Equivariant K-theory, groupoids and proper actions

EQUIV ARIANT K-THEOR Y, GR OUPOIDS AN D PR OPER ACTIONS JOSE CANT ARERO Abstract. In this pap er w e de fine c o mplex e quiv ariant K -theory for actions of Lie group oids using finite-dimensional v ector bundles. F or a Bredon-compa tible Lie gr o upo id G , this de- fines a p e rio dic cohomolo gy theory on the category of finite G -CW-complexes. W e als o establish an analogue of the completion theore m of Atiy ah and Segal. So me examples are discussed. Key words: K-theory , group oids, prop er actions, completion theorem. Mathematics Sub ject Classification 2010: 19L47, 55R91. 1. Intr oduction The recen t theorem of F reed, Hopkins and T eleman [8, 9, 10, 11] relates the complex equiv aria nt t wisted K -theory of a simply-connected compact Lie group acting on itself by conjugation to the V erlinde algebra. This result links information abo ut the conjugation action with the action of the lo op group on its univ ersal space f o r prop er actions. If we use the language of g roup oids, the t w o asso ciated group oids are lo cally equiv alen t. The in v ariance of orbifold K -theory under Morita equiv alence [1 ] also seems to suggest that t he language o f group oids is an appropriate fr a mew ork to w ork with prop er actions. The complex represen t a tion ring of a compact Lie group G can b e iden tified w ith the G - equiv aria nt complex K -theory of a p oint. Equiv ariant complex K - theory is defined via equi- v arian t comple x bundle s, but this procedure does not giv e a cohomology theory for prop er actions of non-compact Lie groups in general, as sho wn in [19]. Using infinite-dimensional complex G -Hilb ert bundles, Phillips [19 ] constructs a n equiv ar ia n t c ohomology theory for an y second countable lo cally compact gro up G on the category of prop er lo cally compact G -spaces that agrees with equiv arian t K -theory for actions of compact L ie groups. But in some cases it is enough to use finite -dimensional v ector bundles. F or example, L ¨ uc k and Oliv er sho w tha t they suffice for discrete groups [16]. Inspired b y metho ds from that pap er, The author is par tially suppor ted by FEDER/MEC gran t MTM2010-2 0692 . Revised version p ublished a t Journal of K-theory: K-theory and its Applica- tions to Algebra, Geometry and T op ology , V olume 9, Iss ue 3 (2012 ), 475-501. http:/ /jour nals.cambridge.org/abstract_S1865243309998816 . F ull cop yright belo ngs to ISOPP .. 1 EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 2 w e will construct a v ersion of complex equiv a rian t K -theory for actions of a Lie group oid G = ( G 0 , G 1 ) (see the next section for some background on g r oup oids) by using extendable complex equiv arian t bundles, defined b elow. These bundles are finite- dimensional, but are required to satisfy an additiona l conditio n whic h will make sure w e hav e a May er-Vietoris sequence . Definition 1.1. Let X b e a G -space, π : X − → G 0 its anc hor map and V − → X a complex G -v ector bundle. W e say V is extendable if there is a complex G -ve ctor bundle W − → G 0 suc h that V is a direct summand of π ∗ W . The Grothendiec k construction then gives a cohomology theory o n the category o f G - spaces. F o r an y G -space X , K ∗ G ( X ) is a mo dule o ve r K ∗ G ( G 0 ) and the latter can b e iden tified with K ∗ or b ( G ) when G is an orbifo ld. But this theory do e s not necessarily satisfy Bo t t p eri- o dicit y . In fact, it may no t agree with classical equiv ariant K - theory when the action on the space is equiv alen t to the action of a compact Lie group. F or our purp oses w e only need to study G -spaces that ar e built out of G -cells, whic h are compact G -spaces whose G - action is equiv alent to the action o f a compact Lie group on a finite complex. Definition 1.2. A group oid G is Bredon-compatible if g iven an y G -cell U , a ll G -v ector bundles on U are extendable. Bredon-compatibilit y makes sure that G - equiv ariant K -theory agr ees with classical equi- v arian t K - theory on the G -cells. This condition also implies Bott p erio dic it y for finite G - CW-pairs, prov ed by a May er-Vietoris arg umen t and induction o v er the cellular structure. These a re tw o of the results in Section 3 neede d to pro ve Theorem 3.22, whic h w e repro duc e here: Theorem 1.3. If G is a Br e don-c omp atible Lie gr oup oid, the gr oups K n G ( X , A ) defin e a Z / 2 -gr ade d multiplic ative c ohom o lo gy the ory on the c ate gory of finite G -CW-p airs. The purp ose of this v ersion of equiv arian t K -theory for actions of Lie group o ids is t he existence of a completion theorem for prop er actions of Lie groups. There are more general v ersions o f equiv arian t K -theory for actions of lo cally compact group o ids a v ailable in the literature using C ∗ -algebras a nd K K -theory , suc h as the ones dev elop ed in [7] and [14], but no completion theorems are expected to hold in suc h generalit y . Emerson and Mey er cons ider the question o f when equiv arian t K - t heory for a ctions o f lo cally compact group oids o n lo cally compact spaces can b e defined using finite-dimensional v ector bundles in [7]. F or instance , Theorem 6.4 in [7] sho ws that equiv ariant K -theory defined in terms of equiv arian t v ector bundles is isomorphic to equiv arian t K -theory define d EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 3 using C ∗ -algebras for actions of a second countable, lo cally compact, Hausdorff group o id G with Haa r system on a prop e r, G -compact, second countable G -space X as long as the C ∗ - algebra of the action group oid has an approximate unit of pro jections. This last condition holds if and only if for eac h x ∈ X , and each irreducible represen tation V of the stabilizer of x , there is a G -v ector bundle on X whose fib er ov er x con tains V . W e sho w in Section 3 t ha t equiv aria nt K -theory defined using extendable ve ctor bundles coincides with the v ersions defined in D efinitions 2.2 and 2.3 of [7]. The follo wing theorem corresp onds to Theorem 3.23: Theorem 1.4. L et G b e a Br e don-c omp atible Lie gr oup oid such that for e ach x ∈ G 0 , an d e ach irr e ducible r epr esentation V of the stabilizer of x , ther e is a G -ve ctor bund le on G 0 whose fib er over x c ontain s V . Th en e quivariant K -the ory define d in terms o f extendab le G -ve ctor bund les c oincides with e quivariant K -the ory define d in terms of C ∗ -algebr as and K K -the ory on the c ate gory of fi n ite G -CW-c omplexes. W e intro duce a univ ersal G -space E G a s the limit of a sequence of free G -spaces E n G as in the case of compact Lie gro ups. The quotien t of E G b y the G -a ctio n is B G , the classifying space of G . W e f o rm t he fib ered pro d uct X × π E n G ov er G 0 and prov e a generalization of the completion theorem of A tiy ah and Sega l [4] when G is finite. Definition 1.5. A Lie group oid G is finite if G 0 is a finite G -CW-complex and the spaces B n G = E n G / G are compact. This v ersion of G -equiv ar ian t K -theory is in v arian t under we ak equiv alence. The naturalit y of completion maps a nd the A tiyah-Segal completion t heorem for a ctions o f compact Lie groups imply the existence of a completion theorem for G -cells. A sp ectral sequence argumen t is t hen used to sho w that the completion map constructed at the end of Section 4 is an isomorphism. This