Restriction maps in equivariant $KK$-theory

RESTRICTION MAPS IN EQUIV ARIANT K K -THEOR Y. OTGONBA Y AR UUYE Abstra ct. W e extend McC lure’s results on the restriction maps in equiv arian t K -theory to b iv ariant K -theory: Let G b e a compact Lie group and A and B b e G - C ∗ -algebras. Sup- p ose that K K H n ( A, B ) is a finitely generated R ( G )-mo dule for every H ≤ G closed and n ∈ Z . Then, if K K F ∗ ( A, B ) = 0 for all F ≤ G finite cyclic , then K K G ∗ ( A, B ) = 0. 0. Introduction Let G b e a compact Lie group and let X b e a finite G -CW-complex. F or an y closed sub group H ≤ G , we ha ve a restriction functor in equiv ariant K -theory: (0.1) res G H : K G ( X ) → K H ( X ) . As an app lication of the generalized A tiyah-Sega l completion th eorem of [AHJM88a], McClur e p ro v ed th e follo wing. Theorem (McClure [McC86, Theorem A]) . If x ∈ K G ( X ) r estricts to zer o in K H ( X ) for e v ery fi nite sub gr oup H of G , then x = 0 . Com bining with Jack o wski’s result [Jac77, Corollary 4.3], one obtains the follo win g. Theorem (Jac k o wski-McClure [McC86, C orollary C]) . If K ∗ F ( X ) = 0 for al l F ≤ G finite cyclic, then K ∗ G ( X ) = 0 . W e extend these to biv arian t K -theory as f ollo ws. Theorem 0.1. L et G b e a c omp act Lie gr oup and A and B b e G - C ∗ -algebr as. Supp ose that K K H n ( A, B ) is a finitely gener ate d R ( G ) -mo dule for eve ry H ≤ G close d and n ∈ Z . (1) Then, if K K F ∗ ( A, B ) = 0 for al l F ≤ G fin ite cyclic , then K K G ∗ ( A, B ) = 0 . (2) Supp ose, in add ition, that K K F n ( A, B ) is a finitely gener ate d gr oup for al l F ≤ G finite and n ∈ Z . Then, if x ∈ K K G ( A, B ) r estricts to zer o in K K H ( A, B ) for al l H ≤ G finite , then x = 0 . Date : December 13, 2010. 2010 Mathematics Subje ct Classific ation. Primary (19K35); Secondary (46L80). Key wor ds and phr ases . K -theory , K K -theory . 1 2 OTGONBA Y A R UUYE Remark 0.2. (1) See [MM04] for a dual result for restriction m ap s in K -h omology of sp aces with actio ns of d iscrete group s. (2) Th eorem 0.1 is in stark cont rast to the results of Heath Emerson, where he sh o wed that ev en for circle actions, noncomm utativ e alge- bras can b eha v e very differently fr om commutat iv e ones. [Eme10] In fact, we p ro v e th e follo w in g. This is done mainly for clarit y , but as an ad d ed b onus, we see that Th eorem 0.1 holds for equiv arian t E -theory as w ell. Theorem 0.3. L et G b e a c omp act Lie gr oup and let e E ∗ G b e an R O ( G ) - gr adable mo dule the ory over e K ∗ G . Supp ose that e E n H ( S 0 ) is a finitely gener ate d R ( G ) -mo dule for e v ery H ≤ G close d and n ∈ Z . L et X b e a finite b ase d G -CW-c omplex. (1) Then, if e E ∗ F ( X ) = 0 for al l F ≤ G finite cyclic , then e E ∗ G ( X ) = 0 . (2) Supp ose, in addition, that e E n F ( S 0 ) is a finitely gener ate d gr oup for al l F ≤ G finite and n ∈ Z . Then, if x ∈ e E ∗ G ( X ) r estricts to zer o in e E ∗ H ( X ) for al l H ≤ G finite , then x = 0 . The pro of follo ws [McC86] ve ry closely . In Section 1, we sh ow that The- orem 0.3 implies Theorem 0.1. In S ection 2, we extend the generalized A tiy ah-Segal co mpletion theorem of [AHJM88a ], supp