olfang Lück and Bob O liv er in [23] (see §3 .5 for the details) where G is a discrete gro up and E G is its classifying space for pro p er actions. W e explain exac tly wha t the conditions ar e and how they are related to embedding theorems gener alizing the em bedding theorem o f Mos tow alluded to above. The c lassical Lefsc hetz ﬁxed- po in t theorem r elates ﬁxed-points of a map and the homolo gical inv ar iant o f the map o btained by taking the graded trace of the induced map o n homology , called the Lefschetz num ber. W e ﬁnish this s urvey by in- tro ducing some global, homological in v ariants o f corresp ondences whic h g eneralize the Lefsc hetz n um ber, at least in the cas e when the g roup oid G is trivial. Ro ughly sp eaking, a Kaspa rov self-mor phism, and in particular a self-corr esp ondence should be c onsidered as determining a linear op er ator on homology instead o f just a num- ber . W e call it the Lefsc hetz operator . W e will e xplain ho w to extend the classical Lefschet z ﬁxed-p oint theorem to corr esp ondences by ident ifying the Lefschetz op er - ator with the oper ator of pairing with th e Lefschetz inv ariant. In situations where a l o cal index formula is a v aila ble, this results in a description of the Lefschetz op erator in lo cal, geometric terms. It rema ins to des crib e the Lefschetz map in global, homo logical terms – as in the cla ssical formula, in whic h ﬁxed-p oints are related to tra ces on homology . This problem seems quite delicate, ho w ev er. One wa y of pro ceeding is to replace K- theory groups b y G -equiv arian t K-theory mo dules ov er Rep( G ) and replace the ordinary trace by the Ha ttori-Stallings trace. How ever, this is only deﬁned under quite s tringent conditions: suc h mo dules are only ﬁnitely presented in general o nly for g roups for whic h the r epresentation rings hav e ﬁnite cohomo logical dimension, and this requires additional h ypotheses on G . This w ork is still in prog ress. 1. Abstract d uality and the Lefschetz map Throughout this pap er, gr oup oid shall mean lo cally compact Hausdor ﬀ gr oupo id with Haar s ystem. All topolog ical spaces will be assumed paracompact, lo cally compact and Hausdorﬀ. F or the material in this section, see [11]. F or source material on equiv ariant KK-th eory for group oids, see [22] . O ne seems to b e forced to consider group oids, as opp os ed to groups, in equiv arian t Kasparov theory , even if one is ultimately only in terested in g roups. This will b e explained later. Therefore we will work mo re or less uniformly with gr oup oids when discus sing general theor y . When we discuss topo logical equiv arian t Kaspar ov theory , we will DUALITY AND CORRESPONDENCES 5 further a ssume that all gro upo ids are pr op er . This r estriction is needed for v ari- ous geometric co nstructions. The additional a ssumption of prop ernes s in v olves no serious loss of ge nerality for our purp oses beca use the Baum-Connes isomorphism, when it applies, giv es a method of replacing non-prop er group oids b y proper ones. 1.1. Equiv ariant Kasparo v theo ry for g roup oids. Let G b e a gr oup oid. W e let Z denote th e base space. A G -C* -algebra is in pa rticular a C*-algebra ov er Z . This means that there is g iven a non-degener ate equiv a riant *-homomorphism from C 0 ( Z ) to central multipliers of A . This identiﬁes A with the section alg ebra of a contin uous bundle of C* -algebras over Z . F or a group oid action we require in addition an isomorphism r ∗ ( A ) → s ∗ ( A ) whic h is compatible with the structure of r ∗ ( A ) and s ∗ ( A ) as C*-alg ebras over G . Here r : G → Z and s : G → Z are the range and source map of the group oid, a nd r ∗ (and simil arly s ∗ ) deno tes t he usual pullback op era tion of bundles. F rom the bundle p oint of view, all of this means that group oid element s g with s ( g ) = x and r ( g ) = y induce * -homomorphisms A x → A y betw een the ﬁbres of A at x and y . In particular, if A is commutativ e, t hen A is the C* -algebra of con tin uous func- tions on a loca lly compact G - space X , equipped wit h a map  X : X → Z called the anchor map for X , and a homeomorphism G × Z,s X → G × Z,r X , ( g , x ) 7→ ( g , g x ) where the domain and r ange of this ho meomorphism (by abuse of notation) are resp ectively G × Z,s X : = { ( g , x ) ∈ G × X | s ( g ) =  X ( x ) } , and similarly for G × Z,r X using r instead of s . 1.2. T ensor pro ducts. The category of G -C* -algebra s has a symmetric monoidal structure g iven b y tensor pro ducts. W e des crib e this very brieﬂy (see [1 1][Section 2] for details). Let A a nd B b e t w o G - C*-algebr as. Since they a re each C*- algebras ov er Z , their externa l tensor pro duct A ⊗ B is a C*-a lgebra over Z × Z . W e restrict this to a C*-algebra ov er the diag onal Z ⊂ Z × Z . The result is c alled the tensor pr o duct of A and B over Z . The tenso r pr o duct of A and B ov er Z car ries a diagonal action of G . W e lea v e it to the reader to c hec k t hat we obtain a G -C* -algebra in th is wa y . In order not to co mplicate notation, we write just A ⊗ B for the tensor pr o d uct of A and B in the c ate gory of G -C*-algebr as. W e emphasize that the tensor pro duct is over Z ; this is not the same as t he tensor pro duct in the categ ory of C*-algebras. F or commutativ e C*-alg ebras, i.e. for G -spa ces, say X a nd Y , with anchor maps as usual denoted  X : X → Z a nd  Y : Y → Z , the tenso r pro duct is Gelfand dual to the opera tion whic h forms f rom X and Y the ﬁbre pro duct X × Z Y : = { ( x, y ) ∈ X × Y |  X ( x ) =  Y ( y ) } . The req uired a nc hor map  X × Z Y : X × Z Y → Z is o f cours e the comp osition of the ﬁrst co or dinate pr o jection and the anchor map for X (or the ana logue using Y ; they are equal). Of co urse g roup oid elements a ct dia gonally in the ob vious way . Suc h c oincidenc e sp ac es as the one just describ ed will app ear aga in and again in the theory of corresp ondences. Finally , for t he record, w e supply the follo wing important deﬁn ition. Deﬁnition 1. 1. Let G be a gr oup oid. A G -space X is pr op er if the map G × Z X → X , ( g, x ) 7→ ( g x, x ) is a prop er map, where G × Z X : = { ( g , x ) ∈ G × X | s ( g ) =  X ( x ) } . 6 HEA TH EM ERSON A group oid is itself called pr op er if it acts pr op erly on its base space Z . Explicitly , the map G → X × Z X , g 7→ ( r ( g ) , s ( g )) is required to be pro per . 1.3. Equiv ariant Kasparo v theory . Le Ga ll has deﬁned G - equiv ar iant KK-theor y in [22]. W e brieﬂy sketc h the deﬁnitions. Let A and B b e (pos sibly Z / 2 -graded) G -C*-algebr as. Then a cycle for K K G ( A, B ) is g iven by a Z / 2-g raded G -equiv arian t Hilb ert B - mo dule E , together with a G -equiv ariant gra ding-preserving *-homomo rphism from A to the C*-algebr a of bounded, adjoin table operator s on E , and a n es sent ially G -equiv a riant self-a djoint o per ator F on E whic h is graded o dd and sa tisﬁes [ a, F ] and a ( F 2 − 1) a re compact operator s (essen tially zero op er ators) for all a ∈ A . Mo dulo an appropriate equiv alence relation, the set of equiv alence classes of cycles can b e identiﬁed with the morphism set KK G ( A, B ) in an additive, symmetric monoidal category . Higher KK-groups are deﬁned using Cliﬀord algebras, and since these are 2-p erio dic, there are o nly tw o up to isomor phism. W e deno te by KK G ∗ ( A, B ) the sum of these t wo g roups. If A a nd B a re G -C* -algebras , then the gr oup RKK G ( X ; A, B ) is b y deﬁnition the group oid-equiv arian t Kasparov group KK G ⋉ X ( C 0 ( X ) ⊗ A , C 0 ( X ) ⊗ B ) . The tensor pro ducts are in the category of G -C* -algebra s. This gro up d iﬀers from KK G ( A, C 0 ( X ) ⊗ B ) o nly in the supp ort condition on cycles. F or example if G is trivial and A = B = C then KK G ( A, C 0 ( X ) ⊗ B ) is the or dinary K-theo ry o f X and RKK G ( X ; A, B ) is the representable K-theor y of X (a non-compactly supp orted theory .) W e discuss these groups in more detail in the next section. Of co urse similar remarks hold for higher RKK G -groups. 1.4. Equiv ariant K -theory. In this section, w e pres en t an exceeding ly brief ov erview of equiv a riant K-theory , roughly suﬃcient for the theory of equiv arian t corre spo n- dences. F o r more details see [12]. Let X be a pr op er G -spa ce. Recall tha t a G ⋉ X -space consists of a G -spac e Y together with a G -equiv ariant map  Y : Y → X serving as the a nc hor map for the G ⋉ X -a ction. Deﬁnition 1.2 . Let Y b e a G ⋉ X -spac e. The G -e quiv ariant r epr esentable K -the ory of Y with X -c omp act supp o rts is the group RK −∗ G ,X ( Y ) : = KK G ⋉ X ∗  C 0 ( X ) , C 0 ( Y )  . The G -e quivariant r epr esent able K -t he ory of Y is RK ∗ G ( Y ) : = RK ∗ G ,Y ( Y ) . Cycles for KK G ⋉ X  C 0 ( X ) , C 0 ( Y )  consist of pa irs ( H , F ) where E is a co un tably generated Z / 2-gr aded G ⋉ X -equiv ariant rig h t Hilb ert C 0 ( X )-mo dule equippe d with a G ⋉ X -equiv arian t no n-degenerate *-ho momorphism fro m C 0 ( X ) to the C*-algebr a of b ounded, adjointable opera tors on E , and F is a bounded, o dd, self-adjoint , essentially G -equiv ariant adjoint able op era tor on H such that f ( F 2 − 1) is a compact op erator, for all f ∈ C 0 ( X ). The prop er ness of G implies that F may b e averaged to be actually G -equiv ariant , so w e assume t his in the following. The Hilbert C 0 ( Y )-mo dule E is the spa ce of con tin uous sections of a co nt inuous ﬁeld of Z / 2-gr aded Hilb ert spa ces {H y | y ∈ Y } ov er Y . Since F must b e C 0 ( Y )- linear, it consists of a co nt in uous family { F y | y ∈ Y } of o dd op er ators on these graded Hilbert spaces such that F 2 y − 1 is a compact op erator on H y for all y ∈ Y . By G ⋉ X -equiv ariance, the representation of C 0 ( X ) on E must factor through the *-homomor phism C 0 ( X ) → C 0 ( Y ) Gelfand dua l to the anchor map  Y : Y → DUALITY AND CORRESPONDENCES 7 X . Therefore F comm utes with the a ction of C 0 ( X ) as well; in fact the induced representation of C 0 ( X ) on eac h Hilbert space H y sends a contin uous function f ∈ C 0 ( X ) to the opera tor of multiplication b y the complex n um ber f   Y ( y )  . In particular, the only role of the repr esentation of C 0 ( X ) is to r elax the support condition on the compact-o per ator v alued-function F 2 − 1 from r equiring it to v anish at ∞ of Y to only requiring it to v anish at inﬁnit y along the ﬁbres of  Y : Y → X . If  Y : Y → X is a pr op er map then RK ∗ G ,X ( Y ) = RK ∗ G ( X ) : = RK ∗ G ,X ( X ); these t wo g roups ha v e exactly t he same cycles. Example 1.3 . Any G -equiv arian t complex v ector b undle V on Y yields a cycle for RK 0 G ( Y ) b y c hoosing a G -inv ariant Her mitian metric on V and forming the corresp onding G ⋉ Y -equiv ariant Z / 2-g raded right Hilber t C 0 ( Y ) mo dule of sections, where the grading is th e trivial one. W e s et the oper ator equal to zero. Example 1.4 . Let X b e a G - space a nd let V b e a G -equiv arian tly K-o riented vector bundle ov er X of (real) dimension n . The G -equiv arian t vector bundle pro jection π V : V → X g ives V the structure of a space over X , so tha t V b ecomes a G ⋉ X - space. Then the Thom isomorphism provides an inv ertible T hom class t V ∈ RK dim V G ,X ( V ) : = K K G ⋉ X dim V  C 0 ( X ) , C 0 ( V )  . In the case G = Spin c ( R n ) and X = ⋆ and V : = R n with the represen tation Spin c ( R n ) → Spin( R n ) → O( n, R ) the class t R n is the ‘Bott’ class ﬁguring in equiv a riant Bott Perio dicit y . Certain further normaliza tions can be made in order to describe the groups RK ∗ G ,X ( Y ). A standar d o ne is to repla ce the Z / 2 -grading o n E by the standar d even grading, s o that E consists of the sum of tw o copies of the same Hilb ert mo dule. This means that F can b e ta ken to b e of the form  0 F ∗ 1 F 1 0  and the co nditions in volving F are replaced by o nes inv olving F 1 and F ∗ 1 ; we may as well r eplace F b y F 1 . With this co nv en tion, the F redholm conditions are that f ( F F ∗ − 1) and f ( F ∗ F − 1 are compact fo r all f ∈ C 0 ( X ). In other words, y 7→ F y takes essentially unitary v alues in B ( H y ) for all y ∈ Y and the co mpact-op erator -v a lued functions F F ∗ − 1 a nd F ∗ F − 1 v anish at inﬁnit y along the ﬁbres of  : Y → X . The equiv ariant stabilization theorem for Hilb ert mo dules implies that we may take H to hav e the s pec ial for m L 2 ( G ) ∞ ⊗ C 0 ( Z ) C 0 ( Y ), where L 2 ( G ) is the G - equiv a riant right Hilbert C 0 ( Z )- Hilb ert mo dule deﬁned using the Ha ar s ystem of G , a nd the sup erscript indica tes the sum of countably many copies of L 2 ( G ). The corresp onding ﬁeld of Hilbert space ha s v alue L 2 ( G y ) ∞ at y ∈ Y where G y denotes all G ∈ G ending in y , on which w e have a given measure sp eciﬁed by the Ha ar system of G . This leads to a descr iption of RK 0 G ,Y ( X ) as the gr oup o f ho motopy-classes of G -equiv a riant cont inu ous ma ps fr om Y to the s pace F G of F redholm o per ators on the Hilber t spaces L 2 ( G y ) ∞ , but top olog izing the spa ce F G is s omewhat delicate. Similarly , the r elative gr oups RK ∗ G ,X ( Y ) ar e maps to F redholm op erator s with com- pact v ertical supp or t with resp ect to the ma p Y → X , where the supp ort of a map to F redholm op erators is by deﬁnition the complement of the set where the ma p takes inv ertible v alues. R emark 1.5 . If G acts pro per ly and co -compactly on X , A is a trivial G -C* -algebra and B is a G ⋉ X -C*-alge bra, then there is a canonical isomorphism KK G ⋉ X ( C 0 ( X ) ⊗ A, B ) ∼ = KK( A, G ⋉ B ) . In pa rticular, the G -eq uiv ar iant repr esentable K-theor y of X ag rees with the K- theory of the corr esp onding cross-pro duct. Und er this iden tiﬁcation, cla sses in 8 HEA TH EM ERSON RK 0 G ( X ) which are represe n ted b y equiv a riant v ector bundles on X corresp ond to classes in K 0 ( G ⋉ C 0 ( X )) which are repr esented b y pro jections in the stabilisation of G ⋉ C 0 ( X ). See [1 2] for more infor mation. Thu s, even if the re ader is only int erested in g roups, or the trivial group, it is conv enien t t o in tro duce group oids to some exten t in order to describ e cohomology theories with diﬀeren t support conditions. 