result corresponds to Theorem 5.6, the main goal of Section 5: Theorem 1.6. L et G b e a Br e don -c omp atible, finite Lie gr oup oid and X a fi nite G -CW- c omplex. The n we h a v e an is o morphism of pr o-rings { K ∗ G ( X ) / I n G K ∗ G ( X ) } − → { K ∗ G ( X × π E n G ) } . In p articular, for X = G 0 we obtain an isomorphi s m of pr o-rings { K ∗ G ( G 0 ) /I n G K ∗ G ( G 0 ) } − → { K ∗ ( B n G ) } . Finally , Section 6 studies the p osibilit y of a pplying these result to prop er actions of Lie groups whic h are not necessarily compact. These actions hav e an asso ciated Lie group oid whic h enco des the action of the group on its classifying space for pro p er actions. Actions EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 4 of finite gro ups and compact Lie gro ups and prop er actions of discrete groups, pro-discrete groups, almost compact g roups and matrix g roups give rise to Bredon-compatible group oids . The finiteness condition is not auto matic, but when it holds a completion theorem follows. Ac kno wledgemen t s. The author w o uld lik e to thank the referee, whose commen ts and suggestions really help ed improv e the quality and readibilit y of t he pap er. 2. Back g r ound on groupoids In this section w e review some basic facts ab out gr o up oids. All this material can b e found in [1] and [17]. Definition 2.1. A top olog ical group oid G consists of a space G 0 of ob jects a nd a space G 1 of a r ro ws, together with five contin uous structure maps, listed b elow . • The source map s : G 1 → G 0 assigns to eac h arro w g ∈ G 1 its source s ( g ). • The targ et map t : G 1 → G 0 assigns to eac h ar r ow g ∈ G 1 its target t ( g ). F or tw o ob jects x , y ∈ G 0 , one writes g : x → y to indicate that g ∈ G 1 is an a rro w with s ( g ) = x a nd t ( g ) = y . • If g and h are ar ro ws with s ( h ) = t ( g ), o ne can form t heir comp osition hg , with s ( hg ) = s ( g ) and t ( hg ) = t ( h ). The comp osition map m : G 1 × s,t G 1 → G 1 , defined b y m ( h, g ) = hg , is th us defined on the fib ered pro duct G 1 × s,t G 1 = { ( h, g ) ∈ G 1 × G 1 | s ( h ) = t ( g ) } and is required to be asso ciativ e. • The unit map u : G 0 → G 1 whic h is a tw o-sided unit for the comp osition. This means that su ( x ) = x = tu ( x ), and that g u ( x ) = g = u ( y ) g for all x , y ∈ G 0 and g : x → y . • An in v erse map i : G 1 → G 1 , written i ( g ) = g − 1 . Here, if g : x → y , then g − 1 : y → x is a tw o-sided in vers e for the comp osition, whic h means that g − 1 g = u ( x ) and g g − 1 = u ( y ). Definition 2.2. A Lie group oid is a top ological g roup oid G for whic h G 0 and G 1 are smo o th manifolds, and suc h that the structure maps a r e smo oth. F urt hermore, s and t are required to b e submersions so tha t the domain G 1 × s,t G 1 of m is a smooth manifold. Example 2.3. Supp ose a Lie group K acts smo othly on a manifold M . One defines a L ie group oid K ⋊ M by ( K ⋊ M ) 0 = M and ( K ⋊ M ) 1 = K × M , with s the pro jection and t the a ction. Comp o sition is defined from t he multiplication in the group K . This group oid is called the action group oid. EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 5 Definition 2.4. Let G b e a Lie g roup oid. F or a p oint x ∈ G 0 , the set of all arrow s from x to itself is a Lie group, denoted by G x and called the isotro py group a t x . The set ts − 1 ( x ) of targets of ar r ows out of x is called the or bit of x . The quotien t | G | of G 0 consisting of all the orbits in G is called the orbit space. Con v ersely , w e call G a group oid presen tation of | G | . Definition 2.5. A Lie gro up oid G is prop er if ( s, t ) : G 1 → G 0 × G 0 is a prop e r map. Note that in a prop er Lie gro up oid, eve ry isotropy group is compact. Definition 2.6. Let G and H b e Lie gr o up oids. A strict homomorphism φ : H → G consists of t w o smo oth maps φ : H 0 → G 0 and φ : H 1 → G 1 that comm ute with a ll the structure maps f o r the t w o gr o up oids. Definition 2.7. A strict homomorphism φ : H → G b etw een Lie group o ids is called an equiv alence if: • The map tπ 1 : G 1 × s,φ H 0 → G 0 is a surjectiv e submersion, where G 1 × s,φ H 0 = { ( g , y ) ∈ G 1 × H 0 | s ( g ) = φ ( y ) } and π 1 : G 1 × s,φ H 0 → G 1 is the pro jection to the first factor. • The square H 1 φ / / ( s,t )   G 1 ( s,t )   H 0 × H 0 φ × φ / / G 0 × G 0 is a fib ered pro duct of manifolds. The first conditio n implies tha t eve ry ob ject x ∈ G 0 can b e connected b y an a rro w g : φ ( y ) → x to an ob ject in the image of φ , that is, φ is essen tially surjectiv e as a functor. The second condition implies that φ induces a diffeomorphism H 1 ( y , z ) → G 1 ( φ ( y ) , φ ( z )) from the space of all arr ows y → z in H 1 to the space of all arr ows φ ( y ) → φ ( z ) in G 1 . In particular φ is full and fa ithful as a functor. A strict homomorphism φ : H → G induces a contin uous map | φ | : | H | → | G | . Moreov er, if φ is an equiv alence, | φ | is a homeomorphism. Definition 2.8. A lo cal equiv alence φ : H − → G is an equiv alence with the additio nal prop ert y that each g 0 ∈ G 0 has a neighbourho od U a dmitting a lift to ˜ H 0 in the diagram EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 6 ˜ H 0   / / H 0 φ 0   G 1 s   t / / G 0 U G G ✎ ✎ ✎ ✎ ✎ ✎ ✎ ✎ / / G 0 , in whic h the square is a pullback square. Definition 2.9. Tw o Lie group oids G and G ′ are w eakly equiv alen t if there exists a third group oid H and t w o lo cal equiv alences G ← H → G ′ . 3. Groupoid actions and K-theor y Definition 3.1. Let G b e a gro up oid. A (right) G -space is a manifold E equipp ed with an action of G . Suc h an action is giv en by t w o maps π : E → G 0 (called the anchor map) and µ : E × G 0 G 1 → E . The latter map is defined on pa ir s ( e, g ) with π ( e ) = t ( g ) and written µ ( e, g ) = e · g . They m ust satisfy π ( e · g ) = s ( g ) , e · 1 π ( e ) = e and ( e · g ) · h = e · ( g h ). The space of orbits of this action is denoted by E / G . Example 3.2. If G is a group oid, G 1 is a G -space with the anc hor map giv en b y s : G 1 → G 0 , so that G 1 × G 0 G 1 = G 1 × s,t G 1 . The a ctio n map is the comp osition m : G 1 × s,t G 1 → G 1 from Definition 2.1. Consider the fo llo wing maps: α : G 1 / G → G 0 α ([ g ]) = t ( g ) β : G 0 → G 1 / G β ( x ) = [ u ( x )], where [ g ] denotes the orbit of g ∈ G 1 . These maps define a homeomorphism b et w een G 1 / G and G 0 . Example 3.3. Let M b e a G -space. W e can construct t he action gr oup oid H = G ⋊ M whic h has space o f ob jects M and morphisms M × G 0 G 1 . This group oid generalizes the earlier EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 7 notion of action g r oup oid for a group a ction and the structure maps are fo rmally the same as in that case. Definition 3.4. Let G b e a group oid and let X , Y b e G -spaces with a nchor maps π X and π Y , resp ectiv ely . A map f : X − → Y is G - equiv aria n t if it satisfies π Y ◦ f = π X and f ( x · g ) = f ( x ) · g for all g ∈ G 1 with t ( g ) = π X ( x ). • G 0 is a final ob ject in the category of G -spaces with the action give n b y e · g = s ( g ) a nd anc hor map giv en by the iden tity . The space of or bits for this a ction is | G | (Definition 2.4). • If X and Y are G -spaces, the fib ered pro duct o ver G 0 , X × π Y = { ( x, y ) | π ( x ) = π ( y ) } b ecomes a G -space with co ordinate-wise action. In part icular X × π G 0 = X . • Similarly , if X is a G -space and Y is any other space, X × Y is a G -space with trivial action on the second factor. In fact, X × Y = X × π ( Y × G 0 ). All v ector bundles in this pap er are complex v ector bundles. Definition 3.5. Let G b e a group oid. A G -v ector bundle on a G -space X is a v ector bundle p : E − → X suc h that E is a G -space with fib erwise linear action and p is a G -equiv ariant map. Definition 3.6. Let X b e a G -space and V − → X a G -ve ctor bundle. W e sa y that V is G -extendable if there is a G - v ector bundle W − → G 0 suc h that V is a direct summand of π ∗ W . • D irect sum of G -extendable vector bundles induces an op eration on the set o f isomor- phism classes of G -extendable v ector bundles on X , making this set a monoid. W e can also tensor G -extendable vec tor bundles. • The pullbac k of a G -extendable ve ctor bundle b y a G -equiv a r ia n t map is a G -extendable v ector bundle. • All G -v ector bundles o n G 0 are G -extendable. When G is an orbifo ld, G -vec tor bundles on G 0 corresp ond to G -v ector bundles o ver the orbifo ld G a ccording to Definition 2.25 of [1 ]. Definition 3.7. Let X b e a G -space. Then let V ect G ( X ) b e the set of isomorphism classes of extendable G -v ector bundles on X a nd K G ( X ) = K (V ect G ( X )), where K ( A ) denotes the Grothendiec k group of a monoid A . W e call K G ( X ) the extendable G -equiv arian t K -theory of X . EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 8 Let X b e a G -space and A a closed subspace of X whic h is G -in v ariant. W e can now define the exten dable K -groups as in D efinition 3.1 of [16]: K − n G ( X ) = Ker [ K G ( X × S n ) i ∗ − → K G ( X )] , K − n G ( X , A ) = Ker[ K − n G ( X ∪ A X ) j ∗ 2 − → K − n G ( X )] , where i : X → X × S n is the inclusion g iven b y fixing a p oint in S n and j 2 : X → X ∪ A X is one of the maps from X to the pushout. W e equip X × S n with a G -action by taking as the anc hor map the comp o sition of the pro jection onto the first co ordinate and the anc hor map for X . Then w e let the group oid act on t he first co o rdinate. The anch or map for X ∪ A X is the o nly map to G 0 making the pushout dia g ram comm utativ e. The action is induced by the action of G on X . F rom no w on G will b e a Lie group oid. The fo llo wing lemma follows easily from the definitions: Lemma 3.8. L et ( X , A ) b e a G -p air. Supp ose that X = ∐ i ∈ I X i , the disjoint union of op en G -invariant subsp ac es X i , and set A i = A ∩ X i . The n ther e is a n atur al isomorphis m K − n G ( X , A ) − → Y i ∈ I K − n G ( X i , A i ) . Definition 3.9. A smo o th left Haar sys tem for a Lie group oid G is a family { λ a | a ∈ G 0 } , where eac h λ a is a p o sitiv e, regular Borel measure on the manifold t − 1 ( a ) suc h that: • If ( V , ψ ) is an op en c hart of G 1 satisfying V ∼ = t ( V ) × W , and if λ W is the Leb e sgue measure on R k restricted to W , then for each a ∈ t ( V ) , the measure λ a ◦ ψ is equiv alen t to λ W , and the map ( a, w ) 7→ d ( λ a ◦ ψ a ) /dλ W ( w ) b elong s to C ∞ ( t ( V ) × W ) and is strictly p os itiv e. • F or an y x ∈ G 1 and f ∈ C ∞ c ( G 1 ), w e ha v e Z t − 1 ( s ( x )) f ( xz ) dλ s ( x ) ( z ) = Z t − 1 ( t ( x )) f ( y ) dλ t ( x ) ( y ) . Prop osition 3.10. Every Lie gr oup oid ad mits a sm o o th left Haar system. The pro of can b e found in [18], Theorem 2.3.1. Note that w e can use a smo oth left Haar system to construct equiv arian t sections of G - v ector bundles f r o m nonequiv arian t sections. More precisely , let f : X → E b e a nonequiv ari- an t section of a G -ve ctor bundle E → X . If w e define h ( x ) = R t − 1 ( π X ( x )) f ( xg ) g − 1 dλ π X ( x ) ( g ), EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 9 then h ( xk ) = Z t − 1 ( π X ( xk )) f ( xk g ) g − 1 dλ π X ( xk ) ( g ) = Z t − 1 ( s ( k )) f ( xk g ) g − 1 dλ s ( k ) ( g ) = Z t − 1 ( s ( k )) f ( xk g ) g − 1 k − 1 dλ s ( k ) ( g ) · k = Z t − 1 ( t ( z )) f ( xz ) z − 1 dλ t ( z ) ( z ) · k = Z t − 1 ( π X ( x )) f ( xz ) z − 1 dλ π X ( x ) ( z ) · k = h ( x ) · k . Corollary 3.11. I f f 0 , f 1 : ( X , A ) − → ( Y , B ) a r e G -homotopic G -map s b etwe en G -p airs, then f ∗ 0 = f ∗ 1 : K − n G ( Y , B ) − → K − n G ( X , A ) for al l n ≥ 0 . Pr o of. Since w e can construct equiv arian t sections from nonequiv aria n t sections, w e can use the same a rgumen t f r o m Prop o sitions 1.1, 1.2 a nd 1.3 of [24]. The relative case follows fr om the definition.  The following lemma is a straightforw ard generalization of Lemma 1.5 in [16 ]: Lemma 3.12. L et φ : ( X 1 , X 0 ) − → ( X, X 2 ) b e a ma p of G -sp ac es, set φ 0 = φ | X 0 , a n d assume that X ∼ = X 2 ∪ φ 0 X 1 . L et p 1 : E 1 − → X 1 and p 2 : E 2 − → X 2 b e G -extendab l e ve ctor bund les, let φ 0 : E 1 | X 0 − → E 2 b e an isomorphism of G -ve ctor bund les c overing φ 0 , and set E = E 2 ∪ φ 0 E 1 . Then p = p 1 ∪ p 2 : E − → X is a G -extendable ve ctor bund le o v e r X . Pr o of. W e ha v e to sho w t hat p : E → X is lo cally trivial. Since E 1 is lo cally trivial, so is E | X − X 2 ∼ = E | X 1 − X 0 . So it remains to find a neigh b o urho o d of X 2 o ver which E is lo cally trivial. Cho ose a closed neigh b ourhoo d W 1 of X 0 in X 1 for whic h there is a strong deformation retraction r : W 1 → X 0 . By the homotopy in v ariance for nonequiv ariant v ector bundles ov er para compact spaces, r is co v ered by an isomorphism of vec tor bundles ¯ r : E 1 | W 1 → E 0 whic h extends ¯ i 1 . Set W = X 2 ∪ φ 0 W 1 . Then ¯ r extends, via the pushout, to a map of v ector bundles E | W → E 2 whic h extends ¯ i 2 and hence E | W is lo cally trivial.  The next lemma is a fundamental piece in the pro o f of the existence of a May er-Vietoris long exact sequence for G -equiv ariant K -theory . It w as inspired by Lemma 3.7 of [16]. EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 10 Lemma 3.13. L et φ : X − → Y b e a G -e quivariant map and let E ′ − → X b e a G -extend a ble ve ctor bund le. Then ther e is a G -extendable ve ctor bund le E − → Y such that E ′ is a summand of φ ∗ E . Pr o of. Consider π : Y − → G 0 and π φ : X − → G 0 . Since E ′ is extendable, there is a G -v ector bundle V on G 0 suc h that E ′ is a direct summand of ( π φ ) ∗ V . Let E = π ∗ V . Then E is a G -v ector bundle on Y and it is the pullback o f a n extendable G -v ector bundle, hence it is extendable. And w e hav e that E ′ is a direct summand of ( π φ ) ∗ V = φ ∗ E .  