lying the missing in- gredien t n eeded to fin ish the pr o of in Section 3. Ac knowledgmen ts. T he author thanks the Centre f or S ymmetry and De- formation at the Univ ersit y of Cop enh agen and the Danish Natio nal R e- searc h F oun dation for s upp ort. 1. RO ( G ) -graded cohomology theories Let G b e a compact Lie group. A based G -space is a G -space with a G - fixed base p oint . In the rest of the pap er, we assume that all G -spaces are G -CW-complexes and all cohomology theories are equiv ariant and reduced cohomology th eories. F or a finite-dimensional r ep resen tatio n V of G , w e write S V for the one- p oint compactification of V , considered a based G -space with base p oint the p oint at infin it y . 1.1. R O ( G ; U ) -gradable theories. W e fix a complete u niv erse U . (cf. [Ma y96 , Definition IX.2.1]). Definition 1.1. A n R O ( G ) -gr ade d cohomolog y theory is an R O ( G ; U )- graded cohomology theory in the sense of [Ma y96, Definition XI I I.1.1]. A Z -gr ade d cohomolog y theory is an R O ( G ; U G )-graded cohomology theory (an y trivial universe would wo rk). W e sa y that a Z -graded cohomology theory is RO ( G ) -gr adable if it is the Z -graded part of an RO ( G )-graded theory . RESTRICTION MAPS IN EQUIV ARIANT K K -THEOR Y. 3 Let e E ∗ G b e a Z -graded cohomolog y theory . F or a closed su bgroup H ≤ G and a based H -CW-complex X , we defin e (1.1) e E ∗ H ( X ) : = e E ∗ G ( G + ∧ H X ) . Then e E ∗ H is a Z -graded cohomology theory on based H -spaces. If X is ac- tually a based G -CW-complex, then we h a ve a natural G -equiv ariant iden- tification (1.2) G + ∧ H X ∼ = G/H + ∧ X and the collapse map G/H → ∗ giv es rise to a natural transformation (1.3) res G H : e E ∗ G → e E ∗ H called the r e striction map. 1.2. Biv ariant K -theory. The follo w ing is the main example w e hav e in mind. First note that e K ∗ G is an RO ( G )-graded comm utativ e ring theory with e K V G ( X ) = K K G ( C 0 ( S V ) , C 0 ( X )) an d R ( G ) ∼ = e K G ( S 0 ). Prop osition 1.2. L et G b e a c omp act Lie gr oup and let A and B b e G - C ∗ - algebr as. F or a finite b ase d G -CW -c omplex X and finite- dimensional r e al r epr esentation V of G , we define (1.4) e E V G ( X ) : = K K G ( A ⊗ C 0 ( S V ) , B ⊗ C 0 ( X )) . Then the fol lowing holds. (i) e E ∗ G defines an RO ( G ) -gr ade d c ohomolo g y the ory on the c ate gory of finite b ase d G -CW-c omplexes. (ii) e E ∗ G extends to an RO ( G ) -gr ade d c ohomolo g y the ory on the c ate gory of b ase d G -CW-c omplexes. (iii) e E ∗ G is a mo dule the ory over e K ∗ G . Pr o of. (i) See [K as88]. (ii) By Adams’ r epresen tation theorem [Ma y96, T he- orem XI I I.3.4], e E ∗ G is r epresen ted by an Ω- G -presp ectrum, hence extends to an RO ( G )-g raded cohomology theory on the catego ry of G -CW-complexes. See [Sch92]. (iii) The mo dule structure (1.5) e E V G ( X ) × e K W G ( Y ) → e E V + W G ( X ∧ Y ) . is give n by the Kasparo v pro duct K K G ( A ( S V ) , B ( X )) × K K G ( C 0 ( S W ) , C 0 ( Y )) (1.6) → K K G ( A ( S V + W ) , B ( X ∧ Y )) . (1.7)  It is well-kno wn that for H ≤ G , (1.8) K K G ( A, B ⊗ C 0 ( G/H + )) ∼ = K K H ( A, B ) and the restriction map is induced b y G/H + → S 0 . Hence we obtain the follo win g corolla ry . 