1.5. T ensor and forgetful functors. The following simple functor will play an impo rtant ro le. If P is a G ⋉ X -a lgebra, w e deno te b y T P the map RKK G ( X ; A, B ) : = K K G ⋉ X ( C 0 ( X ) ⊗ A, 1 X ⊗ B ) → K K G ( P ⊗ A, P ⊗ B ) which s ends a G ⋉ X - equiv ar iant rig h t Hilbert C 0 ( X ) ⊗ B - Hilbert mo dule E to E ⊗ X P , the tensor pro duct b eing in the c ate gory of G ⋉ X - algebr as (w e acco rdingly use a subscript for emphasis) and sends F ∈ B ( E ) to the op erator F ⊗ X id P . This deﬁnition makes sense since F co mm utes with the C 0 ( X )-structure on E . The functor T P is the comp osition of external product ␣ ⊗ X 1 P : KK G ⋉ X ( C 0 ( X ) ⊗ A, C 0 ( X ) ⊗ B ) → KK G ⋉ X ( A ⊗ X P, B ⊗ X P ) (where the X -structure on A ⊗ P et c . is on the P factor), and the forgetful map KK G ⋉ X ( A ⊗ X P, B X ⊗ P ) → K K G ( A ⊗ X P, B ⊗ X P ) which maps a G ⋉ X -alge bra or Hilb ert module to the underlying G -a lgebra, or Hilbert mo dule, thus forgetting th e X -structure. 1.6. Kasparo v duals . W e b egin our discussion of dualit y by by f ormalizing some dualit y calculations of Kaspa rov, c.f. [20, Theorem 4.9]. Explicit examples will be discussed later. F or con v enience of notation w e will often write 1 : = C 0 ( Z ). This notation expresses the fact t hat C 0 ( Z ) is the tensor unit in the tenso r ca tegory of G -C*- algebras . Similarly , if G acts on a space X then we sometimes denote b y 1 X the G -C*-alg ebra C 0 ( X ); thus 1 X is the tensor unit in the catego ry o f G ⋉ X C*- algebras , X b eing the bas e o f G ⋉ X . This notation is cons isten t with the so urce of this material (see [11].) Deﬁnition 1.6. Let n ∈ Z . An n -dimensional G -e quivari ant K asp ar ov dual for the G -space X is a triple ( P, D, Θ), wher e • P is a (pos sibly Z / 2-graded) G ⋉ X -C ∗ -algebra , • D ∈ KK G − n ( P, 1 ), and • Θ ∈ RKK G n ( X ; 1 , P ), subject to the following conditions: (1) Θ ⊗ P D = id 1 in RKK G 0 ( X ; 1 , 1 ); (2) Θ ⊗ f = Θ ⊗ P T P ( f ) in RKK G ∗ + n ( X ; A, B ⊗ P ) for all G -C ∗ -algebra s A and B and all f ∈ RKK G ∗ ( X ; A, B ); (3) T P (Θ) ⊗ P ⊗ P Φ P = ( − 1) n T P (Θ) in KK G n ( P, P ⊗ P ), where Φ P is the ﬂip automorphism on P ⊗ P . The follo wing theorem is pro v ed in [1 1]. Theorem 1.7. Le t n ∈ Z , let P b e a G ⋉ X - C ∗ -algebr a, D ∈ KK G − n ( P, 1 ) , and Θ ∈ RKK G n ( X ; 1 , P ) . Deﬁne two natur al tr ansformatio ns PD : KK G i − n ( P ⊗ A, B ) → RKK G i ( X ; A, B ) , f 7→ Θ ⊗ P f , PD ∗ : RKK G i ( X ; A, B ) → KK G i − n ( P ⊗ A, B ) , g 7→ ( − 1) in T P ( g ) ⊗ P D , DUALITY AND CORRESPONDENCES 9 These two ar e inverse t o e ach other if and only if ( P, D , Θ) is an n -dimensio nal G -e quiva riant K asp ar ov dual for X . 1.7. Abstract duals. The reader may hav e noticed that the o nly place the C 0 ( X )- structure o n P co mes into play in the co nditions listed in Deﬁnition 1.6, and in the statement of Theorem 1 .7, is via the functor T P . In pa rticular, if o ne has a Kasparov dua l ( P , D , Θ) and if one changes the C 0 ( X )-structure o n P , for example b y compo sing it with a G -equiv arian t ho meomorphism of X , then the map PD of Theo rem 1.7 do es not change; since by the theor em P D ∗ is its in verse map, it would not change either, strangely , s ince its deﬁnition uses T P . In fact it turns out that the functor T P can be rec onstructed from P D if one k nows that PD is an isomorphism. This is an imp or tant idea in connection with the Lefschet z map a nd suggests the follo wing useful deﬁnition. Deﬁnition 1 .8. An n - dimensional abstr act dual for X is a pair ( P, Θ), where P is a G -C ∗ -algebra and Θ ∈ RKK G n ( X ; 1 , P ), such tha t the map PD deﬁned as in Theorem 1.7 is an isomorphism for all G -C ∗ -algebra s A and B . This deﬁnition is s horter, and, as mentioned, is useful for t heoretical rea sons, but it seems lik e it should be diﬃcult to c heck in practise. In a ny case, it is clear from Theo rem 1 .7 that a pair ( P , Θ) is an abstr act dual if it is part of a K asparov dual ( P , D , Θ). Prop osition 1 . 9. A n abstr act dual for a sp ac e X is unique u p to a c anonic al KK G -e quiva lenc e if it exists, and even c o variantly functorial in the fol lowing sense. L et X and Y b e two G -sp ac es and let f : X → Y b e a G -e quivariant c ontinuous map. L et ( P X , Θ X ) and ( P Y , Θ Y ) b e abstr act duals for X and Y of d imensions n X and n Y , r esp e ctively. Then ther e is a u nique P f ∈ KK G n Y − n X ( P X , P Y ) with Θ X ⊗ P X P f = f ∗ (Θ Y ) . Given two c omp osable maps b etwe en thr e e sp ac es with duals, we have P f ◦ g = P f ◦ P g . If X = Y , f = id X , and ( P X , Θ X ) = ( P Y , Θ Y ) , then P f = id P X . If only X = Y , f = id X , then P f is a KK G -e quiva lenc e b etwe en the t wo duals of X . Although the map f : X → Y a ppe aring in Pro po sition 1.9 does not hav e t o b e prop er, it nonetheless yields a m orphism P f in KK G . 1.8. Dualit y co-algebra. Let ( P , Θ) b e an n -dimensional a bstract dual for a G -space X . By the Y o neda Lemma, another abstract dual ( P ′ , Θ ′ ) also for X and say of dimension n ′ is related to ( P, Θ) by an in v ertible element (1.10) ψ ∈ KK G n ′ − n ( P, P ′ ) , suc h that Θ ⊗ P ψ = Θ ′ . W e repeat for emphasis that since ( P , Θ) is only an abstr act dual, we are n ot assuming that there is a G ⋉ X -structure on P . Ho w ever, we are going to attempt to reco nstruct what we might consider to b e a G ⋉ X - structure on P at the level of KK -t he ory . Along the wa y we will keep track of how the change in dual from ( P, Θ ) to ( P ′ , Θ ′ ) aﬀects our constructions. Deﬁne D ∈ K K G − n ( P, 1 ) by the req uirement (1.11) PD( D ) : = Θ ⊗ P D = 1 1 in RKK G 0 ( X ; 1 , 1 ). as in the ﬁrs t condition in Deﬁnition 1.6). The class D should th us play the role of the class named D in a K asparov dual. It is routine to check that when we change the dual, as abov e, D is replaced by ψ − 1 ⊗ P D . W e call D c o unit of the duality b ecause it pla ys the algebraic role of a counit in the theory of adjoin t functors (see [1 1] and also Remark 3.3 below). Deﬁne ∇ ∈ KK G n ( P, P ⊗ P ) b y the requiremen t that PD( ∇ ) : = Θ ⊗ P ∇ = Θ ⊗ X Θ in RKK G 2 n ( X ; 1 , P ⊗ P ). 10 HEA TH EM ERSON W e call ∇ the c omultiplic a tion of the duality . When we change the dual, ∇ is replaced b y ( − 1) n ( n ′ − n ) ψ − 1 ⊗ P ∇ ⊗ P ⊗ P ( ψ ⊗ ψ ) ∈ KK G n ′ ( P ′ , P ′ ⊗ P ′ ) . R emark 1.12 . If n = 0 then the object P of KK G with counit D and comulti- plication ∇ is a co commut ative, counital c oalgebra ob ject in the tenso r catego ry KK G : ∇ ⊗ P ⊗ P ( ∇ ⊗ 1 P ) = ∇ ⊗ P ⊗ P (1 P ⊗ ∇ ) , (1.13) ∇ ⊗ P ⊗ P Φ P = ∇ , (1.14) ∇ ⊗ P ⊗ P ( D ⊗ 1 P ) = 1 P = ∇ ⊗ P ⊗ P (1 P ⊗ D ) . (1.15) Equation (1 .13) holds in KK G 2 n ( P, P ⊗ 3 ), equa tion (1.14) holds in K K G n ( P, P ⊗ P ), and (1.15) holds in KK G 0 ( P, P ). Now, for G -C ∗ -algebra s A and B , we deﬁne T ′ P : RKK G ∗ ( X ; A, B ) → KK G ∗ ( P ⊗ A, P ⊗ B ) , f 7→ ∇ ⊗ P PD − 1 ( f ) , where PD is the dualit y isomorphism, ∇ is the comu ltiplication of th e dualit y , and ⊗ P op erates on the se c o nd copy of P in the tar get P ⊗ P of ∇ . A computation yields that (1.16) PD  T ′ P ( f )  = Θ ⊗ X f in RKK G i + n ( X ; A, P ⊗ B ) for all f ∈ RK K G i ( X ; A, B ). It follows that T ′ P ( f ) = T P ( f ) if ( P, Θ) is pa rt of a Kasparov dual, and thus T P is in fact indep enden t of the G ⋉ X -str ucture on P , v erifying our guess abov e. When w e c hange the du al, we r eplace T ′ P b y the map (1.17) RKK G i ( X ; A, B ) ∋ f 7→ ( − 1) i ( n − n ′ ) ψ − 1 ⊗ P T P ( f ) ⊗ P ψ ∈ KK G i ( P ′ ⊗ A, P ′ ⊗ B ) . In fact, one can c hec k tha t the maps T ′ P ab ov e deﬁne a functor T ′ P : RKK G ( X ) → KK G . This is a KK G -functor in the sens e that it is compatible with the tensor pro ducts ⊗ , and it is left adjoint to the functor p ∗ X : KK G → RKK G induced from the group oid homomorphism G ⋉ X → G . It follows that we can write t he inv erse dua lit y map inv olved in an abstra ct dual ( P, Θ ) as: (1.18) PD − 1 ( f ) = ( − 1) in T ′ P ( f ) ⊗ P D in K K G i − n ( P ⊗ A, B ) for f ∈ RKK G i ( X ; A, B ). By the abov e discussion this fo rmula agr ees with the m ap PD ∗ when w e ha ve a Kaspar ov dual. 1.9. The Lefsc hetz map. The formal co mputations summarized in the previo us section allows us to single o ut an interesting inv ariant of a G -space X , at lea st under the h ypo thesis that X has some abstra ct dual. F or an y G -spa ce X the dia gonal embedding X → X × Z X is a prop er map and hence induces a ∗ -homomorphism 1 X ⊗ 1 X ∼ = C 0 ( X × Z X ) → C 0 ( X ) = 1 X . This map is G ⋉ X -equiv a riant and hence yields ∆ X ∈ RKK G ( X ; 1 X , 1 ) ∼ = KK G ⋉ X  C 0 ( X × Z X ) , C 0 ( X )  . DUALITY AND CORRESPONDENCES 11 W e call this the d iagonal r estrictio n class . It yields a canonica l ma p (1.19) ␣ ⊗ 1 X ∆ X : KK G ( 1 X ⊗ A, 1 X ⊗ B ) → RK K G ( X ; 1 X ⊗ A, B ) . In particular, this con tains a map K K G ( 1 X , 1 X ) → RKK G ( X ; 1 X , 1 ). Example 1.20 . If f : X → X is a prop er, contin uous, G -equiv ariant map, then [ f ] ⊗ 1 X ∆ X ∈ RK K G ( X ; 1 X , 1 ) is the class of the ∗ -homomorphism induced b y (id X , f ) : X → X × Z X . Now dro p the assumption that f b e proper . Then (id X , f ) is still a prope r, contin uous, G -equiv a riant map. The class of the ∗ -homomorphism it induces is equal to f ∗ (∆ X ), where w e use the maps f ∗ : RKK G ∗ ( X ; A, B ) → RKK G ∗ ( X ; A, B ) for A = 1 X , B = 1 induced b y f : X → X (the functor X 7→ RKK G ( X ; A, B ) is functorial with resp ect to arbitrar y G -maps, not just prop er ones .) This s uggests that we can think o f RKK G ( X ; 1 X , 1 ) a s g eneralized, p ossibly non-pr op er s elf-maps of X . In fact if the a nc hor map X → Z is a prop er map, so that X is a bundle of compact spa ces ov er Z , then ␣ ⊗ 1 X ∆ X is an iso mophism (an easy exercis e in the deﬁnitions.) Now let T ′ P be the tensor functor a nd ∆ X the diago nal restriction clas s of an abstract dual. W e deﬁne the multiplic ation class of P by (1.21) [ m ] : = T ′ P (∆ X ) ∈ KK G 0 ( P ⊗ 1 X , P ) . A c hange of dual as in (1.10) r eplaces [ m ] b y ψ − 1 ⊗ P [ m ] ⊗ P ψ . Lemma 1.22. L et ( P, D , Θ ) b e a K asp ar o v dual. Then [ m ] is the class in KK G of the mu ltiplic ation homomorphism C 0 ( X ) ⊗ Z P → P that describ es t he X -structu r e on P ( up to c ommuting the t ensor factors ) . W e now have enough theoretical developmen t to deﬁne the Lefschetz map and sketc h the pro o f of it s homotopy in v ariance. Let X be a G -space a nd ( P , Θ) an n -dimensional abstrac t dual for X , PD and PD − 1 the duality isomo rphisms. As b efore, w e wr ite 1 : = C 0 ( Z ), 1 X : = C 0 ( X ) and ∆ X ∈ RK K G ( X ; 1 X , 1 ) = KK G ⋉ X ( 1 X ⊗ 1 X , 1 X ) the diag onal restriction clas s and ¯ Θ : = for get X (Θ) ∈ KK G n ( 1 X , P ⊗ 1 X ) . Deﬁnition 1. 23. The equiv a riant L efsch etz map Lef : RKK G ∗ ( X ; 1 X , 1 ) → KK G ∗ ( 1 X , 1 ) for a G -spa ce X is deﬁned as the compo site map RKK G i ( X ; 1 X , 1 ) PD − 1 − − − − → KK G i − n ( P ⊗ 1 X , 1 ) ¯ Θ ⊗ P ⊗ 1 X ␣ − − − − − − − → KK G i ( 1 X , 1 ) . The equiv a riant Euler char acteristic of X is Eul X : = Lef (∆ X ) ∈ KK G 0 ( 1 X , 1 ) = KK G 0  C 0 ( X ) , C 0 ( Z )  . Let f ∈ RK K G i ( X ; 1 X , 1 ). Equations (1.18) a nd (1.21) yield Lef ( f ) = ( − 1) in ¯ Θ ⊗ P ⊗ 1 X T ′ P ( f ) ⊗ P D , (1.24) Eul X = ( − 1 ) in ¯ Θ ⊗ P ⊗ 1 X [ m ] ⊗ P D . (1.25) W e have alrea dy established that if ( P, Θ ) is part of a K asparov dual, then T ′ P = T P and [ m ] is the KK-class o f the mu ltiplication ∗ -homomorphism C 0 ( X, P ) → P , 12 HEA TH EM ERSON so that (1.24) yields explicit form ulas f or Lef ( f ) and Eul X . This is extremely impo rtant b ecause otherwise it would no t be p oss ible to compute these inv ar iants. Let X and X ′ be G -spaces, and let f : X → X ′ be a G -homotopy eq uiv alence. Then f induces an equiv alence of ca tegories RKK G ( X ′ ) ∼ = RKK G ( X ), that is, we get in v ertible maps f ∗ : RKK G ∗ ( X ′ ; A, B ) → RKK G ∗ ( X ; A, B ) for all G -C ∗ -algebra s A and B . Now assume, in addition, that f is prop er; we do not need the in verse map o r the homotopies to b e prop er. Then f induces a ∗ -homomorphism f ! : C 0 ( X ′ ) → C 0 ( X ), whic h yields [ f ! ] ∈ KK G  C 0 ( X ′ ) , C 0 ( X )  . W e write [ f ! ] instead of [ f ∗ ] to better distinguish this from the map f ∗ ab ov e. Unless f is a pr o p er G -homotopy equiv alence, [ f ! ] need not be inv ertible. Theorem 1.26. L et X and X ′ b e G -sp a c es with abstr act duals, and let f : X → X ′ b e b o th a pr op er map and a G -homotopy e quivalenc e. Then [ f ! ] ⊗ C 0 ( X ) Eul X = Eul X ′ in KK G 0 ( C 0 ( X ′ ) , 1 ) and the L efschetz maps for X and X ′ ar e r elate d by a c ommuting diagr am RKK G ∗ ( X ; C 0 ( X ) , 1 ) Lef X   RKK G ∗ ( X ′ ; C 0 ( X ) , 1 ) f ∗ ∼ = o o [ f ! ] ∗ / / RKK G ∗ ( X ′ ; C 0 ( X ′ ) , 1 ) Lef X ′   KK G ∗ ( C 0 ( X ) , 1 ) [ f ! ] ∗ / / KK G ∗ ( C 0 ( X ′ ) , 1 ) , wher e [ f ! ] ∗ denotes c omp osition with [ f ! ] . In p articular, Eul X and the map Lef X do not dep end on the chosen dual. The proo f relies on the discus sion preceding the theorem. Theorem 1.26 implies that the Lefschetz maps for prop erly G -homotopy equiv a- lent space s ar e equiv alen t beca use then [ f ! ] is inv ertible, so that all horizontal maps in the diagr am in Theorem 1 .26 ar e inv ertible. In this sense, the Lefschetz map and the Euler class are in v ariants of the prop er G -homotopy type o f X . The construction in Example 1.20 ass o ciates a class [∆ f ] ∈ RK K G 0 ( X ; C 0 ( X ) , 1 ) to any con tin uous, G -equiv ariant map f : X → X ; it do es not ma tter whether f is prop er. W e abbrev iate Lef ( f ) : = Lef ([∆ f ]) and call this the Lefschetz inv ariant of f . Of co urse, equiv ariantly homotopic self- maps induce the sa me class in RK K G ( X ; C 0 ( X ) , 1 ) and therefore hav e the same Lefschet z in v ar iant . W e hav e Lef (id X ) = Eul X . More generally , s pec ializing (1.19) gives a ma p ␣ ⊗ 1 X ∆ X : KK G ∗  C 0 ( X ) , C 0 ( X )  → RKK G ∗ ( X ; C 0 ( X ) , 1 ) , which we comp ose with the L efsch etz ma p; abusing no tation, we still denote this comp osition b y Lef : KK G ∗  C 0 ( X ) , C 0 ( X )  → KK G ∗ ( C 0 ( X ) , 1 ) Finally , we recor d that Lefschetz inv ariants for elements of RKK G ∗ ( X ; C 0 ( X ) , 1 ) can b e arbitra rily complicated: the Lefschetz map is ra ther easily s een to be split surjective. The s plitting is giv en b y s pec ializing the inﬂation map (1.27) p ∗ X : KK G ∗ ( A, B ) → KK G ⋉ X ( 1 X ⊗ A, 1 X ⊗ B ) to A : = 1 X and B : = 1 . The fundamental example of a Kasparov dual is provided by the vertical tangent space to a bundle of s mo oth manifolds over the base Z of a g roup oid, in whic h morphisms act smoo thly . W e come back to this in §4. DUALITY AND CORRESPONDENCES 13 2. Examples of comput a tions of the Lefschetz map In thi s section w e will give some examples of computations of the Lef schetz map for v arious instances of spaces with duals and for equiv ariant self-morphisms coming fro m ac tual maps . The pr oblem of computing the Lefschetz inv ariants o f more g eneral Kas parov self-morphisms for the next sec tion is a central problem for us, a nd will b e treated in §4 once we have av a ilable the theory of equiv a riant corresp ondences . Most of the exa mples are quite clos e to prop er a ctions, but they do not quite hav e to b e prop er . The p oint is that an a bstract (or Kasparov) dual for a G -space X also yields one for X regar ded as a G ′ -space where G ′ is a (not necessa rily closed) subgroup oid o f G . Even if X is pro p er as a G -spa ce, it need not b e prop er a s a G ′ -space since G ′ may not b e closed. Known exa mples of duals are of tw o types; they a re bo th K asparov duals. If G is a lo ca lly compac t gro up acting as simplicial automorphisms o f a ﬁnite-dimensional simplicial complex then X has a Kas parov dual by [13]. This does not quit e imply that the action of G is pr op er, since the action o f G co uld b e trivial. Neither do es it quite imply that G m ust b e disc rete, though since the connected comp onent of the iden tit y of G must act t rivially , the only in teresting exa mples m ust in v olv e dis - connected groups. F or instance, the non-disc rete group SL(2 , Q p ) acts simplicially on a tree. If X is a complete complete Riemannian manifold then X is a pr op er G -spa ce where G is the Lie g roup of isometries of X , and eith er the Cliﬀord algebra of X or the C*-alg ebra o f C 0 -functions o n the tangent bundle T X of X is pa rt of a Kasparov dual for X (see e.g. [13] or [11]), and also §4. Hence if G is any g roup of isometries of a Riemannian manifold X with ﬁnitely many co mpo nen ts, then the G -s pace X also has a Kasparov dual. If the tangent bundle T X admits an equiv a riant K-orientation, then C 0 (T X ) can b e r eplaced by C 0 ( X ). F or instance , the cir cle wit h t he group Z acting by an ir rational rotation has a Kaspa rov dual of of dimension 1, given by ( C ( T ) , D, Θ ) w here D is the class of the Dira c op er ator on the circle. The L efsch etz inv ariants of equiv ariant self-maps of X will in bo th these situa- tions turn out to b e in s ome sense zer o -dimensional in th e sense t hat they ar e built out of p oint-ev alua tion class es. As we will s ee later, the Lefschetz inv ariants of more gener al Kasparov self-mo rphisms are more complicated, higher-dimensio nal ob jects. 2.1. The com binatorial Lefsc hetz map. Let X be a ﬁnite-dimensional simpli- cial co mplex and let G b e a lo cally compact group a cting smo o thly a nd simplicially on X (that is, stabilisers o f p oints are op en). W e fo llow [13] a nd [15]. Assume that X admits a co louring (that is, X is typed) a nd that G preserves the colo ur- ing. This ensures that if g ∈ G maps a simplex to itself, then it ﬁxes that simplex po in twise. Let S X b e the set o f (non- degenerate) simplices of X and let S d X ⊆ S X b e the subset of d -dimensional s implices. The g roup G acts on the discrete set S X preserving the deco mpo sition S X = F S d X . Decompo se S X in to G -o rbits. F or each orbit ˙ σ ⊆ S X , choose a representativ e σ ∈ S X and let ξ σ ∈ X be its barycentre and Stab( σ ) ⊆ G its stabiliser. Restriction to the orbit ˙ σ deﬁnes a G -equiv ariant ∗ -homomorphism (2.1) ξ ˙ σ : C 0 ( X ) → C 0  G/ Stab( σ )  → K  ℓ 2 ( G/ Stab σ )  , where the second map is the r epresentation b y p oint wise m ultiplication o per ators. W e let [ ξ ˙ σ ] be its cla ss in KK G 0 ( C 0 ( X ) , 1 ). 14 HEA TH EM ERSON Let ϕ : X → X b e a G -equiv arian t self-map of X . Since ϕ is G -equiv a riantly ho- motopic to a G -equiv arian t cellular map, w e may assume without lo ss of generalit y that ϕ is itself cellular. Hence it induces a G -equiv aria nt chain map ϕ : C • ( X ) → C • ( X ) , where C • ( X ) is the chain complex o f oriente d s implices of X . A basis for C • ( X ) is given by the set of (un-)or ien ted simplices, by ar bitrarily choo sing an o rientation o n each s implex. W e ma y describ e the chain map ϕ by its matrix co eﬃcients ϕ στ ∈ Z with resp ect to this basis; thus the subscripts a re unoriented simplices. F or exa mple, if ϕ maps a simplex to itself, a nd reverses orien tation, then ϕ σ,σ = − 1. Since ϕ is G -equiv a riant, ϕ g ( σ ) ,g ( σ ) = ϕ σσ . So the following ma kes sens e. Notation 2.2. F or ˙ σ ∈ G \ S d X , let n ( ϕ, ˙ σ ) : = ( − 1 ) d ϕ σσ ∈ Z for any choice of representativ e σ ∈ ˙ σ . The following theorem is prov ed in [15] using the simplicial dual developed in [13], and inspired b y ideas of K asparov and Ska ndalis in [21] . Theorem 2.3. L et X b e a ﬁnite-dimensio nal c olour e d simplicial c omplex and let G b e a lo c al ly c omp act gr oup that a cts smo othl y and simplic ial ly on X , pr eserving the c o louring. L e t ϕ : X → X b e a G -e quivariant self-map. Deﬁne n ( ϕ, ˙ σ ) ∈ Z and [ ξ ˙ σ ] ∈ KK G ( C 0 ( X ) , 1 ) for ˙ σ ∈ G \ S X as ab ove. The n Lef ( ϕ ) = X ˙ σ ∈ G \ S X n ( ϕ, ˙ σ )[ ξ ˙ σ ] . In the non-equiv ariant situation, if X is connected and has o nly a ﬁnite n um ber of simplices, th en the form ula j ust giv en reduces to the ordinary Lefsc hetz num ber of G b y the (standard) argumen t that p rov es that the Euler characteristic of a ﬁnite simplicial complex can b e co mputed either by count ing ranks of simplicial homology g roups, o r by dire ctly counting the num ber of s implices in the complex. In the noncompa ct case this do esn’t make sense anymore since the num ber o f orbits of simplices may b e inﬁnite, but our deﬁnition of the Lefschetz inv aria nt still makes sense. This illustrates the adv an tage of considering K-homology clas ses instead o f n umbers as ﬁxed-po in t data. 2.2. The sm o oth Lefsc hetz map. Let X b e a s mo oth manifold a nd G a g roup acting by isometric diﬀeomor phisms of X . Then we ca n build a Kasparov dual for X using the Cliﬀord a lgebra. The real (resp. complex) Cliﬀord algebra of a Euclidean vector space V is g enerated a s a real (resp. complex) unital *-algebra b y a n or thonormal basis { e i } for V with the relations that e i is self-a djoint and e i e j + e j e i = 2 δ ij . In the complex case this pr o duces a ﬁnite-dimensional Z / 2-gr aded C*-algebr a. More g enerally if V is a E uclidean vector bundle, this constr uction applies and pro duces a lo cally trivia l bundle of ﬁnite-dimensional Z / 2 -graded C* - algebras ov er X . If V : = T X for a Riemannian manifold X then the Cliﬀor d a lgebr a of X is the correspo nding C*- algebra of sections v anishing a t inﬁnit y . It is deno ted C τ ( X ). This C*-algebr a ca rries a canonical action o f the group of isometries of X and hence lik ewise for an y subgroup. W e discuss the sp eciﬁc mec hanics o f the Cliﬀord K asparov dual to the following extent . Let d b e the de Rham diﬀeren tial on X , acting on L 2 -forms o n X . This Hilbert spa ce carr ies a unitar y ac tion of G and b oth d a nd its adjoint d ∗ are G - equiv a riant, and d + d ∗ is a n elliptic o per ator called the Euler (or de Rha m ) op era tor on X . If ω is a d iﬀerential form on X v anishing at inﬁnit y t hen the operato r λ ω of ex terior pro duct with ω deﬁnes an op erator o n L 2 -forms which is b ounded and the assignment ω 7→ λ ω + λ ∗ ω determines a representation of C τ ( X ) whic h DUALITY AND CORRESPONDENCES 15 graded commutes mo dulo bounded o p erators with d + d ∗ . Hence we get a cycle for KK G ( C τ ( X ) , C ). It repr esents the class D a ppea ring in the Kasparov dual. W e sta rt by recording one o f the easiest co mputations of the Lefschetz map. The in terested reader can ea sily prov e it for his or herself using after lo oking brieﬂy at the deﬁnition of the clas s Θ (see [13] ) and rev iewing the deﬁnition o f the Lefschetz map. Recall that the Euler class of X is the Lefschetz inv aria n t of the identit y map of X . Prop osition 2 . 4. The Euler class of X is the class of the Euler op er ator on X . The functoriality result 1 .26 combines with Pr op osition 2.4 to imply the homoto p y- in v a riance of the class o f the Euler oper ator: f ∗ (Eul X ) = Eul X ′ for a proper G -equiv ariant homotopy-equiv alence f : X → X ′ betw een smo oth and prop er G -manifolds. W e now describ e the Lefschetz inv ariant o f a mo re genera l smo oth G -eq uiv ar iant self-map of X . This require s a pr eliminary discussion. Let Y be a lo cally compact space and G be a locally compact group acting contin uously on Y , and let π : E → Y b e a G -eq uiv ar iant Euclidean R -vector bundle ov er E . L et A : E → E be a G -equiv ariant vector bundle automo rphism, that is, a contin uous map E → E ov er Y that r estricts to R -v ector spa ce iso morphisms on the ﬁbres of E . W e are going to deﬁne a G -equiv ariant Z / 2 -graded r eal line bundle sign( A ) o v er Y . (Since w e w ork w ith complex K-theory w e will only use its complexiﬁcation.) If Y is a p oint, then G -e quiv ar iant real v ector bundles o v er Y co rresp ond to real o rthogonal r epresentations of G . The endomor phism A b ecomes in this case an inv ertible linea r map A : R n → R n commut ing with G . The sign is a vir tual 1-dimensional r epresentation o f G a nd hence cor resp onds to a pair ( χ, n ), where n ∈ { 0 , 1 } is the parity ( we are referring to t he gr ading, either ev en or odd) of the line bundle a nd χ : G → {− 1 , +1 } is a real-v alued character. The overall pa rity will turn out to be 0 if A preserves orien tation a nd 1 if A reverses o rientation (see Example 2.6). In this sense, our inv ariant will reﬁne the orientation of A . As ab ove, let Cliﬀ( E ) b e the bundle of r e al Cliﬀord algebras asso ciated to E . W e can also deﬁne in a n analogo us w ay Cliﬀ( E ) if E carries an indeﬁnite bilinear form and it is a well-known fact from a lgebra that if the index of the bil inear form on E is divisible by 8, t hen t he ﬁb res of Cliﬀ( E ) are isomorphic to matrix a lgebras. In this ca se, a G -e quiva riant spinor bund le for E is a Z / 2- graded rea l vector bun- dle S E together with a g rading pr eserving, G -equiv ariant ∗ -algebra isomorphism c : Cliﬀ( E ) → End( S E ). This repr esentation is determined uniquely by its r estric- tion to E ⊆ Cliﬀ( E ), which is a G -equiv ariant map c : E → End( S E ) such that c ( x ) is odd and sy mmetric and satisﬁes c ( x ) 2 = k x k 2 for all x ∈ E . The spinor bundle is unique up to tensor ing with a G -equiv a riant r eal line bun- dle L : if c t : E → S t for t = 1 , 2 are tw o G -equiv ariant s pinor bundles for E , then we deﬁne a G -equiv arian t real line bundle L ov er Y by L : = Hom Cliﬀ( E ) ( S 1 , S 2 ) , and the ev aluation iso morphism S 1 ⊗ L ∼ = − → S 2 in tertwines the representations c 1 and c 2 of Cliﬀ( E ). Deﬁnition 2.5. Let A : E → E b e a r eal G -equiv ariant vector bundle automor- phism a nd let A = T ◦ ( A ∗ A ) 1 / 2 be its p olar decompo sition with an orthogo nal vector bundle automor phism T : E → E . Let F be another G -equiv ar iant vector bundle o ver Y with a non-degener ate bilinear form, such that the sig nature of E ⊕ F is divisible by 8, so that Cliﬀ( E ⊕ F ) 16 HEA TH EM ERSON is a bundle of matrix alg ebras o ver R . W e assume that E ⊕ F has a G -equiv ariant spinor bundle, that is, there exists a G -equiv arian t linea r map c : E ⊕ F → E nd( S ) that induces an isomorphism of graded ∗ -algebra s Cliﬀ( E ⊕ F ) ∼ = End( S ) . Then c ′ : E ⊕ F → End( S ) , ( ξ , η ) 7→ c ( T ( ξ ) , η ) yields another G -equiv aria nt s pinor bundle for E ⊕ F . W e let sign( A ) : = Hom Cliﬀ( E ⊕ F )  ( S, c ′ ) , ( S, c )  . This is a G -equiv ariant Z / 2-graded r eal line bundle o ver Y . It is not hard to c heck that sign( A ) is well-deﬁned and a homotopy inv ariant. F ur- thermore, sign( A 1 ◦ A 2 ) ∼ = sign( A 1 ) ⊗ sign( A 2 ) for t w o equiv aria nt a utomorphisms A 1 , A 2 : E ⇒ E of the same bundle, a nd sig n( A 1 ⊕ A 2 ) ∼ = sign( A 1 ) ⊗ sign( A 2 ) for t wo equiv ariant vector bundle automorphisms A 1 : E 1 → E 1 and A 2 : E 2 → E 2 . If Y is a po int a nd G is trivial, then sign( A ) = R for or ien tation-preserv ing A and sign( A ) = R op for orientation-rev ersing A , as claimed ab ov e. Example 2.6 . Consider G = Z / 2. Let τ : G → { 1 } b e the trivial character and let χ : G → { +1 , − 1 } be the non-trivia l ch aracter . Let R χ denote the real r epresenta- tion o f G on R with character χ . This ca n b e consider ed trivially gra ded; let R op χ denote the same representation but with the opp osite g rading (the who le vector space is considered odd.) Consider A : R χ → R χ , t 7→ − t , s o A commutes with G . Then sign( A ) ∼ = R op χ carries a non-trivial represent ation. T o see this, let F be R χ with neg ative deﬁnite metric. Th us the Cliﬀor d alge bra of R χ ⊕ R χ is Cliﬀ 1 , 1 ∼ = M 2 × 2 ( R ). Explicitly , the map c ( x, y ) =  0 x − y x + y 0  induces the isomor phism. W e equip R 2 with the repr esentation τ ⊕ χ , so that c is equiv a riant. T wisting b y A y ields another representation c ′ ( x, y ) : = c ( − x, y ) = S c ( x, y ) S − 1 with S = S − 1 =  0 1 − 1 0  . Since S reverses the grading and exc hanges the representations τ and χ , it induces an isomorphism ( R τ ⊕ R op χ ) ⊗ R op χ ∼ = − → R τ ⊕ R op χ . Hence s ign( A ) = R op χ . Now let X be a smo o th Riemannian manifold and ass ume that G acts o n X isometrically and cont inuously . Let ϕ : X → X be a G -eq uiv ar iant s elf-map of X . In o rder to write down a n explicit lo cal formu la for Lef ( ϕ ), we imp ose the following restrictions o n ϕ : • ϕ is smooth; • the ﬁxed point subset Fix( ϕ ) o f ϕ is a submanif old of X ; • if ( p, ξ ) ∈ T X is ﬁxed by the der iv ative Dϕ , then ξ is tangent t o Fix( ϕ ). The la st tw o conditions are automatic if ϕ is iso metric with resp ect to some Rie- mannian metric (not necessarily the given one) and this o f course a pplies by av er- aging the giv en metric if ϕ has ﬁ nite order. In the simplest case, ϕ and id X are transverse, that is, id − D ϕ is inv ertible at each ﬁxed p oint of ϕ ; t his implies that ϕ has isolated ﬁ xed p oints. T o describ e the Lefschetz inv aria nt , we abbreviate Y : = Fix( ϕ ). This is a closed submanifold o f X by assumption. Let ν b e the nor mal bundle of Y i n X . Since DUALITY AND CORRESPONDENCES 17 the tangent spa ce of Y is left ﬁxed by the deriv ativ e D ϕ , it induces a linear map D ν ϕ : ν → ν . By assumption, the map id ν − D ν ϕ : ν → ν is in v ertible. Theorem 2. 7. L et X b e a c omplete smo oth Riemannian manifold, let G b e a lo c al ly c o mp a ct gr oup that acts on X smo othly and by isometries, and let ϕ : X → X b e a s elf-map satisfying the thr e e c onditions enumer ate d ab ove. L et ν b e the normal bund le of Y in X and let D ν ϕ : ν → ν b e induc e d by the derivative of ϕ as ab ove . L et r Y : C 0 ( X ) → C 0 ( Y ) b e the r estrictio n map and let Eul Y ∈ KK G 0 ( C 0 ( Y ) , 1 ) b e the e quivariant Euler char acteristic of Y . Then Lef ( ϕ ) = r Y ⊗ C 0 ( Y ) sign(id ν − D ν ϕ ) ⊗ C 0 ( Y ) Eul Y . F urthermor e, Eul Y is the e quivariant K -homolo gy class of the de Rham op er ator on Y . In brief, the Lefschetz inv ariant of G is the E uler characteristic of the ﬁxed- po in t set, t wisted by an appropriate e quiv ar iant line b undle dep ending on orien tation data. If ϕ and id X are transverse then the ﬁxed p oint subset Y is discrete. A dis- crete s et is a manifold and its Euler characteristic – a degenera te case o f Pro po - sition 2.4 – is represented b y the Kasparov cycle in whic h the Hilbert space is L 2 (Λ ∗ C ( T ∗ Y )) : = ℓ 2 ( Y ) equipp ed with the representation C 0 ( Y ) → K ( ℓ 2 Y ) by po in twise mult iplication oper ators, and the zero opera tor. The group permutes the points of Y and so acts by unitaries on ℓ 2 ( Y ). The normal bundle ν to Y in X in th is case is the restriction of the v ector bundle T X to the subset Y . F or p ∈ Y , let n p be +1 if id T p X − D p ϕ preserves orientation, and − 1 other wise. The gr aded equiv arian t line bundle sign(id ν − D ν ϕ ) in Theo rem 2.7 is determined by pairs ( n p , χ p ) for p ∈ Y , wher e n p is the parity of the r epresentation at p and χ p is a cer tain real-v alued c haracter χ p : Stab( p ) → {− 1 , + 1 } that dep ends on id T p X − D p ϕ a nd the represen tation of the s tabiliser Stab( p ) ⊆ G on T p X . Equiv ariance implies that n p is constan t along G -o rbits, whereas χ p behaves lik e χ g · p = χ p ◦ A d( g − 1 ). Let ℓ 2 χ ( Gp ) b e the representation of the cross -pro duct G ⋉ C 0  G/ Stab( p )  obtained by inducing the representation χ p from Stab( p ), a nd let C 0 ( X ) act on ℓ 2 χ ( Gp ) by restriction to G/ Stab( p ). This deﬁnes a G -equiv ar iant ∗ -homomorphism ξ Gp,χ : C 0 ( X ) → K ( ℓ 2 χ G ) . Theorem 2.7 asserts the follo wing: Corollary 2 .8. If the gr aph of ϕ is t r ansverse to the diagonal in X × X then, Lef ( ϕ ) = X Gp ∈ G \ Fix( ϕ ) n p [ ξ Gp,χ ] wher e [ ξ Gp,χ ] ∈ KK G 0 ( C 0 ( X ) , 1 ) and the multiplicities n p ar e explaine d ab ove. F urthermor e, the char acter χ : Stab G ( p ) → {− 1 , +1 } at a ﬁxe d p oint p has the explicit formula χ ( g ) = sign det  id − D p ϕ  · sign det  id − D p ϕ Fix( g )  . If, in addition, G is trivia l and X is connected, then ξ Gp,χ = ev p for all p ∈ Y ; moreov er, all p oint ev a luations hav e the same K- homology cla ss be cause they ar e homotopic. Hence we get the clas sical Lefschetz da ta multiplied by the K-homolog y class of a point Lef ( ϕ ) =  X p ∈ Fix( ϕ ) sign(id T p X − D ϕ p )  · [ev] as asserted above. This sum is ﬁnite if X is compact. 18 HEA TH EM ERSON W e include the following for the b eneﬁt of the rea der; it enables her or she to verify o ur computations by direct inspection. Lemma 2.9. L et H ⊆ G b e c o mp a ct and op en, let p, q ∈ X H b el ong to the same p ath c omp onent of the ﬁxe d p oi nt subsp ac e X H , and let χ ∈ Rep( H ) . Then  ξ Gp, Ind Stab( Gp ) H ( χ )  =  ξ Gq, Ind Stab( Gq ) H ( χ )  in K K 0 ( C 0 ( X ) , 1 ) . R emark 2.10 . If the iden tit y map id : X → X can be equiv ariantly p erturb ed to be in gener al po sition in the sens e explained ab ove, then combining Pr op osition 2.4 and Cor ollary 2.8 pr ov es that the class of the de Rham operator in KK G ( C 0 ( X ) , C ) is a sum o f p oint-ev aluation classes. Lück and Rosenberg show in [24] that this can alwa ys b e a c hieved when G is discrete and acts proper ly on X . The next t w o explicit examples in v olv e isolated ﬁxed p oints. Example 2.11 . Let G ∼ = Z ⋉ Z / 2 Z b e the inﬁnite dihedra l group, identiﬁed with the g roup of aﬃne tra nsformations of R generated by u ( x ) = − x and w ( x ) = x + 1. Then G has exactly t w o conjugacy classes of ﬁnite subgroups, each isomorphic to Z / 2. Its ac tion o n R is pr op er, and the closed interv al [0 , 1 / 2 ] is a fundamental domain. Ther e are tw o orbits of ﬁxed point in R , those of 0 and 1 / 2 , and their stabilisers represent the tw o conjugacy classes of ﬁnite subgr oups. Now we use some notation from E xample 2.6. E ach copy of Z / 2 acting on the tangent space at the ﬁxed po in t a cts by multip lication by − 1 on tangent vectors. Therefore, the computations in Example 2 .6 show that for any no nzero real num- ber A , viewed as a linea r transformation of the tangent space th at comm utes with Z / 2, w e ha ve sign( A ) = ( R op χ if A < 0, and R τ if A > 0. Let ϕ be a s mall G -equiv ariant per turbation o f the identit y map R → R with the following pr op erties. Fi rst, ϕ maps the int erv al [0 , 1 / 2 ] to itself. Secondly , its ﬁxed po in ts in [0 , 1 / 2 ] are the end p oints 0, 1 / 2 , and 1 / 4 ; thirdly , its deriv ativ e is bigge r than 1 a t b oth endp oints and b etw een 0 and 1 at 1 / 4 . Suc h a map clearly ex ists. F urthermore, it is homotopic to the identit y map, so that Le f ( ϕ ) = Eul R . By construction, there are three ﬁxed p oints mo dulo G , namely , the orbits of 0, 1 / 4 and 1 / 2 . The isotropy gr oups of the ﬁrst and third or bit ar e non-co njugate subgroups isomorphic to Z / 2; from Example 2.6, each of them con tributes R op χ . The po int 1 / 4 co nt ributes the trivial c haracter of t he trivial subgroup. Hence Lef ( ϕ ) = − [ ξ Z ,χ ] − [ ξ Z + 1 / 2 ,χ ] + [ ξ Z + 1 / 4 ] . On the other hand, suppo se we change ϕ to ﬁx the same p oints but to ha v e zero deriv a tive at 0 and 1 / 2 and large deriv ativ e at 1 / 4 . This is ob viously possible. Then we get co n tributions of R τ at 0 and 1 / 2 a nd a con tribution of − [ ξ 1 / 4 ] at 1 / 4 . Hence Lef ( ϕ ) = [ ξ Z ,τ ] + [ ξ Z + 1 / 2 ,τ ] − [ ξ Z + 1 / 4 ] . Combinin g bo th form ulas yields the iden tit y (2.12) [ ξ Z ,τ ] + [ ξ Z + 1 / 2 ,τ ] − [ ξ Z + 1 / 4 ] = − [ ξ Z ,χ ] − [ ξ Z + 1 / 2 ,χ ] + [ ξ Z + 1 / 4 ] . By the w a y , the left-hand side is the description of Eul R we get f rom the com- binatorial dual with the ob vious G -in v ariant triangulation of R with v ertex set Z + 1 / 2 Z ⊂ R . Using Lemma 2.9 one can chec k (2.12) by direct co mputation. DUALITY AND CORRESPONDENCES 19 3. Geometric KK-theor y In the previous chapter w e explained our computation of the Lefschetz map for self- maps of a G -space X in several rela tiv ely simple s ituations. In these cases , G was in each case a gro up (a gro upo id with trivial base). Obviously , not all equiv ar iant Kasparov self-mo rphisms KK G ∗  C 0 ( X ) , C 0 ( X )  are r epresented by maps. W e hav e organized this surv ey around the problem of computing Lefsc hetz inv ariants of more genera l equiv a riant Kas parov self-morphisms. This requires describing the morphisms themselves in some geo metric wa y . The theory of c orr esp ondenc es of Baum, Connes and Skandalis (see [3] [8]) w ould seem ideal f or this purp ose. Since we are w orking in the equiv ariant setting, to use it w ould necessitate c hec king that the pseudodiﬀerential calculus which pla ys suc h a pr ominen t r ole in [8] works equiv a riantly with resp ect to a an a ction o f a group (or gro upoid), as well as proving the ma in functoriality r esult for K -oriented maps. Although it seems plausible that such an extension could be carried out, a major problem arises in connnection with c ompo sing corresp ondences using tr ansversalit y in the equiv a riant situation (w e explain this below.) A trick of Ba um and Blo ck (see §3.8 a nd [3]) is useful in this co nnection, but in or der for it to work, some hypo theses on vector bundles are necessary . Our approach is to build in the vector bundle requir emen ts in to the deﬁnitions. This is o nly r easonable if G is pro per ; we now show how to reduce to this case using the Baum-Connes conjecture. 3.1. Using B aum- Connes to reduce to prop er group o ids. Let G b e a lo cally compact group (or group oid). The classifying sp ac e E G for pr op er actions of G is the prop er G -space with the universal prop erty that if X is any pr op er G -space, then there is a G -equiv ariant classifying map χ : X → E G which is unique up to G -homotopy . If G is a pr op er groupo id t o begin with, then E G = Z gives a simple mo del for E G , for it is pr op er a s a G -space and has the r equired universal prop erty . In particular, if G is a compact g roup, then E G is a po int . F or G a nd G -spaces X and Y , the inﬂation map (1.27) (3.1) p ∗ E G : KK G ∗ ( C 0 ( X ) , C 0 ( Y )) → RKK G ∗ ( E G ; C 0 ( X ) , C 0 ( Y )) : = K K G ⋉ E G ∗ ( C 0 ( X × E G ) , C 0 ( Y × E G )) is an isomorphism as so o n as the G action on X is to po logically amenable, and in particular as soo n as it is proper . An a bstract (resp ectively Kaspar ov) dual for the G -space X pulls back to one f or X × E G as a G ⋉ E G -space, and the diagram (3.2) KK G ∗ ( C 0 ( X ) , C 0 ( X )) ∼ = p ∗ E G   Lef / / KK G ∗ ( C 0 ( X ) , C ) ∼ = p ∗ E G   KK G ⋉ E G ∗  C 0 ( X × E G ) , C 0 ( X × E G )) Lef / / KK G ⋉ E G ∗  C 0 ( X × E G ) , C 0 ( E G )  commut es. Hence the Lefschet z ma p for G acting o n X is is omorphic to the Lef- schetz map of G ⋉ E G a cting on X × E G . This replac es the non-pro per group oid G by the prop er gr oup oid G : = G × E G at no loss of information. In terms o f this situation, our deﬁnitions ar e g oing to yield a a theory of G ⋉ E G - equiv a riant corre spo ndences base d o n K-oriented G ⋉ E G -equiv a riant vector bundles (equiv alently , G -equiv ariant v ector bundles on X × E G ) a nd G ⋉ E G -equiv a riant open embeddings. Such corresp ondences will yield analytic Kaspar ov morphisms since op en embeddings do , while zer o sections a nd pr o jections of G ⋉ E G -equiv ar iantly 20 HEA TH EM ERSON K-oriented vector bundles yield analytic K asparov mor phisms for the Thom iso - morphism fo r K-or ient ed vector bundles over a space with an ac tion of a pro per group oid, whic h is pro ved in [22]). In order to comp ose co rresp ondences we need a suﬃcient supply of ‘trivia l’ vector bundles, but for this to o, t he fa ct that w e hav e a pr op er gr oupo id makes a big diﬀerence. If G is a gr oup oid acting on a s pace, then a G -eq uiv ar iant vector bundle (see §3.2) over that space is trivial if it is pulled back from the unit spac e of G using the a nc hor map for the space. In the case we a re discus sing, where G : = G ⋉ E G , a G ⋉ E G -vector bundle on X × E G is the same as a G -v ector bundle on X × E G , and a trivial G ⋉ E G -vector bundle is a G -vector bundle on X × E G whic h is pulled back fro m E G under the co or dinate p ro jection. In gener al, if a gro upo id is not prop er, it may hav e no equiv a riant vector bundles on its bas e, e.g. if our gro upo id is a gr oup G , then its base is a p oint, so that a trivial G -vector bundle ov er X is equiv alen t to a ﬁnite-dimensional re presentation of G . 