The existen ce of a Ma ye r-Vietoris sequence is sho wn next. The pro of is an adaptation of Lemma 3 .8 in [16]. Lemma 3.14. L et A i 1 / / i 2   X 1 j 1   X 2 j 2 / / X b e a pushout squar e of G -sp ac es, w h er e i 1 and i 2 ar e c ofib r ations. Then ther e is a natur al exact se quenc e, infinite to the left . . . d − n − 1 − → K − n G ( X ) j ∗ 1 ⊕ j ∗ 2 − → K − n G ( X 1 ) ⊕ K − n G ( X 2 ) i ∗ 1 − i ∗ 2 − → K − n G ( A ) d − n − → . . . . . . − → K − 1 G ( A ) d − 1 − → K 0 G ( X ) j ∗ 1 ⊕ j ∗ 2 − → K 0 G ( X 1 ) ⊕ K 0 G ( X 2 ) i ∗ 1 − i ∗ 2 − → K 0 G ( A ) . Pr o of. W e first sho w that the sequence (1) K G ( X ) j ∗ 1 ⊕ j ∗ 2 − → K G ( X 1 ) ⊕ K G ( X 2 ) i ∗ 1 − i ∗ 2 − → K G ( A ) is exact, and hence the long sequenc e in the statemen t of the lemma is exact at K − n G ( X 1 ) ⊕ K − n G ( X 2 ) for all n . Clearly the comp osite is zero. So fix an elemen t ( α 1 , α 2 ) ∈ Ker( i ∗ 1 − i ∗ 2 ). By the previous lemma, w e can add an elemen t of the for m ([ j ∗ 1 E ′ ] , [ j ∗ 1 E ′ ]) fo r some G -v ector bundle E ′ → X , and arrange that α 1 = [ E 1 ] and α 2 = [ E 2 ] fo r some pair of G -ve ctor bundles E k → X k . Then i ∗ 1 E 1 and i ∗ 2 E 2 are stably isomorphic, and after adding the restrictions of another bundle o v er X , w e can arra nge that i ∗ 1 E 1 ∼ = i ∗ 2 E 2 . Lemma 3.12 now applies to sho w that there is a G -v ector bundle E o v er X such that j ∗ k E ∼ = E k for k = 1 , 2, and hence that ( α 1 , α 2 ) = ([ E 1 ] , [ E 2 ]) ∈ Im( j ∗ 1 ⊕ j ∗ 2 ). Assume now that A is a retract of X 1 . W e claim that in this case the ma p (2) Ker[ K G ( X ) j ∗ 2 − → K G ( X 2 )] j ∗ 1 − → Ker[ K G ( X 1 ) i ∗ 1 − → K G ( A )] EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 11 is an isomorphism. It is surjectiv e b y the exactness of (1). So fix an elemen t [ E ] − [ E ′ ] ∈ Ker( j ∗ 1 ⊕ j ∗ 2 ). T o simplify t he notation, w e write E | A = i ∗ 2 j ∗ 2 E and E | X i = j ∗ i E for i = 1 , 2. Let p 1 : X 1 → A b e a retraction, and let p : X → X 2 b e its extension to X . By the previous lemma, w e can arrange that E | X k ∼ = E ′ | X k for k = 1 , 2. Applying the same lemma t o the retraction p : X → X 2 , w e o bta in a G -vec tor bundle F ′ → X 2 suc h that E ′ is a summand of p ∗ F ′ . Stabilizing a g ain, w e can assume that E ′ ∼ = p ∗ F ′ and hence that F ′ ∼ = E ′ | X 2 and E ′ | X 1 ∼ = p ∗ 1 ( F ′ | A ) ∼ = p ∗ 1 ( E ′ | A ). Fix isomorphisms ψ k : E | X k → E ′ | X k co vering the iden tity on X . The automorphism ( ψ | A ) ◦ ( ψ 1 | A ) − 1 of E ′ | A pulls back, under p 1 , to an automorphism φ of E ′ | X 1 . By r eplacing ψ 1 b y φ ◦ ψ 1 w e can arrange that ψ 1 | A = ψ 2 | A . Then ψ 1 ∪ ψ 2 is an isomorphism from E to E ′ , and this pro ve s the exactness. No w for eac h n ≥ 1, K − n G ( A ) = Ker[ K G ( A × S n ) → K G ( A )] ∼ = Ker[ K G ( X ∪ A × pt ( A × S n )) incl ∗ − → K G ( X )] ∼ = Ker[ K G (( X 1 × D n ) ∪ A × S n − 1 ( X 2 × D n )) ( − ,pt ) ∗ − → K G ( X )] , the last step since ( X 1 × pt ∪ A × D n ) is a strong defo r ma t ion retract of X 1 × D n . Denote Y = ( X 1 × D n ) ∪ A × S n − 1 ( X 2 × D n ) and define d − n : K − n G ( A ) → K − n +1 G ( X ) to be the homomorphism whic h mak es the following diagra m comm ute: 0 / / K − n G ( A ) / / d − n   K G ( Y ) ( − ,pt ) ∗ / / incl ∗   K G ( X ) / / I d   0 0 / / K − n +1 G ( X ) / / K G ( X × S n − 1 ) ( − ,pt ) ∗ / / K G ( X ) / / 0 . W e hav e already sho wn that the desired lo ng sequence is exact at K − n G ( X 1 ) ⊕ K − n G ( X 2 ) for all n . Let us denote Z = ( X 1 × D n ) ∐ ( X 2 × D n ) and W = ( X 1 × D n ) ∪ A × pt ( X 2 × D n ). T o pro ve exactness at K − n +1 G ( X ) and K − n G ( A ) for any n ≥ 1, apply the exactness of (2) to the follo wing split inclusion of pushout squares: X 1 ∐ X 2 / /   X 1 ∐ X 2   ( X 1 ∐ X 2 ) × S n − 1 / /   Z   X / /   X incl / /   X × S n − 1 / /   Y   X / / X X / / W . EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 12 The upp er pair of squares induces a split surjection of exact sequence whose ke rnel yields the exactne ss of the long sequenc e at K − n +1 G ( X ). And since Ker[ K G ( W ) → K G ( X )] ∼ = Ker h K G ( Z ) → K G ( X 1 ∐ X 2 ) i ∼ = K − n G ( X 1 ) ⊕ K − n G ( X 2 ) b y (2), the lo wer pair of squares induces a split surjection of exact sequence s whose k ernel yields t he exactness of the long sequence at K − n G ( A ).  W e now consider pro ducts on K ∗ G ( X ) and o n K ∗ G ( X , A ). W e follow the analogous con- struction from Section 3 of [16]. T ensor pro ducts of G -extendable v ector bundles mak es K G ( X ) into a commu tativ e ring, and all induced maps f ∗ : K G ( Y ) − → K G ( X ) are ring homomorphisms. F or eac h n, m ≥ 0, K − n − m G ( X ) ∼ = Ker[ K − m G ( X × S n ) − → K − m G ( X )] = = Ker[ K G ( X × S n × S m ) − → K G ( X × S n ) ⊕ K G ( X × S m )] , where the first isomorphism f ollo ws fro m the usual Ma yer-Vietoris sequences . Hence the comp osition K G ( X × S n ) ⊗ K G ( X × S m ) p ∗ 1 ⊗ p ∗ 2 − → K G ( X × X × S n × S m ) − → K G ( X × S n × S m ) restricts to a homomorphism K − n G ( X ) ⊗ K − m G ( X ) − → K − n − m G ( X ) . By applying the ab o v e definition with n = 0 o r m = 0, the multiplicativ e iden tit y f or K G ( X ) is seen to b e a n iden tit y f or K ∗ G ( X ). Asso ciativit y of t he graded pro duct is clear and g r a ded comm utativit y follows up on showing that comp osition with a degree − 1 map S n → S m induces m ultiplication by − 1 o n K − n ( X ). This pro duc t mak es K ∗ G ( X ) in to a graded ring. Clearly , f ∗ : K ∗ G ( Y ) − → K ∗ G ( X ) is a ring homomorphism for an y G -map f : X − → Y . This makes K ∗ G ( X ) in t o a K ∗ G ( G 0 )-algebra, since G 0 is a final ob ject in the category of G -spaces. It r emains to prov e Bott p erio dicit y . Recall that ˜ K ( S 2 ) = Ker[ K ( S 2 ) − → K ( pt )] ∼ = Z , and is generated by the Bot t elemen t B ∈ ˜ K ( S 2 ), the elemen t [ S 2 × C ] − [ H ] ∈ ˜ K ( S 2 ), where H is the canonical complex line bundle ov er S 2 = C P 1 . F o r an y G - space X , t here is an obvious pairing K − n G ( X ) ⊗ ˜ K ( S 2 ) ⊗ − → Ker[ K − n G ( X × S 2 ) − → K − n G ( X )] ∼ = K − n − 2 G ( X ) induced b y external tensor pro du ct of bundles. Ev aluation at the Bott elemen t now defines a homomor phism b = b ( X ) : K − n G ( X ) − → K − n − 2 G ( X ) , EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 13 whic h b y construction is natural in X . And this extends to a homomorphism b = b ( X , A ) : K − n G ( X , A ) − → K − n − 2 G ( X , A ) defined for any G -pair ( X , A ) and all n ≥ 0. Definition 3.15. An n -dimensional G -cell is a space o f the form D n × U where U is a compact G -space suc h that G ⋊ U is weakly equiv alen t to a n action group o id corresp onding to a n action of a compact Lie g roup G on a finite G -CW-complex. Definition 3.16. A G -CW-complex X is a G -space together with an G -inv ariant filtration ∅ = X − 1 ⊆ X 0 ⊆ X 1 ⊆ . . . ⊆ X n ⊆ . . . ⊆ ∪ n ≥ 0 X n = X suc h tha t X n is obtained from X n − 1 for eac h n ≥ 0 b y atta ching equiv ariant n -dimensional cells and X carries the colimit t op ology with resp ect to this filtratio n. That is, for eac h n ≥ 0 there is a collection o f n - cells { U i × D n | i ∈ I n } for some index set I n , along with attac hing maps q n i : U i × S n − 1 → X n − 1 , Q n i : U i × D n → X n , and a G -pushout ` i ∈ I n U i × S n − 1 ` i ∈ I n q n i − − − − → X n − 1   y   y ` i ∈ I n U i × D n ` i ∈ I n Q n i − − − − → X n . F urthermore, w e sa y that X is a finite G -CW-complex if it is constructed with a finite n umber of G -cells, and similarly , ( X , A ) is a finite G -CW-pair if X can b e constructed from A by attac hing a finite n umber of G -cells. Definition 3.17. A group oid G is Br edon- compatible if given an y G -cell U , all G -vec tor bundles on U ar e extendable. Example 3.18. An example o f a Bredon-compatible group oid is G = G ⋊ M , where G is a compact Lie group a nd M is a finite G -CW-complex. A G -cell U is a finite G -CW-complex with an equiv arian t map to M and G -vec tor bundles on U are just G - v ector bundles. By Prop osition 2.4 in [24], for an y G - vector bundle A on U , there is another G -vec tor bundle B suc h that A ⊕ B is a trivial bundle, that is, the pullbac k of a G - v ector bundle V o v er a p oint. Consider the only map from M to a p o int. The pullbac k of V ov er this ma p is a G -v ector bundle on M . If w e pull it back to U we reco ver A ⊕ B and therefore G is Bredon-compatible. EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 14 Prop osition 3.19. S upp os e that F : G − → H is a lo c al e quivalenc e. The n the pul lb ack functor F ∗ : { Fib er bund les on H } − → { F i b er bund les on G } is an e q uiva l e nc e of c ate gories. Pr o of. See Prop osition A.1 8 of [9].  