4 OTGONBA Y A R UUYE Corollary 1.3. Su pp ose that The or em 0.3 holds. Then The or em 0.1 holds.  2. A tiy a h-Segal Completion First w e abs tract the main fin iteness condition from Theorem 0.3. Definition 2.1. Let R b e a unital comm utativ e rin g and let e E ∗ G b e a Z - graded cohomology theory with v alues in R -mo dules. W e sa y that e E ∗ G is finite o v er R if e E n G ( X ) is a fi nitely generated R -mo du le f or ev ery finite based G -CW-complex X and n ∈ Z . Clearly , this is equiv alen t to asking that e E k − n H ( S 0 ) ∼ = e E k G ( G/H + ∧ S n ) is a fin itely generated R -mo dule for H ≤ G . Lemma 2.2. L e t G b e a c omp act Lie gr oup and let R b e a unital c ommuta- tive ring. L et e E ∗ G b e a Z -gr ade d c ohomolo gy the ory with values in R -mo dules. Supp ose that R is No etherian and e E ∗ G is finite over R . Then for any family I of ide als in R , the fol lowing defines a Z -g r ade d c oho molo gy the ory with values in pr o- R -mo dules: (2.1) e E ∗ G ( X ) ∧ I : = { e E ∗ G ( Y ) /J · e E ∗ G ( Y ) } . wher e Y ⊆ X runs over the finite b ase d G -CW-sub c omplexes of X and J runs over the finite pr o ducts of ide als in I . Note that in this lemma, it is enough to hav e e E ∗ G to b e a cohomolog y theory on finite based G -CW-complexes (only fi n ite wedges are considered in the additivit y axiom). Pr o of. Exactness follo ws fr om the Artin-Rees lemma. See the pro of of [AHJM88b, Lemm a 2.1].  2.1. Bott P erio dicit y . Let V b e a complex G -repr esentati on. By Bott p e- rio dicit y [Ati6 8 , Theorem 4.3], e K 0 G ( S V ) is a free e K 0 G ( S 0 )-mo dule generated b y the Bott elemen t λ V ∈ e K 0 G ( S V ). The Euler class of V is defin ed to b e e ∗ ( λ V ) ∈ e K 0 G ( S 0 ), where e : S 0 → S V is the obvious m ap . Lemma 2.3. L et e E ∗ G b e an RO ( G ) -gr ade d mo dule the ory over e K ∗ G . Then for any c omplex r epr esentation V , multiplic ation by the B ott element λ V ∈ e K 0 G ( S V ) gi ves an isomorph ism (2.2) e E 0 G ( S 0 ) ∼ = e E 0 G ( S V ) . If V ⊆ W ar e c ompl ex r epr esentations and i : S V → S W is the inclusion, then the fol lowing diagr am c ommutes (2.3) e E 0 G ( S 0 ) / / · χ W − V   e E 0 G ( S W ) i ∗   e E 0 G ( S 0 ) / / e E 0 G ( S V ) . RESTRICTION MAPS IN EQUIV ARIANT K K -THEOR Y. 5 Pr o of. Let λ − 1 V ∈ e K V G ( S 0 ) denote the inv erse Bott elemen t: it h as the pr op- ert y th at (2.4) λ V · λ − 1 V = λ − 1 V · λ V = 1 ∈ e K V G ( S V ) ∼ = e K 0 G ( S 0 ) . Then multiplicatio n by λ − 1 V giv es the in v er s e map (2.5) e E 0 G ( S V ) → e E V G ( S V ) ∼ = e E 0 G ( S 0 ) . The second statemen t is sho wn for e E ∗ G = e K ∗ G in [AHJM88a, page 4]. The general case follo ws by functorialit y .  