3.2. Equiv ariant v ector bundles. As p er ab ov e, G shall be a pr op er gro upo id un til further no tice. If G is a gr oup, this means that it must b e compac t. W e will frequentl y consider this case in examples. If X is a G -space, we remind the reader that the anchor map for the action is denoted  X : X → Z . A G - e quivariant ve ctor bund le on X is a vector b undle on X which is also a G -space suc h that elemen ts o f G map ﬁbres to ﬁbres linearly . There is an obvious notion of isomorphic G -e quiv ar iant vector bund les. If G is a compact group, then a G -equiv ariant vector bundle on a po in t is a ﬁnite-dimensional linear representation of G . Notation 3.3 . If V is a n equiv ariant vector bundle over X then we denote b y π V : V → X the vector bundle pro jection and ζ V : X → V the zero section. W e frequentl y denote by | V | the total spa ce of V , and denote by VK G ( X ) the Grothen- dieck group of the monoid of isomor phism classe s of G -equiv ariant vect or bundles ov er X . Given that G is a ssumed prop er, equiv arian t vect or bundles b ehav e in some ways just like or dinary vector bundles. F or example, if Y ⊂ X is a clos ed, G -inv ar iant subset of a G -s pace X , and if V is a G -equiv ariant vector bundle on X , then any equiv a riant section of V deﬁned on Y can b e extended to a n equiv ariant section deﬁned on a n o pen G - in v ar iant neighbour ho o d of Y . This involv es an av eraging pro - cedure (see [16]). As a consequence, if f i : X → Y , i = 0 , 1 are tw o G -equiv ariantly homotopic maps, and if V is a G -equiv ariant vector bundle on Y , then f ∗ 0 ( V ) and f ∗ 1 ( V ) are G -equiv a riantly is omorphic. On the other hand, some new pr oblems app ear in connection with equiv ariant vector bundles. W e ﬁrs t formalize o ur notion of triviality and the cor resp onding notion of subtriviali ty in a deﬁnition. Deﬁnition 3.4. Let X b e a G -space. A G -vector bundle over X is trivial if it is pulled back from Z under the anchor ma p  X : X → Z . W e denote the pull-back of a G -vector bundle E ov er Z to the G - space X b y E X . A G -equiv arian t vector bundle is subtrivial if it is a direct summand of a tr ivial G -vector bundle. Example 3 .5 . If G is a compact gr oup, then a tr ivial G -equiv ariant vector bundle ov er X has the form X × R n where R n carries a linear representation of G , and where G acts on X × R n diagonally . W e will make r ep eated use of the following basic fac t ab out repres en tations of compact groups (see [27]). DUALITY AND CORRESPONDENCES 21 Lemma 3.6 . L et G b e a c omp act gr o up and G ′ ⊂ G b e a sub gr oup. Then any ﬁnite-dimensional r epr esentation of G ′ is c ontaine d in the r estrictio n t o G ′ of a ﬁnite-dimensional r epr esent ation of G . Even if G is a co mpact gr oup, d ue to the notion o f ‘trivia l’ vector bundle w e are using, no t every G -vector bundle is lo cally trivial in the categ ory of G -vector bundles, i.e. lo ca lly is omorphic t o a t rivial G -vector bundle. But it is not hard to prov e the following and it is a go o d exercise for understanding equiv ariant vector bundles. W e leav e the pr o of to the rea der, but see §3.3 for some imp ortant ing redients in the argument. Lemma 3.7. Every G -e quivaria nt ve ctor bund le on X is lo c al ly subtrivial in the sense that for every x ∈ X ther e is a G -e quivariant ve ctor bund le on Z , a G - e quiv ariant neighb ourho o d U of x , and an emb e d ding ϕ : V | U → E U . Improving this lo cal result to a glo bal one is not p ossible, how ev er, without a n appropriate compactness assumption. Example 3.8 . Let X : = Z with the trivia l action o f the compact gr oup G : = T . Then the 1-dimensional complex vector bundle Z × C with the a ction of z ∈ G in the ﬁbre ov er n by the character z 7→ z n is not subtrivial, since it co nt ains inﬁnitely many distinct irreducible representations of G . Deﬁnition 3. 9. Let X b e a G -space. • The space X has enough G -e quivari ant ve ctor bund les if whenever we a re given x ∈ X and a ﬁnite-dimensional re presentation of the compact isotropy group G x x of x , there is a G -equiv arian t vector bundle ov er X whose restric- tion to x con tains the giv en r epresentation of G x x . • The space X ha s a ful l vector bundle if there is a v ector bundle V ov er X such that an y irreducible repres en tation of G x x is con tained in the represen- tation of G x x on V x (and w e say that such a vector bundle V is full.) It is the co n ten t of Lemma 3.6 that a c ompact gro up a cting on a p oint has enough vector bundles. It do es no t hav e a full vector bundle unless it is ﬁnite, b ecause a compact group with a ﬁnite dual ha s ﬁnite-dimensional L 2 ( G ) by the Peter-W eyl theorem and m ust then be ﬁnite. It is e asy to c hec k th at if f : X → Y is a G -equiv ariant ma p then X has enough equiv a riant vector bundles if Y do e s, and f ∗ ( V ) is a f ull vector bundle on X if V is a full v ector bundle on Y . Bo th of these assertions us e the basic fact Lemma 3.6. Example 3 .10 . The G -space describ ed in E xample 3.8 do es not hav e a full vector bundle, although it ob viously has enough vector bundles. The follo wing example is m ore subtle. It is due to Julianne Sauer (see [30]). Example 3.11 . Let X = R and K b e the compact group K : = Q n ∈ Z Z / 2 acting trivial ly o n R . Since X is K - equiv ar iantly contractible, and b y the homotop y- in v a riance o f e quiv ar iant vect or bundles (mentioned ab ov e at the b eg inning of §3.2), any K -equiv ariant v ector bundle V on X is trivia l, a nd hence a ll the represe n tations of K on the ﬁbres V x are equiv alent. Now let σ : K → K b e the shift automorphism and co nsider the group G : = K ⋉ σ Z ; it acts on X by letting σ ( t ) : = t + 1. This is a proper action. Of course as a group oid G is not proper, but we r epair this below. W e claim that the only tr ivial G -vector bundles on X yield trivial K -r epresentations in their ﬁbres. This will show th at X do es not have enough G -vector bundles. The pro of is as follows: any G -vector bundle V on X must b e tr ivial as a K - vector bundle, as ab ov e. On the other hand, the co v a riance rule for the semi-direct 22 HEA TH EM ERSON pro duct impli es that represent ations o f K on V x and V x +1 are mapped to eac h other (up to e quiv alence) by the action of σ , and therefore ˆ σ : Rep( K ) → R ep( K ) ﬁxes the p oint [ V x ]. But Rep( K ) is the direct sum o f d Z / 2’s and b σ acts as th e shift. The only ﬁxed po in t then is the zer o sequence. This corresp onds to a triv ial representation. This means that at every point x ∈ X the representation of K w e get on V x is trivial. T o repair the non-prop erness of G , r eplace it by G : = G ⋉ E G a nd replace X b y X × E G as explained at the b eg inning o f this section, then we get an example of a prop er group oid G and a G -space X × E G such which do es not hav e eno ugh equiv a riant vector bundles. This is b ecause X × E G and X are G -equiv arian tly homotopy-equiv alent anyw a y , b eca use the action of G on X is p rop er. Hence these spaces hav e canonically isomorphic monoids o f is omorphism cla sses o f equiv ariant vector bundles. A Morita- equiv alent approach is via a mapping cyclinder construction and pro- duces a compa ct g roup oid a cting on a compact space without enough vector bundles. T ake [0 , 1] × K mo dulo the rela tion (1 , k ) ∼ (0 , σ ( k )). T his results in a bundle of compact g roups ov er the c ircle which can shown to b e lo cally compact g roup oid with Haar system. Let G be this gr oup oid: it is pro per . Its base Z is the circle. By a ho lonomy argument similar to the one just given, an y G -equiv arian t vector bundle over Z m ust restr ict in ea ch ﬁbre to a trivia l representation of K . Thus, there a re not enough G - equiv a riant vector bundles on Z . Example 3 .12 . If G is a dis crete group with a G -co mpact mo del for E G , then Lück and Oliver hav e s hown in [23] that t here is a full G -e quiv ar iant vector bundle on Z , where G : = G ⋉ E G , (so that Z = E G .) 3.3. The top olog ical index of A tiy ah-Singer. W e now indicate wh y the condi- tion of having enough v ector bundles, or having a full vector bundle, is impo rtant for describing analytic equiv aria nt K K-groups topolo gically . W e start with the problem, famously trea ted by Atiy ah and Singer , of descr ibing the equiv arian t (ana lytic) index of a G -equiv arian t elliptic op erato r in topo logical terms, where G is a c ompact gro up, k eeping in mind that a n equiv ariant elliptic op erator is an importa n t example of a cycle for equiv ariant KK-theory . Let X b e a smooth manifold with a s mo oth action of the co mpact group G . The symbol o f an equiv a riant elliptic ope rator o n X is an equiv a riant K- theory class for T X . The idea o f Atiy ah and Singer for deﬁning the top ological index o f the op erator is to smoo thly em bed X in a ﬁnite-dimensional (linear) repr esentation of G on R n . The deriv ativ e of this embedding gives a smo o th embedding of T X in R 2 n , where T X has the induced actio n of G . Since T X is an e quiv ar iantly K-o riented manifold, the norma l bundle ν to the embedding is a G -equiv ariant ly K -oriented vector bundle on T X . The tubular neighbourho o d em bedding iden tities it with an op en, G -equiv arian t neighbourho o d of the image of T X in R 2 n . W e now obtain a comp osition K ∗ G (T X ) → K ∗ G ( N ) → K ∗ G ( R 2 n ) → K ∗ G ( ⋆ ) ∼ = Rep( G ) , where the ﬁrst map is the Thom isomorphism for the eq uiv ar iant ly K-oriented G - vector bundle N , the seco nd is the map on equiv arian t K-theory induced b y the op en inclusio n N ֒ → R 2 n and the third is equiv arian t Bott Perio dicity ( R 2 n ∼ = C n with the given action of G has an eq uiv ar iant complex structure, so a G -e quiv ar iant spin c -structure. The spinor bundle is the trivial G -vector bundle Λ ∗ C ( C n ) o v er C n .) The conten t of the index theorem is that this comp osition agrees with the map K ∗ G (T X ) → Rep( G ) obtained b y ﬁrst in terpreting cycles for K ∗ G (T X ) a s sym bols of DUALITY AND CORRESPONDENCES 23 equiv a riant ellipt ic op erators on X , making these elliptic op era tors into F redholm op erators , a nd taking their equiv ariant indices. But how do we get a smo o th, eq uiv ar iant e m bedding of X in a ﬁnite-dimensional linear repre sent ation of G in the ﬁr st place? Since it inv olves impo rtant ideas for us, we will sketc h the pro of. The result seems due to Mostow (see [2 7]). V ery similar arguments also prove Lemma 3.7. First o f a ll, we may assume (or av erage using the Ha ar system o n G ) that X has an in v ar iant Riemannian metric. Now the orbit G x is a smo oth em bedded submanifold of X isomor phic to G / G x x . The tangent space of X a t x splits int o the or thogonal sum of the tangent space to the orbit and its orthog onal comple- men ts N x : = T x ( G x ) ⊥ . The latter is a ﬁni te-dimensional representation of G x x , and inducing it results in a G -equiv ariant vector bundle N : = G × G x x N x : = G × N x / ( g , n ) ∼ ( g h, h − 1 n ) for h ∈ G x x on the orbit which is precisely the nor mal bundle to the embedded submanifold G x . By expo nent iating we o btain an equiv arian t diﬀeomorphism b etw een the total space o f N and an in v a riant op en neighbourho o d U of the orbit. W e embed this neighbourho o d as follo ws. By Lemma 3.6, the represen tation of G x x on N x is con tained in the restriction o f some repre sent ation of G o n some ﬁnite-dimensional vect or spa ce ˜ N x . The na tu- rality of induction implies that we hav e an inclusion of vector bundles N ⊂ ˜ N : = G ⋉ G x x ˜ N x . But since ˜ N x is the restriction o f a G - representation, ˜ N is a pro duct bundle, i.e. a trivial G vector bundle on the orbit. This pr ovides a G -equiv ariant map U → ˜ N x , ex plicitly , by mapping [( g , n )] ∈ U ∼ = N ⊂ ˜ N to the p oint g n ∈ ˜ N x . It is of co urse not necessar ily an embedding; to improv e it to an embedding, ﬁx a vector v ∈ ˜ N x whose isotropy in G is exactly G x x (for this see also [27]) and se t ϕ : U ∼ = G × G x x N x → ˜ N x ⊕ ˜ N x , ϕ ([ g , n ]) : = ( g n, g v ) . The map ϕ is an equiv ariant embedding as requir ed. As mentioned, if X is compact, then we can then we can (carefully) paste tog ether the lo cal embeddings to get an e m bedding of X ; see [16], o r the sour ce [2 7]. The reader sho uld notice that the ass umption that X has enough vector bundles is used implicitly to show that the r epresentation o f G x x on N x can b e extended to a G - equiv a riant vector bundle o n the orbit o f x (the vector bundle ˜ N induced from ˜ N x ). This w as the statemen t of Lemma 3 .6, and is just an explicit wa y o f saying that G has enough v ector bundles on it s one-p oint ba se space. 