Corollary 3.20. I f G i s Br e don-c omp atible and U is a G -c el l, then K ∗ G ( U ) ∼ = K ∗ G ( M ) for some c omp act Lie gr oup G and a finite G -C W-c om plex M . Theorem 3.21. If G is Br e don-c omp atible, the Bott h o momorphism b = b ( X , A ) : K − n G ( X , A ) − → K − n − 2 G ( X , A ) is an i s omorphism for any finite G -CW-p air ( X , A ) and al l n ≥ 0 . Pr o of. Assume first that X = Y ∪ φ ( U × D m ) where U × D m is a G -cell. Assume inductiv ely that b ( Y ) is an isomorphism. Since K − n G ( U × S m − 1 ) ∼ = K − n G ( M × S m − 1 ) a nd K − n G ( U × D m ) ∼ = K − n G ( M × D m ), the Bott homomorphisms b ( U × S m − 1 ) and b ( U × D m ) are isomorphisms b y the equiv ariant Bot t p erio dicity theorem for actions o f compact Lie groups. The Bott map is natural and compatible with the b oundary o p erators in the May er-Vietoris sequence for Y , X , U × S m − 1 and U × D m and so b ( X ) is a n isomorphism b y the 5-lemma. The pro of that b ( X, A ) is an isomor phism follo ws immediately from the definitions of the relativ e groups.  Based on the Bott isomorphism w e just pro v ed, w e can no w redefine for a ll n ∈ Z K n G ( X , A ) = ( K 0 G ( X , A ) if n is ev en K − 1 G ( X , A ) if n is o dd. F or any finite G -CW-pair ( X, A ), define the b oundary op erator δ n : K n G ( A ) − → K n +1 G ( X , A ) to b e δ : K − 1 G ( A ) − → K 0 G ( X , A ) if n is o dd, and to b e the comp osite K 0 G ( A ) b − → K − 2 G ( A ) δ − 2 − → K − 1 G ( X , A ) if n is ev en. W e collect no w all the information w e ha ve a b out G -equiv ariant K -t heory in the followin g theorem: Theorem 3.22. If G is a Br e don- c o m p atible Lie gr oup oid, the gr oups K n G ( X , A ) define a Z / 2 -gr ade d multiplic ative c ohom o lo gy the ory on the c ate gory of finite G -CW-p airs. EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 15 There are more general vers ions of equiv ar ia n t K - t heory for actions of lo cally compact group oids a v ailable in the lit era t ure using C ∗ -algebras a nd K K -theory , suc h a s the o nes dev elop ed in [7] and [14], but no completion theorems a re exp ected to ho ld for them. F or example, Emerson and Mey er define tw o v ersions o f equiv ariant K - theory for actions of a second coun ta ble, lo cally compact, Hausdorff group oid G in [7]. The first definition uses the C ∗ -algebra of the action group oid and it is giv en b y C K ∗ G ( X ) = K ∗ ( C ∗ ( G ⋊ X )). W e use this no tation to av oid confusion with our version of G -equiv arian t K -theory . The second definition is a Kasparov type K -theory , called G - equiv aria n t represen table K -theory and giv en by RK ∗ G ( X ) = K K G ⋊ X ∗ ( C 0 ( X ) , C 0 ( X )). These definitions are extended to pairs and sho wn to be cohomolo gy theories on a suitable category of G -spaces. They ar e isomorphic if X is G -compact, that is, if X/ G is compact. Emerson and Mey er consider the question of when equiv arian t K -theory for gro up oid actions can b e defined using finite-dimensional v ector bundles. Let G b e a second coun table, lo cally compact, Hausdorff group o id with Haar system, X a prop er, G -compact, second coun table G - space and A a closed G -in v arian t subset of X . Assume that for eac h x ∈ X , and eac h irreducible represen tat io n V of t he stabilizer of x , there is a G -v ector bundle o n X whose fib er o v er x con ta ins V . Note that since the anc hor ma p π : X → G 0 is equiv arian t, it is sufficien t that this condition holds fo r the a ction of G on G 0 . Then Theorems 6.4 and 6.14 in [7] show that equiv ariant K -t heory defined in terms o f G -v ector bundles V K ∗ G ( X , A ) is isomorphic to C K ∗ G ( X , A ) a nd RK ∗ G ( X , A ). If moreov er, G is a Bredon-compatible Lie group oid and U is a G -cell, all G -v ector bundles on U are extendable and therefore K ∗ G ( U ) ∼ = V K ∗ G ( U ). Assume that for each x ∈ G 0 , and eac h irreducible represen tation V of the stabilizer of x , there is a G -ve ctor bundle on G 0 whose fib er ov er x con ta ins V , so that V K ∗ G ( Y ) = C K ∗ G ( Y ) for any prop e r, G -compact, second coun table G -space Y . The same argument used in the pro of o f Theorem 3.21 implies that K ∗ G ( X ) ∼ = C K ∗ G ( X ) ∼ = RK ∗ G ( X ) for all finite G -CW-complexes . Therefore w e ha v e prov ed the follo wing theorem: Theorem 3.23. L et G b e a Br e don-c omp atible Lie gr oup oid such that for e ach x ∈ G 0 , and e ach irr e ducible r epr esentation V of the stabilizer of x , ther e is a G -ve ctor bund le on G 0 whose fib er over x c ontain s V . Th en e quivariant K -the ory define d in terms o f extendab le G -ve ctor bund les c oincides with e quivariant K -the ory define d in terms of C ∗ -algebr as and K K -the ory ( [7] , D efinitions 2.2 and 2.3) on the c ate gory of finite G -CW-c omp lexes. The condition o n G 0 is satisfied, for instance, b y o r bifolds (Example 6.17 in [7 ]) a nd by the action group oids asso ciated to certain actions of lo c ally compact groups G o n prop er G -compact spaces (Theorem 6.15 in [7]). EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 16 4. The completion map Giv en a Lie gro up oid G , w e can asso ciate an imp ortan t top olo gical space to it , namely its classifying space B G . F or n ≥ 1 , let G n b e the iterated fib ered pro duct G n = { ( g 1 , . . . , g n ) | g i ∈ G 1 , s ( g i ) = t ( g i +1 ) for i = 1 , . . . , n − 1 } . T ogether with the space o f ob jects G 0 , the spaces G n ha ve a simplicial structure called the nerv e of G . Here we are really thinking of G as a to p ological category . F ollowing the usual con ve n tio n, w e define face op erators d i : G n → G n − 1 for i = 0 , ..., n , giv en b y d i ( g 1 , . . . , g n ) =      ( g 2 , . . . , g n ) if i = 0 ( g 1 , . . . , g n − 1 ) if i = n ( g 1 , . . . , g i g i +1 , . . . , g n ) ot herwise for 0 < i < n when n > 1. Similar ly , w e define d 0 ( g ) = s ( g ) and d 1 ( g ) = t ( g ) when n = 1. F or suc h a simplicial space, w e can glue t he disjoin t union o f the G n × ∆ n as follow s, where ∆ n is the top ological n -simplex. Let δ i : ∆ n − 1 → ∆ n b e the linear embedding of ∆ n − 1 in to ∆ n as t he i -th face. W e define the classifying space of G a s the geometric realization of its nerv e as a simplicial space, that is: B G = a n ( G n × ∆ n ) / ( d i ( g ) , x ) ∼ ( g , δ i ( x )) . This is usually called the fat realization of the nerv e, meaning tha t w e hav e c hosen to lea ve out identifications in v olving degeneracies. The t wo definitions will pro duc e homotop y equiv alent spaces provided that the top o logical category has sufficien t ly nice prop erties. Another nice prop erty of the fa t realization is that if ev ery G n has the homo t o p y type o f a CW-complex, then t he realization will also ha v e the homotop y t yp e of a CW-complex. A strict homomorphism of group oids φ : H → G induces a con tin uous map B φ : B H → B G . In particular, an import a n t basic prop ert y is that an equiv alence of group oids induces a homotop