2.2. C ompletion. A class of subgroup s of G closed under su b conjugacy is called a f amily . A family C of subgroups of G determines a class, again denoted C , of ideals of R ( G ) b y the kernels of the restriction maps: (2.6) k er(res G H : R ( G ) → R ( H )) , H ∈ C , hence a top ology on an y R ( G )-mo dule. The follo wing is a straigh tforw ard generalization of [AHJM88a, Theorem 3.1]. Theorem 2.4. L et G b e a c omp act Lie gr oup and let e E ∗ G b e an R O ( G ) - gr adable mo dule the ory over e K ∗ G , which is finite over R ( G ) . L et C b e a family of sub gr oups of G . F or any b ase d G -CW-c omp lex X , if e E ∗ H ( X ) ∧ C | H = 0 for al l H ∈ C , then e E ∗ G ( X ) ∧ C = 0 . Pr o of. By [Seg68, Corollary 3.3], R ( G ) = e K 0 G ( S 0 ) is No etherian. Hence, by Lemma 2.2, e E ∗ G ( X ) ∧ C is a cohomology theory . No w the pr o of of [AHJM88a, Th eorem 3.1] carries o ver ad verbatum, once w e extend Bott p eriod icit y to e E ∗ G as in L emm a 2.3 .  Corollary 2.5. L et E C denote the classifying sp ac e of C . F or any finite b ase d G -CW-c omplex X , the pr oje ction map E C + → S 0 gives c omp letion (2.7) e E ∗ G ( E C + ∧ X ) ∼ = lim e E ∗ G ( Y ∧ X ) ∼ = lim e E ∗ G ( X ) ∧ C , wher e Y runs over finite b ase d sub c omplexes of E C + . Pr o of. The inv erse system e E ∗ G ( X ) ∧ C satisfies the Mittag-Leffler condition and e E ∗ G ( Y ∧ X ) is C -complete for any finite based sub complex Y ⊂ E C + (cf. [AHJM88a, C orollary 2.1]).  3. Proof of Theo rem 0.3 3.1. F - spaces. Let F b e a family of subgroups of G . W e sa y that a based G -CW-complex X is an F -space if all the isotrop y groups , except at the base p oint , are in F . The follo wing lemma says that in the pro of of Theorem 0.3, w e m a y assume that X is an F -space, for any F con taining all finite cyclic subgroups of G . 6 OTGONBA Y A R UUYE Lemma 3.1. L et G b e a c omp act Lie gr oup and let e E ∗ G b e an R O ( G ) - gr adable mo dule the ory over e K ∗ G , which is finite over R ( G ) . L et F b e a family c ontaining al l finite cyclic su b gr oups of G . Then for any finite b ase d G -CW-c omp lex X , the top horizo ntal map in the c ommutative diagr am (3.1) e E ∗ G ( X ) / /   lim Y ⊂ E F + e E ∗ G ( Y ∧ X )   Q F ∈F e E ∗ F ( X ) / / lim Y ⊂ E F + Q F ∈F e E ∗ F ( Y ∧ X ) , is inje ctive. Her e Y runs over the finite b ase d su b c omplexes of E F + , the horizonta l maps ar e induc e d by the pr oje ctions Y ∧ X → X and the vertic al maps ar e r estrictions. Pr o of. The F -top ology on e E ∗ G ( X ) is Hausdorff b y [McC86, Corollary 3.3]. Hence, th e claim follo ws from Corollary 2.5 .  Let C d enote th e family of finite cyclic subgroup s of G . Pr o of of The or em 0.3(1). By assum ption, e E ∗ F ( X ) = 0 for all F ∈ C . Let Y b e a fi n ite based G -CW-complex, wh ic h is a C -space. T hen the zero ske leton Y 0 and the skel etal qu otien ts Y n / Y n − 1 are finite w edges of G -spaces of the form G/F + ∧ S n with F ∈ C . It follo ws that e E ∗ G ( Y ∧ X ) = 0. Hence b y Lemma 3.1, e E ∗ G ( X ) = 0.  3.2. I nduction. W e write O G for the category whose ob jects are orb it spaces G/H , w here H ≤ G is a closed subgroup, and whose morphisms are homotopy classes of G -maps. Recall that a compact Lie group is said to cyclic if it has a top ologi cal generator (an element whose p ow ers are dense) and hyp er elementary if it is an extension of a cyclic group b y a finite p -group. W e write H for th e class of hyp erelemen tary su bgroups of G and let O H denote the full su b category of O G of orb its G/H with H sub conjugate to a subgroup in H . Lemma 3.2. L et G b e a c omp act Lie gr oup and let e E ∗ G b e an R O ( G ) - gr adable mo dule the ory over e K ∗ G . Then, for any b ase d G -CW- c omplex, the r estriction maps induc e an isomorph ism (3.2) e E ∗ G ( X ) ∼ = lim O H e E ∗ H ( X ) . Pr o of. F ollo ws from Prop ositions 2.1 and 2.2 of [McC86].  