3.4. Embedding theorems from [16] . More g enerally , in [16] the following is prov ed. Let X b e a G -space, where G is a prop er groupo id. W e say t hat X is a smo o th G -manifold if we can cover X by c harts of the form U × R n where U ⊂ Z is open, so that with respect to t his pro duct structure the a nc hor map  X : X → Z iden tiﬁes with the ﬁrst co o rdinate pro jection, and such that gro upo id elemen ts and change of co ordinates a re smo oth in the vertical direction. An smo oth op en emb e ddi ng b et ween G -manifolds is a s mo oth equiv arian t map which is a diﬀeomorphism onto a n open subset of its co domain. Theorem 3.13. L et G b e a (pr op er) gr oup oid and X and Y b e smo oth G -manifolds. Supp o se that either A . The G -sp ac e Z has enough ve ctor bund les and G \ X is c omp act, B . Z has a ful l ve ctor bund le and G \ X ha s ﬁnite c ov ering dimension. Then, given a smo oth, G -e quivari ant map f : X → Y , ther e exists • A smo oth G -e quivaria nt ve ctor bund le V over X , 24 HEA TH EM ERSON • A smo oth G -e quivaria nt ve ctor bund le E over Z , • A n smo o th, e quivariant op en emb e dding ϕ : V → E Y , such t hat (3.14) f = π E Y ◦ ϕ ◦ ζ V . F urthermor e, under the union of hyp othe ses A , B , any G - e qui variant ve ctor bund le over X (or Y ) is subtrivial. Recall that the nota tion E Y means the pullback of E to Y using the anchor map  Y : Y → Z . W e call a factorisation of a map f : X → Y of the form (3.14) a normal factori- sation . 3.5. Normally non-sing ul ar maps. As in the previous section, G is a pro per group oid. The co nstructions o f the previous section motiv ate the following deﬁni- tion. Deﬁnition 3 .15. A G -e quiva riant normal ly non-singular map Φ fro m X to Y is a triple ( V , E , ˆ f ) where V is a G -equiv arian t subtrivia l vect or bundle o v er X , E is a G - equiv ar iant vector bundle ov er Z and ˆ f : V → E Y is a G - equiv ar iant op en embedding. • The tr ac e of Φ is the compo sition π E Y ◦ ˆ f ◦ ζ V . • The stable normal bund le of Φ is the class [ V ] − [ E X ] ∈ VK G ( X ). • The de gr e e o f Φ is dim( V ) − dim ( E ). • The normally non-s ingular map Φ is K -oriente d if the V and E a re equiv- ariantly K-oriented. • The normally non- singular map Φ is smo oth if X and Y ar e smo oth G - manifolds and V a nd E ar e smooth equiv ariant vector bundles on whic h G acts smo othly , and if ˆ f is a smo oth embedding. Of co urse the trace of a G -equiv arian t smo oth normally non-s ingular map is itself a smo o th e quiv ar iant map, and the con ten t of Theo rem 3.13 is that, co n versely , any smo oth equiv ariant map bet ween smo oth G -ma nifolds is t he trace of some smo oth normally non-singular map , under some h ypotheses ab out the a v ailability of equiv aria n t v ector bund les. (This statemen t is impr ov ed in Theo rem 3.19.) Example 3.16 . The simplest example of a normally non- singular map is the zer o section and bundle pro jection ζ V : X → V , π V : | V | → X , of a G -equiv ariant vector bundle V over a c omp act G -space X , where G is a compact group. The zero sectio n is the trace of the nor mally non-singular map ( V , 0 | V | , id). The stable nor mal bundle is the class [ V ] ∈ V K G ( X ) ∈ of the vector bundle itself. Since X is compact, V is subtrivial. If X is not compa ct, this c an fail, c.f. Example 3.8. With the sa me (compact) X , V etc , the bundle pro jection π V : | V | → X is the trace of a nor mally no n-singular map ( π ∗ V ( V ′ ) , E , ϕ ), where V ′ ∈ V ect G ( X ) is a choice of G -vector bundl e on X such that V ⊕ V ′ is a trivial bundle E X , and ϕ : | V ⊕ V ′ | ∼ = | π ∗ V ( V ′ ) | ∼ = − → E X is a trivia lisation. The stable normal bundle is π ∗ V ([ V ′ ]) − [ E X ] ∈ VK G ( | V | ) resp ectively . The nor mally non-singula r map just describ ed seems to dep end on the choice of trivialisa tion o f V , but it ca n b e chec ked that a n y tw o choices yield equiv alen t normally non-singular maps in the sense ex plained below. DUALITY AND CORRESPONDENCES 25 Example 3 .17 . Let G b e a compact g roup acting smoothly on manifolds X , Y with X compact. By the discussion in §3.3 we can ﬁx a smo oth, equiv ariant embedding i : X → E in a linear representation of G . Deﬁne V to be t he normal bundle to the em bedding x 7→  f ( x ) , i ( x )  of X in E Y : = Y × E . Let ϕ : V → E Y be the co rresp onding tubular neigh bo urho o d embedding. Then the trace of the comp osition π E Y ◦ ϕ ◦ ζ V of the normally non-s ingular map ( V , E , ˆ f ) is f . Sin ce T X ⊕ V ∼ = f ∗ (T Y ) ⊕ E X , the stable normal bundle is f ∗ ([T Y ]) − [T X ] ∈ VK G ( X ). Example 3 .18 . If G is a dis crete group with a G -co mpact mo del for E G , then Lück and O liver have shown in [23] that there is a full G -equiv a riant vector bundle on E G , where G : = G ⋉ E G , (so that the base of G is Z : = E G .) Let X and Y be smo oth manifolds equipp ed with smo oth actions of G and f : X → Y b e a s mo oth, G -equiv a riant map. As above let G b e the prop er gr oup oid G × E G . Applying the Baum-Co nnes pr o cedure of §3.1 to this situation we get smo o th G -manifolds X × E G and Y × E G and a smo oth G -map f × id E G : X × E G → Y × E G . It is the tra ce of a nor mally non- singular map b ecause of Theor em 3.13 and the result of Lüc k and Oliv er. As we will see in the next section, if f is also K-orientable in an appropr iate sense, then it will give rise to a morphism in K K G ( C 0 ( E G × X ) , C 0 ( E G × Y ) . If X is a topo logically amenable G -space, this gives an element of KK G ( C 0 ( X ) , C 0 ( Y )). T w o normally non-singular map s a re isomorphic if there a re vector bundle iso- morphisms V 0 ∼ = V 1 and E 0 ∼ = E 1 that intert wine the op en embeddings f 0 and f 1 . The lifting of a no rmally non-singula r map Φ = Ψ = ( V , ϕ, E ) a long a n equi- v ar iant vector bundle E + ov er Z is the normally non-s ingular ma p Φ ⊕ E + : = ( V ⊕ ( E + ) X , E ⊕ E + , ˆ f × Z id E + ). T wo normally non-singular map s are stably iso- morphic if there are G -equiv arian t v ector bundles E + 0 and E + 1 such that Φ 0 ⊕ E + 0 is isomo rphic to Φ 1 ⊕ E + 1 . Finally , t w o no rmally non-singula r map s Φ 0 and Φ 1 are isotopic if there is a contin uous 1-pa rameter family of norma lly non-singula r map s whose v alues a t the endp oints are stably isomo rphic to Φ 0 and Φ 1 resp ectively (see [1 6] for the exact deﬁnition), and are e quivalent if they hav e isotopic liftings. There is an obvious notion o f smo oth e quivalenc e of smo oth normally non-singular map s. There a re ob vious K-o riented a nalogues of t he above r elations. F or example, lifting must only use K- oriented trivial bundles, and isomor phism must preserve the g iven K-or ien tations. Referring to this kind of equiv alence we will speak of K -oriente d e quivalenc e o f K-oriented nor mally non-singular maps. 3.6. Manifolds with smo o th normally non-singul ar maps to Z . A useful h yp othesis co vering a n um ber of ge ometric situations is that a given smo o th G - manifold X admits a smo oth normally non-singular map to the o bject space Z of G . By the theorem ab ov e this is the case if A o r B hold. It means explicitly that we hav e a triple (N X , ˆ g , E ) where N X is a smo oth s ubtrivial vector bundle ov er X , E is an equiv ariant vector bundle over Z and ˆ g is a smo oth op en equiv arian t embedding N X → E . Note th at N X ⊕ T X ∼ = E X . Such a normally non-singular map is (smo othly) sta bly isomor phic to a K -oriented normally no n-singular map b eca use we can replace if needed E b y E ⊕ E , which is canonically eq uiv ar iantly K -oriented using the G -equiv ariant co mplex structure, and r eplacing N X b y N X ⊕ E X . If (N X , ˆ g , E ) is a smo oth normal map to Z s uch that E is equiv ariantly K- oriented, then K-orient ations on N X are in 1-1 corre spo ndence with K-orientations on T X b ecause of the 2-out-o f-3 pro per ty . One can pro ve the following. Theorem 3.19. L et X and Y b e smo oth G -manifolds, and assume that X ad mits a smo oth, n ormal G -map to Z and that f ∗ (T Y ) is subtrivial. 26 HEA TH EM ERSON Then any smo oth G -map fr om X to Y is the tr ac e of a smo oth n ormal G -map, and two smo oth normal ly non- singular maps fr om X to Y ar e smo othly e quiva lent if and only if their t r a c es ar e smo o thly homotopic. F urthermor e, smo oth e quivalenc e classes of smo oth K -oriente d normal ly non- singular map s fr om X to Y ar e in 1-1 c o rr esp ondenc e with p airs ( f , τ ) wher e f is a smo o th homotopy class of e quiva riant smo oth map X → Y and τ is an e quiva riant K -orientation on N X ⊕ f ∗ (T X ) . W e sketc h the existence part of this pro of. Fix a smo o th equiv arian t nor mally non-singular map (N X , ˆ g , E ) fro m X to Z . W e can as sume by replacing E by E ⊕ E and N X b y N X ⊕ E X if needed that E is equiv ariant ly K-or ient ed. Let Y b e another smo oth G -manifold and f : X → Y b e a smo o th map. Let g : X ζ N X − − − → N X ˆ g − → E the comp osite smoo th em bedding. One obtains a a smo oth embedding X → Y × Z E = E Y , x 7→  f ( x ) , g ( x )  . It has a (smooth) nor mal bundle V with a smoo th op en embedding in E Y . Since V ∼ = N X ⊕ f ∗ (T Y ), V is subtrivial and ( V , E , ˆ f ) is a smo o th norma lly non-singular map with trace f and stable norma l bundle [ V ] − [ E X ] ∈ VK G ( X ). Note that V ⊕ T X ∼ = f ∗ (T Y ) ⊕ E X . The stable no rmal bundle is [ V ] − [ E X ] = f ∗ ([T Y ]) − [T X ] ∈ VK G ( X ). Equiv a ri- ant K-orientations on V ar e in 1-1 corres po ndence with equiv ar iant K-orientations on N X ⊕ f ∗ (T Y ). 3.7. Corresp ondences. W e are now in a p osition to deﬁne what corresp ondences are. L et G conti nue to denote a prop er group oid. Deﬁnition 3.20 . Let X and Y b e G -spaces. A G -e quivaria nt c orr esp ondenc e fr om X to Y is a qua druple ( M , b, f , ξ ) where M is a G -spa ce, f : M → Y is a G - equiv a riantly K-or ient ed normally non-singular map , b : M → X is an equiv a riant map, and ξ ∈ RK ∗ G ,X ( M ) is a G -equiv arian t K -theory class with X -compact supp ort (see §1.4) wher e the G ⋉ X -structure on M is that determined by the G -equiv ariant map b : M → X . The d e gr e e o f the corresp ondence ( M , b, f , ξ ) is the s um of the degrees of f a nd ξ . R emark 3.21 . Th us a signiﬁca n t diﬀerence from the set-up of Co nnes a nd Skandalis in [8] is that the map b : M → X is not required to be prop er; we hav e replaced this b y a support condition on ξ . Several equiv alence relations on corresp ondences are imposed. The ﬁrst is to consider t w o corresp ondences ( M , b 0 , f 0 , ξ ) a nd ( M , b 1 , f 1 , ξ ) to b e equiv alen t if their nor mally non-s ingular maps are e quiv alent. The seco nd is to co nsider b or dant corresp ondences equiv alen t (w e will not disc uss this at all in this sur vey .) The third is most int eresting, and is called Thom mo diﬁc ation . The Thom mo diﬁcation of a corresp ondence ( M , b, f , ξ ) using a subtrivia l K-or ient ed vector bundle V ov er M is the corresp ondence  V , b ◦ π V , f ◦ π V , τ V ( ξ )  , where τ V : RK ∗ G ,X ( M ) ∼ = − → RK ∗ +dim( V ) G ,X ( | V | ) is the Thom isomorphism, i.e. τ V ( ξ ) : = π ∗ V ( ξ ) · ξ V , wher e ξ V ∈ RK dim( V ) G ,M ( | V | ) is the Thom clas s. W e decla re a cor resp on- dence and its Thom mo diﬁcation to b e Thom equiv alen t. Note that a pplying Thom mo diﬁcation to a corresp ondence does not change its deg ree. The equiv alence relation on corresp ondences is that gener ated by equiv alence o f normally non-singular maps, bordism and Thom equiv alence. DUALITY AND CORRESPONDENCES 27 Deﬁnition 3.22. Let G b e a proper group oid and X and Y b e G -spaces. W e let c kk ∗ G ( X, Y ) denote the Z / 2-gra ded set o f equiv alence c lasses of G -equiv arian t corresp ondences from X to Y , graded b y degree. A corresp ondence ( M , b, f , ξ ) from X to Y is smo oth if X , Y and M are smo o th manifolds, f is a smo oth normally no n-singular map (see §3.3), and b is a smo oth map. There is a rather obvious notion of smo o th e quivale nc e of smo o th corresp on- dences. This gives rise to a parallel theory using o nly smo oth e quiv alence classes of smoo th corresp ondences; w e do not use no tation for this. 3.8. c kk G as a category . Classes of corresp ondences for m a catego ry with analo - gous prop erties to Ka sparov’s equiv a riant KK (that is, to analytic K asparov theory). The comp osition o f corresp ondences is c alled the interse ction pr o duct . F or comp o- sition we use similar nota tion to Ka sparov’s: if Ψ ∈ c kk G i ( X, Y ) is a topologica l morphism, i.e. an eq uiv alence class of equiv a riant corresp ondence from X to Y , and if Φ ∈ c kk G j ( Y , W ) is a nother, then we write Ψ ⊗ Y Φ ∈ c kk G i + j ( X, W ) for their comp osition. W e do not describ e the general intersection product here, bu t will focus instead on the transversalit y metho d of [8]. Recall that tw o smooth G -ma ps f 1 : M 1 → Y and b 2 : M 2 → Y are tr ansverse if for every ( p 1 , p 2 ) ∈ M 1 × M 2 such that f 1 ( p 1 ) = b 2 ( p 2 ), we have D p 1 f 1 (T p 1 M 1 ) + D p 2 b 2 (T M 2 ) = T f 1 ( p 1 ) ( X ). T ra nsversalit y ens ures that the space M 1 × X M 2 : = { ( p 1 , p 2 ) ∈ M 1 × M 2 | f 1 ( p 1 ) = b 2 ( p 2 ) } has the structure of a smooth G -manifold. Theorem 3.23. 