y equiv alence b e t ween classify ing spaces. This follows from the fact that a n equiv - alence induces an equiv alence of categories. F or any g roup G we can construct the univ ersal G -space E G in the sense o f Milnor or Milgram. W e hav e ana lo gous constructions f o r a group oid. The first construction is the analogue t o the univ ersal space of Milgram a nd it is also describ ed in Example 2.36 of [12]. Giv en the group oid G , construct the translation group oid ¯ G = G ⋊ G 1 . This is the group oid that has G 1 as its space of ob jects and G 1 × s G 1 as its space of arrow s, that is, only one arro w EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 17 b et w een tw o elemen ts if t hey hav e the same source and none otherwise. The nerv e of t his category is give n b y: N ¯ G k = { ( g 1 , . . . , g k +1 ) | g i ∈ G 1 for i = 1 , . . . , n , and s ( g 1 ) = · · · = s ( g k +1 ) } . W e consider the nat ural simplicial structure on N ¯ G ∗ as the nerv e of the category ¯ G . There is a natural action of G on N ¯ G k with anc hor map π give n b y the source map and ( f 1 , ..., f k +1 ) · h = ( f 1 h, ..., f k +1 h ). With this action there a r e homeomorphisms N ¯ G n / G ∼ = G n for all n > 0 g iv en b y the maps: Φ 0 : N ¯ G 0 / G → G 0 [ g ] 7→ t ( g ) Φ 2 k + 1 : N ¯ G 2 k + 1 / G → G 2 k + 1 [ g 1 , . . . , g 2 k + 1 ] 7→ ( g 2 g − 1 1 , g 1 g − 1 3 , g 4 g − 1 1 , . . . , g 2 k + 1 g − 1 1 ) Φ 2 k : N ¯ G 2 k / G → G 2 k (if k > 0) [ g 1 , . . . , g 2 k ] 7→ ( g 2 g − 1 1 , g 1 g − 1 3 , g 4 g − 1 1 , . . . , g 1 g − 1 2 k ) Ψ 2 k + 1 : G 2 k + 1 → N ¯ G 2 k + 1 / G ( h 1 , . . . , h 2 k + 1 ) 7→ [ u ( s ( h 1 )) , h 1 , h − 1 2 , h 3 , . . . , h 2 k + 1 ] Ψ 2 k : G 2 k → N ¯ G 2 k / G ( h 1 , . . . , h 2 k ) 7→ [ u ( s ( h 1 )) , h 1 , h − 1 2 , h 3 , . . . , h − 1 2 k ], where [ g 1 , . . . , g n ] denotes the orbit of the elemen t ( g 1 , . . . , g n ) in N ¯ G n / G . These maps com- m ute with the face op e rators d i for the simplicial structures on N ¯ G ∗ and G ∗ , therefore N ¯ G / G and N G are homeomorphic a nd so B ¯ G / G ∼ = B G . It is a lso clear that G a cts freely on B ¯ G . The second construction imitates Milnor’s univ ersal G -space. Consider the spaces : E n G = n n Σ i =1 λ i g i | s ( g 1 ) = · · · = s ( g n ), n Σ i =1 λ i = 1 o with the subspace top ology induced from the natural inclusion in the join of n copies of G 1 and an action of G give n b y the maps π  n Σ i =1 λ i g i  = s ( g 1 ) and  n Σ i =1 λ i g i  · g = n Σ i =1 λ i g i g . There are inclusions of E n G in E n +1 G giv en b y sending Σ n i =1 λ i g i to Σ n +1 i =1 µ i h i , where h i = g i , λ i = µ i if i ≤ n and h n +1 = u ( s ( g 1 )), µ n +1 = 0. No w define E G = lim → E n G . Note tha t G acts EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 18 freely on E n G fo r all n and th us on E G . Let us denote B n G = E n G / G . Since the inclusions of of E n G in E n +1 G desc rib ed ab ov e are equiv arian t, they induce inclusions of B n G in B n +1 G . The fibrant replacemen t of a top ological group oid ([12], Definition 2.45) is w eakly equiv a- len t to the o riginal group oid. It turns out t hat E G is the geometric realization of the fibran t replacemen t of ¯ G ([12], Example 2 .4 7) and therefore E G / G and B ¯ G / G = B G are homoto py equiv alent. F rom this p o in t on w e will use E G as our univ ersal G -space a nd iden t ify E G / G with B G . Definition 4.1. Giv en a G - space X , w e define the Borel construction X G = ( X × π E G ) / G . Example 4.2. Let G = H ⋊ M . Then a G -space is a H -space X with a H - equiv aria nt map to M . In this case, G -equiv ar ia n t v ector bundles on X corresp ond to H - v ector bundles on X a nd so K G ( X ) = K H ( X ). W e hav e E G = M × E H and B G = M H . Example 4.3. Let M b e a G -space. Consider t he group oid H = G ⋊ M . An H - space is a G -space X with a G -equiv ar ian t map to M . As in the previous example, H - equiv ariant v ector bundles on X are just G -equiv arian t v ector bundles on X a nd so K H ( X ) = K G ( X ). It can b e sho wn that E H = M × π E G and B H = M G . Lemma 4.4. L et X b e a G -sp ac e and Y = X × π E n G . I f Y / G is c omp act, then ther e is an isomorphism K G ( Y ) ∼ = K ( Y / G ) . Pr o of. Pulling bac k via the pro jection p 1 : Y → Y / G tak es v ector bundles ov er Y / G to G - v ector bundles o v er Y . On the other hand, if E is a G -v ector bundle o v er Y , w e claim that E / G is a v ector bundle ov er Y / G . T o prov e this claim, we only need to show that E / G is lo cally trivial. And to show this, it suffices to pro v e that p 1 has lo cal sections around any p oin t. Indeed, let [ x, Σ n i =1 λ i g i ] = p 1 ( x, Σ n i =1 λ i g i ), where x ∈ X , g i ∈ G 1 and π X ( x ) = s ( g i ) for i = 1 , . . . , n . There is some j for whic h λ j is not zero. Let U b e a G -in v arian t op en neigh b o urho o d o f Σ n i =1 λ i g i in E n G suc h t ha t ev ery elemen t of U satisfies that λ j is not zero, and consider V = p 1 ( X × π U ). Now consider the map f : V → X × π U that sends [ y , Σ n i =1 µ i h i ] to ( y g − 1 j , Σ n i =1 µ i k i ), where k i = h i g − 1 j if i 6 = j a nd k j = u t ( g j ). This map is w ell defined and it is a section of p 1 . These t wo maps define a bijection b etw een v ector bundles o v er Y / G and G -v ector bundles on Y . Hence an y G -v ector bundle on Y is the pullbac k o f a v ector bundle on Y / G . T o pro ve the lemma w e need t o show that v ector bundles on Y / G pull back to extendable G -ve ctor bundles on Y . The anc hor map π 1 : Y − → G 0 induces a map π 2 : Y / G − → | G | . These maps EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 19 fit in to a comm utativ e dia g ram: Y π 1 / / p 1   G 0 p 2   Y / G π 2 / / | G | , where p 2 is the pro jection g iv en in Definition 2.4. Let r a nd q b e the only maps from Y / G and | G | to a p oint, resp ectiv ely . Giv en a vec tor bundle A o ver Y / G , there is a v ector bundle V o ver a p oint suc h t hat r ∗ V = A ⊕ B for some ve ctor bundle B ov er Y / G . If W = q ∗ V , then w e also hav e π ∗ 2 W = A ⊕ B . Consider W ′ = p ∗ 2 W . W e hav e π ∗ 1 W ′ = π ∗ 1 p ∗ 2 W = p ∗ 1 π ∗ 2 W = p ∗ 1 ( A ⊕ B ) = p ∗ 1 A ⊕ p ∗ 1 B .  Note that if B n G is compact, w e obtain K G ( E n G ) ∼ = K ( B n G ) b y taking X = G 0 . In t he next lemma and in the rest of the pap er w e will use represen ta ble K -theory for spaces that are not compact. Lemma 4.5. Consider ˜ M = n n Σ j =1 λ j g j | g 1 = · · · = g n , λ 1 = · · · = λ n = 1 /n o , which is a G -e quivariant subsp ac e of E n G and M = ˜ M / G ⊂ B n G . Then any p r o duct of n elements in K ∗ ( B n G , M ) is zer o. Pr o of. The result is obv ious if n = 1. Let n > 1 and consider the G -subspaces ˜ A i = n Σ n j =1 λ j g j | λ i ≥ 1 /n o of E n G and the corresp onding subspaces A i = ˜ A i / G of B n G . Give n h ∈ G 1 , let us denote by h ( i ) the elemen t Σ n j =1 λ j g j ∈ E n G with λ i = 1, λ j = 0 if j 6 = i and g j = h for all j . Now , w e define ˜ M i = { h ( i ) | h ∈ G 1 } . The space ˜ M i is a deformation retract of the spaces ˜ A i , as the follo wing maps sho w: ρ i : ˜ A i → ˜ M i ρ i  n Σ j =1 λ j g j  = g i ( i ) H i : ˜ A i × I → ˜ A i H i  n Σ j =1 λ j g j , t  = n Σ j =1 µ j h j , where µ i = tλ i + (1 − t ), µ j = tλ j if j 6 = i and h j = g i for a ll j . Since all these maps ar e G -equiv a r ia n t, M i = ˜ M i / G is a deformation retract of the space A i . In particular K ∗ ( A i , M i ) = 0. The map ˜ Φ : ˜ M i → ˜ M g iven b y Φ( g ( i )) = Σ n j =1 1 n g is a homeomorphism and it is G -equiv arian t, hence it defines a homeomorphism Φ : M i → M . EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 20 Then the relativ e K -theory of the pair ( A i , M ) is isomorphic to the K -theory of the mapping cone fo r the map Ψ : M i → A i that ta kes g ( i ) to Σ n j =1 1 n g ∈ A i . But t he inclusion of M i in A i is ho mo t o pic to Ψ via the homotop y H i : ˜ M i × I → ˜ A i H i ( g ( i ) , t ) = n Σ j =1 λ j g , where λ i = t + 1 − t n and λ j = 1 − t n if j 6 = i . Therefore K ∗ ( A i , M ) ∼ = K ∗ ( A i , M i ) = 0, and so K ∗ ( B n G , M ) ∼ = K ∗ ( B n G , A i ) f or all i via the long exact sequence for the triples ( B n G , A i , M ). Since B n G = A 1 ∪ A 2 ∪ . . . ∪ A n , the relativ e pro duct K ∗ ( B n G , A 1 ) ⊗ K ∗ ( B n G , A 2 ) ⊗ · · · ⊗ K ∗ ( B n G , A n ) → K ∗ ( B n G , A 1 ∪ A 2 ∪ . . . ∪ A n ) is identically zero, and therefore the pro duct of n elemen ts of K ∗ ( B n G , M ) is zero.  