F or an y ab elian group M , let M ∧ Z denote its adic completion lim n M /nM . Pr o of of The or em 0.3(2). Let F denote the f amily of finite s u bgroups of G . By Lemma 3.2, we ma y assume that G is a hyp erelemen tary group and b y Lemma 3.1, w e ma y assume that X an F -space. RESTRICTION MAPS IN EQUIV ARIANT K K -THEOR Y. 7 Let G b e a h yp erelemen tary group and X an F -space. Then the restric- tion map (3.3) e E ∗ G ( X ) ∧ Z → lim F ∈O F e E ∗ F ( X ) ∧ Z . is an isomorphism b y [McC86, T heorem 1.1]. By [McC86, Corollary 3.3], the adic top ologies on e E ∗ G ( X ) and e E ∗ F ( X ) are Hausdorff. T h is completes the p ro of.  Referen ces [AHJM88a] J. F. Adams, J.-P . Haeb erly , S. Jac k o wski, and J. P . May , A gener aliza- tion of the Atiyah-Se gal c ompletion the or em , T op ology 27 (1988), n o. 1, 1–6. MR MR935523 (90e:55026 ) [AHJM88b] , A gener alization of the Se gal c onje ctur e , T op ology 27 (1988), no. 1, 7–21. MR 935524 (90e:55027) [Ati6 8] M. F. Atiy ah, Bott p erio dicity and the i ndex of el li ptic op er ators , Q uart. J. Math. Ox ford S er. (2) 19 (1968), 113–140. MR MR0228000 (37 #3584) [Eme10] H. Emerson, L o c al i zation te chniques in cir cle-e quivariant KK-the ory , A rXiv e-prints (2010). [Jac77] Stefan Jac ko wski, Equivariant K -the ory and cyclic sub gr oups , T ransforma- tion groups (Proc. Conf., Univ. N ewcastle up on Tyne, Newca stle up on Tyne, 1976), Cambridge Univ. Press, Cam bridge, 1977, pp. 76–91. London Math. Soc. Lectu re N ote S eries, No. 26. MR MR0448377 (56 #6684) [Kas88] G. G. Kasparo v, Equivariant K K -the ory and the Novikov c onj e ctur e , Inve nt. Math. 91 (1988), no. 1, 147–201. MR MR 918241 ( 88j:58 123) [Ma y96] J. P . May , Equivariant homotopy and c ohomolo gy the ory , CBMS Regional Conference Series in Mathematics, vol . 91, Published for the Conference Board of the Mathematical Sciences, W ashington, DC, 1996, With con tri- butions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmendorf, J. P . C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. T riantafillou, and S. W aner. MR MR1413302 (97k:55016) [McC86] James E. McClure, R estrict ion maps i n e quivariant K -the ory , T op ology 25 (1986), n o. 4, 399–409. MR 862427 (88f:550 22) [MM04] Mic hel Matthey and Guido Mislin, Equivariant K -homolo gy and r estriction to finite cyclic sub gr oups , K -Theory 32 ( 2004), no. 2, 167–179. MR 2083579 (2005k:1901 2) [Sch92 ] Claude Sc hochet, On e quivariant Kasp ar ov the ory and Sp ani er-Whitehe ad du- ality , K - Theory 6 (1992), no. 4, 363–385. MR MR1193150 (94c:19006) [Seg68] Graeme Segal, The r epr esentation ring of a c omp act Lie gr oup , Inst. Hautes ´ Etudes Sci. Publ. Math. (1968), no. 34, 113–128. MR MR0248277 (40 #1529) Dep ar tme nt of Ma thema tical Sciences, Unive rsity of Copenhagen, Unive r- sitetsp arken 5, DK-2100 Copenhagen E, D enmark E-mail addr ess : otogo@math.ku.d k URL : http://www.math.ku .dk/~otogo