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 o rr esp ondenc e s fr om X to Y and fr om Y t o U , r esp e ctivel y. A ssume t hat b oth M 1 and M 2 admit smo oth n ormal ly non-singular map s to Z (se e §3.5), so t hat we lose nothing if we view f 1 and f 2 as K - oriente d smo oth maps (se e The or em 3.19). A ssume also that f 1 and b 2 ar e tr ansverse, so that M 1 × Y M 2 is a smo oth G -manifold; it has a smo oth n ormal ly non-singular map to Z as wel l, and the interse ctio n pr o duct of Φ 1 and Φ 2 is the class of the c o rr esp ondenc e  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. In the non- equiv ar iant situation, any t w o smo oth maps can b e p erturb ed to b e transverse, and in [8] this is sho wn to give rise to a b or dism o f co rresp ondences. As a result, one can comp ose bo rdism class es of cor resp ondence by the recip e describ ed in Theorem 3.23. How ev er, this fails in the equiv ariant situa tion beca use pairs of smooth maps cannot in general b e p erturb ed equiv ariantly to be tra nsverse; this ha ppsn in even some of the simplest situations. Example 3.24 . Let µ b e the non-trivia l ch aracter of Z / 2 . The cor resp onding o ne- dimensional representation is denoted C µ . W e regar d this as an equiv ariant vector bundle C µ ov er a point. Its total space is | C µ | . The equiv aria nt vector bundle C µ is equiv arian tly K-oriented, since the Z / 2-action pres erves the co mplex structure on C µ . W e therefore o btain a smo oth normal equiv ariant map ⋆ → | C µ | of degree 2. But since the o rigin is the only ﬁxed-p oint of the Z / 2- action, this ma p ca nnot be p er turbed to b e transverse to itself. This means that we cannot compos e, for ex ample, the topo logical morphism x ∈ c kk Z / 2 2 ( ⋆, | C µ | ) represen ted b y the corres po ndence ⋆ ← ⋆ → | C µ | , and t he topo logical morphism y ∈ c kk Z / 2 0 ( | C µ | , ⋆ ) represen ted by | C µ | ← ⋆ → ⋆ in the or der x ⊗ | C µ | y , using the transversalit y recip e of Theorem 3.23. 28 HEA TH EM ERSON How ev er, we may apply Thom mo diﬁcation to the corresp o ndence ⋆ ← ⋆ → | C µ | , using C χ . Mo diﬁcation replace s the middle s pace ⋆ of x by | C χ | and replaces the normally non-sing ular map ⋆ → C by the co mpo sition | C χ | → ⋆ → | C µ | . This latter is obviously is otopic to the identit y map on | C µ | . Finally , o ne adds the Tho m class ξ C µ ∈ K 2 Z / 2 ( | C µ | ). Therefore the morphism x is equiv a len t to the morphism represented by ⋆ ← ( | C µ | , ξ C µ ) id − → | C µ | . The iden tit y map is transverse to any other map, since it is a submersio n. Comp os ing the mo diﬁed cor resp ondence and the o riginal representative of y using the tra nsversalit y r ecipe yields the class o f the degree 2 corresp ondence ⋆ ← ( ⋆, ( ξ C µ ) | ⋆ ) → ⋆, where ( ξ C µ ) | ⋆ denotes the r estriction of the Thom cla ss to the p oint. This equals the diﬀerence [ ǫ ] − [ χ ] ∈ Rep( Z / 2) of the trivial and the non-trivial repres en tation of Z / 2. It is the Euler cla ss of the K -oriented vector bundle C χ , e.g. the re striction of the Thom class to the zero section. With the given architecture of equiv aria n t co rresp ondences, a similar pro ces s can b e ca rried out when comp osing tw o a rbitrary (smo oth) co rresp ondences. Let X b ← − ( M , ξ ) f − → Y be suc h. Let f = ( V , E , ˆ f ). The equiv a riant vector bundles V and E are K-oriented by as sumption. Thom modiﬁcation using V results in the corresp ondence X b ← − ( | V | , ξ V · ξ ) f ◦ π V − − − → Y . An ob vious s mo oth isotopy of normally no n-singular map s replaces this by X b ← − ( | V | , ξ V · ξ ) π E Y ◦ ˆ f − − − − → Y . Since π E Y ◦ ˆ f is a submer sion, it is transverse to an y other smooth map t o Y . Hence this corresp ondence can b e comp osed using the tra nsversalit y r ecip e of Theorem 3.23 with any other one (on the right). An analog ous pro cedure ca n b e used to deﬁne a co mpo sition r ule for ar bitrary (not necessar ily smooth) co rresp ondences. This rule is quite top ological in ﬂav our, of co urse, but is o nly deﬁned up to iso topy and is les s sa tisfying than the sharp formulas one gets in the presence of tra versalit y , which o f course only a pply in the presence o f smo o th structures. W e will only compute comp ositions in this se tting in this survey . 3.9. F urther prop erties of top olo gical KK -theory. W e ha ve said that c kk G is a categor y . It is also additive, with the sum op eration on co rresp ondences deﬁned b y a disjoint union pro cedure. The o ther imp orta n t pro per t y is the existence of external pro ducts. This means that there exists an external pro duct map c kk G i ( X, Y ) × c kk G j ( U, V ) → c kk G i + j ( X × Z U, Y × Z V ) . It leads to the structure on c kk G of a symmetric monoidal category . Finally , there is a natural map c kk G → KK G . This is deﬁned n ot using the pseu- do diﬀerent ial calculus, a s in [8], but by purely top ological considerations . Indeed, b y deﬁnition, a nor mal K-oriented G -map from X to Y facto rs, b y deﬁnition, as a comp osite of a zero section of an equiv ar iantly K-orient ed vector bundle, an equiv ari- ant open embedding, and the pro jection map for another equiv a riantly K-or ient ed vector bundle. Zero sections and bundle pro jections yield elements of KK G beca use of the Tho m isomorphism of [22]. Op en embeddings clearly determine elements morphisms in KK G beca use they determine equiv ar iant *-homomor phisms. The v a rious na turality prop erties of the Thom is omorphism imply corres po nding facts a b out the map c kk G → K K G . Other functorial prop erties o f c kk G , e.g. with resp ect to homomorphisms G ′ → G of group oids, are explained in detail in [16]. DUALITY AND CORRESPONDENCES 29 4. Topological duality and the topological Lefschetz map W e ha v e o rganized th is survey a round the goal of computing the Lefschetz map for smo oth G -manifolds. This problem is in tert wined with that of computing equi- v ar iant KK-gr oups topo logically and we will solve b oth problems at once in this section. The ﬁrst step is to describe a clas s of G -spa ces to which the g eneral theory of dualit y describ ed in §1 .6 applies. Subject to the res ulting constr aint s on X we will obtain a top olo gi c al mo del of the Lefsc hetz map and simultaneously a pro of that the c kk G → KK G to b e is an isomorphism on bo th the do main and range of the Lefschetz map. This will complete our computation of Lef for a fairly wide sp ectrum of G -spa ces X . 4.1. Normally no n-singular G -spaces. Deﬁnition 4 .1. A normal ly non-singu lar G -sp ac e X is a G -space equipped with a G -equiv a riant normally non-singula r map (N X , E , ˆ g ) from X to Z . W e also require of a nor mally non-singular G -space X that every G -equiv a riant vector bundle on X is s ubtrivial. The v ector bundle N X is called the stable normal bund le of X . W e may assume without los s of generality that E is equiv ariantly K-oriented. Since the zero bundle is alwa ys uniquely K-or ien ted, we o btain an equiv a riant K- oriented no rmally no n-singular map (0 N X , E , ˆ g ) from N X to Z . L et D ∈ c kk G (N X , Z ) be the co rresp onding class. Since c kk G ⋉ X has external pro ducts, w e can deﬁne a map c kk G ⋉ X ( X × Z U, Y × Z V ) → c kk G ⋉ X (N X × Z U, N X × Z V ) If every G -eq uiv ar iant v ector bundle over X is subtrivial, then there is a for getful functor c kk G ⋉ X → c kk G and this results in a m ap c kk G ⋉ X (N X × Z U, N X × Z V ) → c kk G ((N X × Z U, N X × Z V ) . Comp osing with the previous one yields a to po logical analogue c kk G ⋉ X ( X × Z U, X × Z V ) → c kk G (N X × Z U, N X × Z V ) of the functor d enoted T P in the discussion of dualit y in §1.6, and, comp osing further with the mor phism D gives a top ologica l a nalogue of the Kaspar ov duality map PD ∗ : c kk G ⋉ X ( X × Z U, X × Z V ) → c kk G (N X × Z U, V ) of Theorem 1.7. Ho w ev er, it is of cour se not the ca se in general that every equi- v ar iant vector bundle ov er X is s ubtrivial, which is why we hav e added this as a h yp othesis. Example 4.2 . Let X be the integers with the trivial action of the circle group G : = T . Then ther e are ( c.f. Example 3 .8) equiv ariant v ector bundles o n X whic h are not subtrivial. This is despite the fact that X admits a norma lly non-singular map to a po in t, s ince it s mo othly embeds in the trivial representation of G on R , with trivial normal bundle. In an y case, X is not normally non-singular. An y smo oth G -ma nifold satisfying one of the h ypo theses of Theorem 3.13 i s normally non-singular. T o deﬁne a top ologica l analog ue o f the map denoted P D w e need a class Θ ∈ c kk G ⋉ X ( X, X × Z N X ). Co m bining compos ition with this class and the map KK G (N X × Z U, V ) → c kk G ⋉ X ( X × Z N X × Z U, X × Z V ) , 30 HEA TH EM ERSON which is deﬁn ed in the topo logical ca tegory in the same wa y as in the analytic one, we will obtain the required top ological map PD : KK G (N X × Z U, V ) → c kk G ⋉ X ( X × Z U, X × Z V ) . W e just describ e Θ in a heuristic fashion. Assume for simplicit y that the base Z of G is a p oint . So E is just a E uclidean space. Cho ose a p oint x ∈ X . Using the zero section o f N X and the ma p ˆ g , we see x as a po int in the op en subset | N X | of E , and hence by r escaling E (ﬁbrewise) into a suﬃciently small op en ball around x we obtain a n op en em bedding o f E into N X . E xplicitly , we use an o pen em bedding of the form ˆ  x : | E | β x − → B ǫ  ˆ g ( ζ N X ( x ))  ⊂ | N X | where the ﬁrst map is th e re-scaling . This yields an ob vious nor mally no n-singular map ( E , 0 | N X | , ˆ  x ) from a p o in t to | N X | It is eas ily chec k ed that this can b e ca rried out contin uously in the para meter x ∈ X , and w e obtain an X -eq uiv ar iant normal K-oriented map δ from X to X × Z | N X | with trace the gra ph o f the zero-s ection Ξ : X → X × Z | N X | , Ξ( x ) : = ( x, ( x, 0)). This yields a n element of c kk X ( X, X × | N X | ). The same can be c hec ked to w ork equiv arian tly , and a ﬁbrewise version works for gr oup oids with nontrivial base. W e let Θ ∈ c kk G ⋉ X ( X, X × Z | N X | ) be the corresp onding class. Theorem 4.3. In the notation ab ove, let X b e a normal G -sp ac e of ﬁnite t yp e, let N X b e the st able n ormal bu nd le and D ∈ K K G (N X , Z ) and Θ ∈ K K G ⋉ X ( X, X × Z N X ) t he classes c onstructe d ab ove. Then (N X , D , Θ) is a K asp ar o v dual for X in c kk G . , and the maps PD and P D ∗ ar e isomorphisms. The same forma l computations as in §1.6 then imply that the maps PD and PD ∗ are isomorphisms. Again, the stronger result is pro ved in [16] that one gets a symmetric Kasparov dual; this, remem ber this is designed to give, a s w ell, an isomorphism o f the form c kk ∗ G ( X × Z U, V ) ∼ = c kk ∗ G ⋉ X ( X × Z U, N X × Z V ) for any pa ir of G - spaces U and V . W e do not g ive the deta ils. As a consequence one deduces the following theorem b y using duality to r educe from the biv aria n t to the monov ariant case. Theorem 4.4. L et X b e a n ormal G -sp a c e of ﬁn ite typ e, and Y b e an arbi tr ary G -sp a c e. Then the natu r a l tr ansformation c kk ∗ G ( X, Y ) → KK G ∗  C 0 ( X ) , C 0 ( Y )  is invertible. Example 4 .5 . Co nsider the spa ce X : = Z with the trivial action of G : = T . This space is no t normally non-singula r, though it is a smo o th G manifold a dmitting a normally non-singular map to a point. The map (4.6) c kk G ( X, ⋆ ) → K K G ( C 0 ( X ) , C ) is not an isomorphism in this case. By dualit y , the elemen ts of KK G ( C 0 ( X ) , C ) are parameterised b y G -equiv arian t co mplex v ector bundles on X . One can c hec k t hat the elements o f c kk G ( X, ⋆ ) are b y contrast parameterised by G -equiv arian t c omplex vector bundles whi ch only in v olve a ﬁnite n um ber of r epresentations of G . In other words, (4.6) is equiv alen t to the em be dding ⊕ n ∈ Z Rep( T ) → Y n ∈ Z Rep( T ) of the direct sum in to the direct pro duct of representation rings. DUALITY AND CORRESPONDENCES 31 If we dropp ed the s ubtriviality requiremen t on vector bundles that we impos ed on cycles for c kk then (4.6) w ould be an isomor phism, but then we would not b e able to deﬁne the in tersection product of corresp ondences in general. R emark 4.7 . Restricting to smo oth corr esp ondences and smo oth equiv alence classes of corres po ndences yields a par allel ‘s mo oth’ theor y . If X is a smo oth normally non- singular G -spa ce and Y is smo o th, then it follows that the s mo oth and non-smooth versions o f c kk ∗ G ( X, Y ) agree . W e ar e now in a p osition to solve the pr oblem w e have b een w orking tow ards: a topo logical co mputation of the Lefschetz map for a normal G -spa ce o f ﬁnite type. Firstly , w e deﬁne t he top olo gic al Le fschetz map Lef : c kk G ⋉ X ∗ ( X × Z X , X ) → c kk ∗ G ( X, Z ) for any normal G - space of ﬁnite type as in Deﬁnition 1.23, using the top olo gical Kas- parov dual constructed ab ov e o ut of the stable no rmal bundle . Since a to po logical dual maps to an analytic d ual, the diagram that the diagram (4.8) c kk G ⋉ X ∗ ( X × Z X , X )   Lef / / c kk ∗ G ( X, Z )   KK G ⋉ X ∗  C 0 ( X × Z X ) , C 0 ( X )  Lef / / KK G ∗  C 0 ( X ) , C 0 ( Z )  . commut es. The ﬁnite-type hypothesis implies (Theor em 4.4) that the vertical ma ps are bo th isomor phisms. Therefore we ha v e obtained a complete des cription of Lef in purely top ological terms. W e now pro ceed to describ e it explicitly for smo oth G -manifolds in terms of transv ersality . 