Definition 4.6. A Lie group oid G is finite if G 0 is a finite G -CW-complex and the spaces B n G are compact. Let G b e a finite group o id. W e ha ve an augmen tatio n homomorphism K ∗ G ( G 0 ) → K ∗ ( G 0 ) giv en by forgetting the G -action. Let I G b e the kerne l o f this map and let us consider the follo wing compo sition: K ∗ G ( G 0 ) π ∗ → K ∗ G ( E n G ) ∼ = → K ∗ ( B n G ) i ∗ → K ∗ ( M ) ∼ = → K ∗ ( G 0 ) , where π : E n G → G 0 is the anchor map, the middle isomorphism is giv en b y Lemma 4.4, i : M → B n G is t he inclusion giv en in Lemma 4.5 and the last isomorphism is giv en by the homeomorphism M → G 0 induced b y the G -homeomorphism ˜ M → G 1 that sends Σ n j =1 1 n g to g . The comp o sition of these four maps coincides with the aug men tation homomorphism, whose whose ke rnel is I G , so the map K ∗ G ( G 0 ) → K ∗ G ( E n G ) factors through K ∗ G ( G 0 ) /I n G . F or an y G -space X , K ∗ G ( X ) is a mo dule ov er K ∗ G ( G 0 ) and by naturality t he homomorphism K ∗ G ( X ) → K ∗ G ( X × π E n G ) factors through K ∗ G ( X ) / I n G K ∗ G ( X ), giving a homomorphism Φ n : K ∗ G ( X ) / I n G K ∗ G ( X ) → K ∗ G ( X × π E n G ). The equiv arian t inclusion E n G → E n +1 G a nd I n +1 G ⊂ I n G induce the following comm utativ e diag ram: K ∗ G ( X ) / I n +1 G K ∗ G ( X )   Φ n +1 / / K ∗ G ( X × π E n +1 G )   K ∗ G ( X ) / I n G K ∗ G ( X ) Φ n / / K ∗ G ( X × π E n G ) , whic h induces a homomorphism of pro-ring s { K ∗ G ( X ) / I n G K ∗ G ( X ) } → { K ∗ G ( X × π E n G ) } . EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 21 Conjecture 4.7. L et G b e a finite Lie gr oup oid and X a G -sp ac e. T hen we have an isomor- phism of pr o-rings { K ∗ G ( X ) / I n G K ∗ G ( X ) } − → { K ∗ G ( X × π E n G ) } . If a g roup oid G satisfies this conjecture for X = G 0 , we will sa y G satisfies the completion theorem. 5. The completion theorem Recall that w e are using represen table K -theory for spaces that are not compact. Lemma 5.1. L et G = G ⋊ X , wher e G is a c omp act Lie gr oup and X is a c omp act G -sp ac e such that K ∗ G ( X ) is finitely gener ate d over R ( G ) . Then G satisfies the c ompletion the or em. Pr o of. Let I X b e the kernel of K ∗ G ( X ) − → K ∗ ( X ). W e would lik e to pro ve that there is an isomorphism of pro-g roups { K ∗ G ( X ) / I n X } ∼ = { K ∗ G ( X × E n G ) } . By the Atiy ah-Segal completion theorem (Theorem 2.1 in [4]), we hav e an isomorphism { K ∗ G ( X ) / I n G K ∗ G ( X ) } ∼ = { K ∗ G ( X × E n G ) } . So it suffices to pro ve that the I G -adic top ology and the I X -adic top ology are the same in K ∗ G ( X ). Since K ∗ G ( X ) is a mo dule ov er R ( G ), w e ha ve I G K ∗ G ( X ) ⊂ I X . Let K n b e the kerne l of α n : K ∗ G ( X ) − → K ∗ G ( X × E n G ). By Corollary 2 .3 in [4], the sequence of ideals { K n } defines the I G -adic top ology on K ∗ G ( X ). In particular, there is m ∈ N suc h that K m ⊂ I G K ∗ G ( X ). Note that K 1 = I X . Consider the comp osition K ∗ G ( X ) − → K ∗ G ( X × E m G ) − → K ∗ G ( X × E 1 G ) ∼ = K ∗ ( X ). Since X × E m G is the union of m op en sets whic h are homotopy equiv alent to X × E 1 G = X × G , w e hav e K ∗ G ( X × E m G, X × E 1 G ) m = 0. Th us the first map factors through I m X , th us I m X ⊂ K m . Hence I m X ⊂ I G K ∗ G ( X ).  Lemma 5.2. L et Θ : H → G b e a lo c al e quivalenc e of finite Lie gr oup oids. Then G satisfies the c ompletion the or em if an d only if H do es. Pr o of. The lo cal equiv alence Θ : H − → G induces an isomorphism f : K ∗ G ( G 0 ) ∼ = − → K ∗ H ( H 0 ). The follo wing diagram is commutativ e: K ∗ G ( G 0 ) f / / α   K ∗ H ( H 0 ) β   K ∗ ( G 0 ) φ / / K ∗ ( H 0 ) . Therefore w e hav e f ( I G ) ⊂ I H . L et g b e the inv erse of f , x ∈ I H and y = g ( x ). Since β ( x ) = 0, w e hav e φα ( y ) = 0 . But α ( y ) = ( n y , a y ) ∈ Z ⊕ ˜ K ∗ ( G 0 ) ∼ = K ∗ ( G 0 ) and so φα ( y ) = ( n y , ˜ φ ( a y )). This implies n y = 0 , that is, αg ( I H ) ⊂ ˜ K ∗ ( G 0 ). EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 22 Since G 0 is compact, there is m ∈ N suc h that ˜ K ∗ ( G 0 ) m = 0. Then w e hav e αg ( I m H ) ⊂ ˜ K ∗ ( G 0 ) m = 0 and so g ( I m H ) ⊂ I G . Th us I m H = f g ( I m H ) ⊂ f ( I G ) and t he top ologies induced b y I G and I H on K ∗ H ( H 0 ) are t he same. Therefore w e hav e an isomorphism of pro-rings { K ∗ ( G 0 ) /I n G } ∼ = { K ∗ ( H 0 ) /I n H } . The lo cal equiv alence Θ also induces a ho mo t o p y equiv alence b et w een B G and B H . If w e consider the asso ciated filtrations to each of these spaces, { B n G } and { B n H } , the lo cal equiv alence mus t tak e B n G to some B n + k H a nd B n H t o some B n + k ′ G . Hence we ha v e an isomorphism of pro-rings { K ∗ ( B n G ) } ∼ = { K ∗ ( B n H ) } . The lemma follows fro m the diagram: { K ∗ ( G 0 ) /I n G } ∼ = / /   { K ∗ ( H 0 ) /I n H }   { K ∗ ( B n G ) } ∼ = / / { K ∗ ( B n H ) } F rom the previous lemma, we obtain the follo wing theorem: Theorem 5.3. L et G a n d H b e we akly e quivalent finite gr oup oids. Then G satisfies the c ompletion the or em if and only if H do es. Note that Theorem 5.3 and Lemma 5.1 imply the follo wing: Corollary 5.4. If G is Br e don- c o m p atible and U is a G -c el l, K ∗ G ( U ) is a finitely gener ate d ab elian gr oup and the g r oup oid G ⋊ U satisfies the c omp l e tion the or em . This corollary tells us that the completion theorem is true for G -cells. No w w e will use this corollary to prov e the completion theorem for finite G -CW-complexes. W e will also need to use the follo wing lemma: Lemma 5.5. Fi x any c ommutative No etherian ring A , and any ide al I ⊂ A . T hen for any exact se quenc e M ′ → M → M ′′ of finitely gener ate d A -mo dules, the se quenc e { M ′ /I n M ′ } − → { M /I n M } − → { M ′′ /I n M ′′ } of pr o-gr oups ( p r o- A -mo dules) is exact. Pr o of. See [16], Lemma 4.1.  