4.2. Explicit computation of Lef . Let X b e a smo o th G -manifold satisfying one of the hypotheses A o r B of T heorem 3.13. F rom that Theorem, and by Remar k 4.7, the map X → Z is the trac e of an essentially unique smo o th norma lly non- singular G -map. The norma l bundle N X is a smo oth G -equiv arian t vector bundle, and ˆ g : N X → E is a smo oth G -equiv arian t em bedding. Moreov er, the general morphism in c kk G ⋉ X ( X × Z X , X ) is represented b y a smo oth, G ⋉ X -equiv arian t correspo ndence Ψ : = ( M , b, f , ξ ) , o r in diagram form X × Z X b ← − ( M , ξ ) f − → X from X × Z X to X . Let Ψ denote its cla ss. Reca ll that the X -structure o n X × Z X is on the ﬁrst co ordinate. R emark 4.9 . The map f : M → X embedded in the corr esp ondence Ψ is assumed a smo oth G ⋉ X -equiv ariant no rmally non-singular map . This pre suppo ses the structure on M of a smo oth G ⋉ X -manifold . Note that this is a stronger co ndition than b eing a smo oth G -manifold: it entails a bundle structure on M with smo oth ﬁbres a nd is equiv alen t to requiring that the smo oth normally non-singular f is a submersion. F ollowin g the deﬁnition of the Lefschet z map in §1.9 we next apply the functor T N X : c kk G → K K G which sends a G ⋉ X -s pace, i.e. a G - space W ov er X , to the G -space W × X N X . The latter is the same as the pullbac k to W of the vector bundle N X using the map W → X ; r ecall that this functor is well-deﬁned provided that every G -equiv ariant vector b undle ov er X is subtrivial. The functor T N X maps a G ⋉ X -equiv a riant ma p from W to V to the G - equiv ar iant map W × X N X → V × X N X given by the obvious formula. Since X × Z X × X N X ∼ = 32 HEA TH EM ERSON X × Z N X via the map which for gets the ﬁrst co or dinate, applying the map T N X to Ψ yields the G -equiv ariant corr esp ondence X × Z N X ¯ b ← − ( M × X N X , ξ ) ¯ f − → N X , where ¯ b ( m, x, ξ ) : =  b ′ ( m ) , x, ξ ), where b ′ : = pr 2 ◦ b is the sec ond co ordinate v alue of b : M → X × Z X (since the ﬁrst co o rdinate v alue is just the anchor map  G ⋉ X M : M → X the map b is determined by b ′ .) W e then comp ose with the class of the s mo oth K-oriented normally non-s ingular map N X → Z , using tra nsversalit y (see Theor em 3.23) in the catego ry of smo oth, G -equiv a riant corr esp ondences. In the nota tion o f the theor em M 1 : = M × X N X and M 2 = N X , and f 1 = f × X id : M × X N X → X × X N X = N X , and b 2 : = id : N X → N X . Since N X → Z and f 1 are smo oth and nor mal by assumption and f 1 and b 2 are obviously transverse s ince b 2 is a submersion, the comp osite using Theorem 3.23 is the class of the smooth G -equiv a riant corr esp ondence (4.10) X × Z N X ¯ b ← − ( M × X N X , ξ ) → Z , and the o nly r emaining step is to comp ose with the dual class , which is the mo st in teresting one from our point of view . Recall that a bar denotes for getting the X -str ucture on a G ⋉ X -equiv a riant morphism. The clas s Θ that w e want to comp ose with is that o f the G -equiv ariant corresp ondence X id ← − X δ − → X × Z N X – which ca n be assumed s mo oth. Since the tr ace of δ is the smooth section Ξ where Ξ( x ) = ( x, ( x, 0)), o ne chec ks that the transversality co ndition needed to co mpos e Θ with (4.10) is that Ξ is transverse to the ma p ¯ b . This can b e chec k ed to b e the case if, for all m ∈ M for whic h  G ⋉ X G ( m ) = b ′ ( m ), the linear map (4.11) T m M → T  M ( m ) X , ζ 7→ D m b ′ ( ζ ) − D m  G ⋉ X M ( ζ ) is non-singular. This thus implies that the c oincid enc e sp a c e F Ψ : = { m ∈ M | ρ G ⋉ X M ( m ) = b ′ ( m ) } , is a smo oth, e quivariantly K -oriente d G -manifold ; in fact, more, by Theorem 3.23 it implies that the projection F Ψ → Z is a smo oth, K-oriented no rmally no n-singular map . Finally , it is easily chec ked that the re striction of ξ to F Ψ has compact vertical suppo rt with resp ect to the map (  M ) G ⋉ X | F Ψ : F Ψ → X , (this is alr eady implied by Theorem 3.23), so w e get a G -equiv arian t c orresp ondence from X to Z : (F Ψ , (  G ⋉ X M ) | F Ψ ,  G F Ψ , ξ ) , in the usual notation for anc hor ma ps. W e ca ll this the c oincidenc e cycle of Ψ. Theorem 4.1 2. Le t Ψ ∈ c kk G ⋉ X ∗ ( X × X , X ) ; let Lef : c kk ∗ G ( X × Z X , X ) → c kk ∗ G ( X, Z ) b e t he L efschetz map in t op olo gic al e quivariant KK -the ory. Then the top olo gic al L ef schetz invari ant of t he cla ss of a c orr esp o ndenc e Ψ in gener al p osition in the sense ex plaine d ab ove, is the class of the c oinci denc e cycle of Ψ , Lef  [( M , b, f , ξ )]  = [(F Ψ , (  M ) G ⋉ X | F Ψ ,  G F Ψ , ξ )] ∈ c kk ∗ G ( X, Z ) . Similar statements fol low in analytic KK . DUALITY AND CORRESPONDENCES 33 Namely , the Lefschetz inv ariant of the KK G ⋉ X ( X × Z X , X )-morphism KK G ⋉ X (Ψ) determined b y Ψ is the class KK G (Lef (Ψ)). F u rthermore, this class is the pushfor- ward under the map (  M ) G ⋉ X | F Ψ : F Ψ → X of the class o f the Dirac opera tor o n the K-or ien ted co incidence manifold F Ψ , twisted by ξ (by a n app eal to the Index Theorem.) W e leave it to the re ader to compute the Lefschetz inv ariant of Ψ in the situation where the transversalit y condition (4.11) fails; in this case it becomes necessa ry to mo dify Θ top using Thom modiﬁca tion and an isotop y as in E xample 3.24. Our computation of the L efsch etz map for s mo oth norma l G -manifolds of ﬁnite t yp e is now complete, in view of ( 4.8) and Theorem 4.4 in combination with Rema rk 4.7. 4.3. Lefsc hetz in v arian ts of se lf-morphisms of X . Let X b e a smooth normal G -manifold of ﬁnite t ype. W e no w consider the composition c kk G ( X, X ) → c kk G ⋉ X ( X × Z X , X ) Lef − − → KK G ( X, Z ) , where the ﬁrst map is t he comp osition of the canonical inﬂation map p ∗ X : c kk G ( X, X ) → c kk G ⋉ X ( X × Z X , X × Z X ) and the map c kk G ⋉ X ( X × Z X , X × Z X ) → c kk G ⋉ X ( X × Z X , X ) of compo sition with the diagona l restrictio n class ∆ X ∈ KK G ⋉ X ( X × Z X , X ) . Recall that the latter is the class of the G ⋉ X - equiv a riant corresp ondence X × Z X δ X ← − − X id − → X where δ X is the diago nal embedding. Let Ψ = ( M , b, f , ξ ) b e a smo oth, G -equiv a riant corresp ondence from X to X . Of course this implies that there is a smo oth normally non-singular map M → Z , by compo sing f : M → X and  X : X → Z . The inﬂation map replaces Ψ by the G ⋉ X -equiv ariant corresp ondence X × Z X id X × Z b ← − − − − − ( X × Z M , ξ ) id × Z f − − − − → X × Z X . The X -structure s a re all o n the ﬁrst v a riable. In order to comp ose (on the right) with the diag onal r estriction class using trans versalit y , we ﬁrst e asily c heck that Theorem 3.23) applies, and deduce that we require that the smo oth maps id X × Z f : X × X M → X × X X and ∆ X : X → X × Z X are transverse in the sense of Theorem 3.23, in the category of G ⋉ X -equiv ariant smo oth maps. They ar e tr ans- verse if and only the smo oth G -map f : M → X is a submersion. If this condition is met, then f : M → X gives M no t just the structure of a smo o th G -manifold, but the structure of a smooth G ⋉ X -manifold, i.e . a bundle of smoo th manifolds o v er X with mo rphisms in G acting by diﬀeomor phisms b etw een the ﬁbres. Co mpos - ing with ∆ X using transversality then yields th e G ⋉ X -equiv arian t corresp ondence X × Z X ( b,f ) ← − − − ( M , ξ ) f − → X where ( b, f )( m ) : = ( b ( m ) , f ( m )). Fina lly , we a pply Theorem 4.12 to obtain the follo wing. Theorem 4. 13. L et X b e a smo oth n ormal ly non-singular G -manifold, let Ψ = ( M , b, f , ξ ) b e a smo oth, G -e quivari ant c orr esp ondenc e fr om X to X such that f is a ﬁbr ewise submersion and such that for every m ∈ M su ch that b ( m ) = f ( m ) the line ar map T m M → T b ( m ) X , ζ 7→ D m b ( ζ ) − D m f ( ζ ) is n on-singular. Then the L ef schetz invariant of Ψ is the class of the smo oth, G - e quiv ariant c orr esp o ndenc e X b ← − (F ′ Ψ , ξ | F ′ Ψ ) → Z 34 HEA TH EM ERSON fr om X to Z , wher e F ′ Ψ : = { m ∈ M | b ( m ) = f ( m ) } with its induc e d K -orientatio n. Example 4.14 . If b : X → X is a smo oth G -equiv ariant ma p, then X b ← − ( X , ξ ) id − → X is a smo oth, zero-dimensional corr esp ondence from X to X . Since id : X → X is obviously a submersion, the transversalit y co ndition a moun ts to saying that for every x ∈ X which is ﬁxed by b , the linear map id − D x b : T x X → T x X is non- singular. This is the classica l condition. The coincidence spa ce is of course the ﬁxed-p oint set of b , suitably K-or ient ed. This means a sign is attac hed to eac h po in t, which can b e chec ked to a gree with the usual ass ignment . Thus the Lefschetz in v a riant is the algebra ic ﬁxed-point set. 4.4. Homolo gical inv arian ts for corresp ondences. In this section we describ e, in the non-equiv arian t case, the pa iring b etw een the index of the Lefschetz inv ari- ant of a Kas parov self-morphism of X , and a K- theory class, in terms of gr aded traces on K-theory . In com bination with the theory of cor resp ondences this yields a generalisation of the classical Lefsc hetz ﬁxed-point for m ula. Recall that the classical Lefsch etz ﬁxed-p oint theorem describ es the algebraic n umber of ﬁxed-p oints of a m ap satisfying a trans versalit y condition, to t he gr aded trace of the induced map on homolo gy , a ‘glo bal’ in v ar iant. The la tter of cour se only ma kes sense when the homolog y g roups of X have ﬁnite rank. Duality a nd the Univ ersal Co eﬃcient Theor em taken together imply this , so in par ticular it is the case if X is a co mpact manif old. All the same remarks hold of course for K-theory as w ell. The r e ader may take al l K - the o ry gr oups to b e tensor e d by Q in the fol lowing . Recall that the K-theory K ∗ ( X ) is a graded ring. This r ing structure is im po rtant for what fo llows. Let L x : K ∗ ( X ) → K ∗ ( X ) denote the additive gro up homomor- phism of K ∗ ( X ) by mult iplication b y x ∈ K ∗ ( X ). F or f ∈ KK ∗ ( C ( X ) , C ( X )), let f ∗ denote the endomorphism o f K-theory induced by f . W e ar e interested in the linear transformation L ( f ) : K ∗ ( X ) → Z , L ( f ) x : = trace s ( f ∗ ◦ L x ) . W e ca ll L ( f ) the L efschetz op er ator o f f . It is a globally deﬁned ob ject, g eneralizing the cla ssical Le fschetz num b er l ( f ) : = trace s ( f ∗ ) o f f in the sense that e v alua ting L ( f ) at the unit [1] ∈ K 0 ( X ) recov ers the Lefschetz n um ber: L ( f )([1]) = l ( f ) . The Lefschetz op erator contains more informa tion; if for example if f is an o dd morphism then ℓ ( f ) = 0 but L ( f ) 6 = 0 except in sp ecial cases. Theorem 4.15. L et X b e a c omp act sp ac e admitting an abstr act dual. L et f ∈ KK ∗  C ( X ) , C ( X )) b e a K asp ar ov morphism. Then (4.16) L ( f ) ξ = h ξ , Lef ( f ) i holds for every ξ ∈ K ∗ ( X ) . By the characteristic-class formulation [2] of the A tiy ah-Singer Index theorem, for ea ch c ompact, smo oth K- oriented manifold M there exists a cohomolog y class I ( M ) on M which w e call the o rientation char acter of M such that Ind( D · ξ ) = Z M I ( M )ch( ξ ) , where D · ξ denotes the class o f the Dira c op era tor on M twisted by ξ . Of course we choose representativ e diﬀerential for ms for I ( M ) and ch( ξ ) b oth ca n b e computed more o r less explicitly from Chern W eil theor y . In com bination with Theorem 4.15 we obtain a lo cal formula for the Lefschetz op era tor of a mor phism represen ted by a corresp ondence, as follo ws. DUALITY AND CORRESPONDENCES 35 Corollary 4. 17. L et X b ← − ( M , ξ ) f − → X b e a smo oth self-c o rr esp ondenc e of X , assume it is of de gr e e d : = dim( M ) − dim ( X ) + dim( ξ ) and denote its class by Ψ . A ssume that the tr ansversali ty a ssumptions in The or em 4.13 ar e met, so that the c o incidenc e sp ac e F ′ Ψ : = { m ∈ M | b ( m ) = f ( m ) } has the structu r e o f a smo o th, K -oriente d manifold of dimension d . Then (4.18) L ( f ) η = Z F Ψ ch  ξ | F ′ Ψ · ( b ∗ η ) | F ′ Ψ  I ( D F Ψ ) , holds, wher e I ( D F ′ Ψ ) is the i ndex char acter of F ′ Ψ , and L ( f ) η : = tra ce s ( f ∗ ◦ L x ) is the L efschetz op er ator applie d to η . In the case where M = X , f = id and b : X → X is a smo oth equiv arian t map in g eneral position, the co incidence manifold F ′ Ψ is a ﬁnite set of p oints, and (4.18) reduces to the traditional Lefsc hetz ﬁxed-point theorem: (4.19) trace s ( b ∗ ) = X x ∈ Fix( b ) sign det(id − D x f ) . Ralf Meyer and I ar e cur rently aiming at an e quiv ar iant version of the ab ov e, but the work is no t yet complete. Let G be compact gro up. Let X b e a compac t G -spac e. Then the G -equiv ariant K-theory K ∗ G ( X )of X is a mo dule ov er the representation ring Rep( G ). The Hatori-Stal lings tr ac e trace Rep( G ) : K ∗ G ( X ) → Rep( G ) is then deﬁned under suitable co nditions. No w assume that X admits a n abstract dual, so that the Lefschetz map is deﬁned. If f ∈ KK ∗ ( C 0 ( X ) , C 0 ( X )) is a morphism, then ca ll t he L efsche tz index ind G ◦ Lef ( f ) ∈ Rep( G ) the pa iring o f the unit clas s [1] ∈ K 0 G ( X ) = KK G ( C , C 0 ( X )) with Lef ( f ) ∈ KK G ( C 0 ( X ) , C ). Our expectation is that the following result holds – we state it as a conjecture since the pro of is no t complete at the time of writing. Conjecture 4.2 0. If f ∈ KK G ∗ ( C 0 ( X ) , C 0 ( X )) , then ind G ◦ Lef ( f ) = trace Rep( G ) ( f ∗ ) , wher e f ∗ denotes the action of f on K ∗ G ( X ) and trace Rep( G ) denotes the Hattori- Stal lings tr ac e. This theorem ca n b e of co urse combined with corres po ndences to achiev e in ter- esting loca l-global equalities of what seem to be r ather subtle in v ariants. The Ha ttori-Stallings trace is d eﬁned for mo dules M ov er rings R whic h a re ﬁnitely pr esente d , i.e. have ﬁnite-length resolutions by ﬁnitely generated pro jective R -mo dules. Therefor e we require this h yp othesis o n the equiv ariant K- theory of X as a module over Rep( G ). This a reas onable ass umption only for compact, connected Lie groups with torsion-free fundamen tal gr oup G . 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Duality, correspondences and the Lefschetz map in equivariant KK-theory: a survey

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