Let X b e a finite G -CW-complex with p -cells { V i = U i × D p | i ∈ I p } (see Definition 3.1 6) and consider the sp ectral sequenc e for this decomp osition in G -equiv ariant K - t heory: E pq 1 = Y i ∈ I p K q G ( V i ) = ⇒ K p + q G ( X ) . EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 23 This is a spectral sequ ence of K ∗ G ( G 0 )-mo dules. Let us assume that G is finite so that K ∗ G ( G 0 ) is a No e therian ring. All eleme n ts in these sp ectral seq uences are finitely generated, and so b y the previous lemma, t he functor taking a K ∗ G ( G 0 )-mo dule M to the pro-group { M /I n G M } is exact and w e can form the followin g sp ectral seq uence of pro-rings: F pq 1 =    Y i ∈ I p K q G ( V i ) /I n G K q G ( V i )    = ⇒ { K p + q G ( X ) / I n G K p + q G ( X ) } . No w consider the G - maps h n : X × π E n G − → X . They giv e us anot her sp ectral sequence of pro -rings: ¯ F pq 1 =    Y i ∈ I p K q G ( h − 1 n V i )    = ⇒ { K p + q G ( X × π E n G ) } . W e also ha v e a map of sp ectral sequences φ : F − → ¯ F induced by the pro jections h − 1 n V i = V i × π E n /G → V i . If G is Bredon-compatible, the group o ids G ⋊ V i satisfy the completion theorem for all i . Since w e are taking quotien t b y the ideal I G and not by I G ⋊ V i , w e need t o c hec k that b oth top ologies coincide. W e consider the long exact sequence s in equiv arian t and non- equiv ariant K -theory for the pair ( C V , V ) where V is any V i and C V is the mapping cylinder of the map π : V − → G 0 . Note that C V is G - homotop y equiv alen t to G 0 and V ⊂ C V . K G ( C V , V ) / /   K G ( G 0 ) / /   K G ( V ) / /   K 1 G ( C V , V )   K ( C V , V ) / / K ( G 0 ) / / K ( V ) / / K 1 ( C V , V ) . It is clear that I G K G ( V ) ⊂ I V . Now let m ∈ N suc h that K ( C V , V ) m = 0 and n ∈ N such that K G ( C V , V ) n = 0. Then I nm V ⊂ I G K G ( V ) a nd so the top ologies coincide. This pro ve s φ is an isomorphism when restricted to any par t icular elemen t F ij and there- fore, it is an isomorphism of sp ectral sequence s. In particular, we ha v e an isomorphism { K p + q G ( X ) / I n G K p + q G ( X ) } ∼ = { K p + q G ( X × π E n G ) } . T o summarize, we hav e pro ve d: Theorem 5.6. L et G b e a Br e don-c omp a tible finite Lie gr oup oid and X a finite G -CW- c omplex. The n we h a v e an is o morphism of pr o-rings { K ∗ G ( X ) / I n G K ∗ G ( X ) } − → { K ∗ G ( X × π E n G ) } . Corollary 5.7. Under the same cir cumstanc es, the homomo rp hism K ∗ G ( X ) → K ∗ ( X G ) in- duc es an is o morphism of the I G -adic c ompletion of K ∗ G ( X ) with K ∗ ( X G ) . EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 24 6. App lica tions Unless o therwise stated, thro ughout this whole section S will b e a Lie group, but not necessarily compact. T o study prop er actions of S , we can consider the group oid G = S ⋊ E S , where E S is the univers al space fo r prop er actions of S ([1 5], D efinition 1.8 ). This space is a prop er S -CW-complex suc h that E S G is contractible for all compact Lie subgroups G of S and suc h that ev ery prop er S -CW-complex has an S -map to E S that is unique up to S -ho mo t op y . The existence of E S is sho wn in Theorem 1.9 of [15]. Some immediate consequenc es fo llow: • Prop e r S -CW-complexes are G -CW-complexes . • Extendable G -bundles on a prop er S - CW-complex X are extendable S -bundles for an y S -map X − → E S , since all of them are S -homoto pic. In order for G to b e finite, we need E S to b e a compact finite S -CW-complex and the spaces B n G t o be compact. Note that this is not necessarily the case, but when these conditions hold w e are under the h yp o theses of Theorem 5.6. Equiv aria nt K -theory for actions of finite groups and compact Lie gro ups hav e b e en studied extensiv ely in the past. It is a w ell- known fact that for these actions, G is Bredon-compatible [24] and finite a nd so they satisfy a completion theorem, first prov ed in [4] using different metho ds. Prop er actions of discrete groups and totally disconnected groups that are pro jectiv e limits of discrete groups also satisfy Bredon-compatibility , as sho wn in [16] and [21 ], and therefore a completion theorem f o llo ws if the finiteness condition holds. A completion theorem for prop er actions of discrete groups is also pro v ed in [16]. In general, vec tor bundles may not b e enoug h to construct an in t eresting equiv aria n t cohomology theory for prop er actions of second coun table lo cally compact groups [19 ], but they suffice for t w o imp ortant families, almost compact groups and matrix groups [20]. Almost compact groups, that is, second countable lo cally compact groups whose gro up of connected comp onents are compact, alwa ys hav e a maximal compact subgroup. Any space with a prop er action of one of these gro ups is the induction of a space with an action of that compact subgroup and so the study of prop er actions of almost compact groups are reduced to studying compact Lie group actio ns. This is carr ied out in [20], where it is prov en that these action g roup oids are Bredon-compatible t ha t a completion theorem ho lds. This is done b y sho wing that the completion ma ps a r e compat ible with the reduction map to the maximal compact subgroup. With differen t tec hniques it is prov en that pro p er actions o f EQUIV ARIAN T K-THEOR Y, GROUPOIDS AND PROPER AC TIONS 25 matrix groups, that is, closed subgroups of GL ( n, R ), are Bredon-compatible. Prop er actions of a b elian Lie groups constitute a particular instance of this case. The g r o up oid G is not necessarily Bredon-compatible. When G is a Bredon-compatible group oid we m ust hav e V ect G ( S/G ) = V ect G ( pt ) for a compact subgroup G o f S . Let S b e a Kac-Mo o dy group and T its maximal torus. Note t hat S is not a Lie group, but all the constructions and r esults in Sections 3 and 4 remain v a lid f or a top olog ical group oid with a veraging of v ector bundles. There is an S -map S/T − → E S whic h is unique, up to homoto p y . G iv en an S -v ector bundle V on E S , the pullback to S/T is given b y a finite-dimensional represen tation o f T in v ariant under t he W eyl group. This r epresen tation giv es rise to a finite-dimensional represen tation of S . All finite-dimensional represen tations of K a c-Mo o dy gro ups whic h are not compact Lie groups are trivial. Therefore V m ust be trivial and so extendable S -vector bundles on S/T only come from trivial represen tations of T . In order to deal with these g roups, it is more con v enien t to use dominan t K -theory , which w as dev elop ed in [13]. K ac-Mo o dy gro ups p o ssess an imp ortant class of represen tations called dominan t represen tations. A dominant represen tation of a Kac-Mo o dy g r o up in a Hilb ert space is one that decomp oses into a sum of highest w eigh t represen tations. Equiv ariant K - theory fo r prop er a ctions of Kac-Mo o dy gro ups is defined as the represen table equiv ariant cohomology theory mo deled on the space of F redholm op era t o rs on a Hilb ert space which is a maximal dominant represen tat io n of t he group. Reference s [1] A. Adem, J. Le ida and Y. Ruan, Orbifolds and stringy top olo gy , Ca mbridge T r acts in Mathematics, V olume 171, 20 07. [2] M. Atiy ah, Char acters and c ohomolo gy of finite gr oups , Ins t. Hautes Etudes Sci. Publ. Ma th. 19 61, no. 9, 23-64 . [3] M. Atiy ah and F. Hirzebruch, V e ctor bund les and homo gene ous sp ac es , Pro c. Symp os. P ure Math. V ol. 3, Amer. Math. So c. 1 9 61, 7 -38. [4] M. Atiy ah and G. Seg a l, Equivariant K -t he ory and c ompletion , J . Diff. Geom. 3 , 1 969, 1-18. [5] M. Atiy ah and G. Seg a l, Twiste d K -the ory , Ukr. Mat. 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Dep ar tment of Ma thema tics, St an f ord University, St an f ord, California 94305, USA. E-mail addr ess : c antare r@stan ford.edu