The Baum-Connes assembly map and the generalized Bass conjecture

THE BA UM-CONNES ASSEMBL Y MAP AND THE G ENERALIZED BASS CONJECTURE C. Ogle (OSU) May 2007 Intr oduction In the early 1 980’s, P . Baum a nd A. Connes defined an assembly map (0.1) A G,a ∗ : K K G ∗ ( C ( E G ) , C ) → K t ∗ ( C ∗ r ( G )) where G denotes a lo ca lly compact group, E G the classifying space for prop er G -ac tions, C ( E G ) the G -algebra of com plex-v a lued functions on E G v anishin g at infinit y , a nd K K G ∗ ( C ( E G ) , C ) the G - equiv a rian t K K -groups of ( E G ) with c o efficients in C , while K t ∗ ( C ∗ r ( G )) represen ts the to po lo gical K -groups of the reduced C ∗ -algebra of G . The original details o f this map appeare d ( a few years later) in [BC1] a nd [BC2], with further elab ora t ions in [B C H]. As shown in [BC 3 ], when G is discrete the left-hand side admits a Chern character which may b e re presen ted as ch B C ∗ ( G ) : K K G ∗ ( C ( E G ) , C ) → ⊕ x ∈ f in ( ) H ∗ ( B G x ; C ) ⊗ H P er ∗ ( C ) where f in ( < G > ) is the set o f c onjugacy classes of G corresp onding to elements of finite order, G x the centralizer of g in G where x = < g > , and H P er ∗ ( C ) the p erio dic cyclic homolog y of C . Note that H ∗ ( B H ; C ) ⊗ H P e r ∗ ( C ) are simply the 2-perio dized complex homology groups of B H , and (via the c las- sical A tiyah-Hirzebru ch Chern character) can be alte rnat ively viewed as the complexified K -homology groups of B H . Up on complexificatio n, the ma p ch B C ∗ ( G ) is a n isomorphism. The original construction of Baum and Connes A G,a ∗ was analytica l. Mo tiv ated by the need to co nstruct a homoto pical analogue to their map, we constructed an assembly map in [O1] which we will denot e here as A G,h ∗ ⊗ C : H ∗ ( a x ∈ f in ( ) B G x ; K ( C )) ⊗ C → K t ∗ ( C ∗ r ( G )) ⊗ C where K ( C ) denotes the 2 -p erio dic top ological K -theo ry sp ectrum of C . The construction of this map amounted to a n e xtension of the cla ssical assembl y map constructed in [L] which was designed to ta ke into acc oun t t he co n tribution coming from the conjugac y c lasses of finite order. The tw o Key wor ds and phr ases. Baum-Connes Assembly map, Baum-Connes C onjecture, Bass C onjecture. Typeset by A M S -T E X 1 essen tial f eatures of A G,h ∗ ⊗ C , shown in [O1], were (i) it fa ctors through K t ∗ ( C [ G ]) ⊗ C (where K t ∗ ( C [ G ]) denotes the Bott-perio dized top ological K -theory of the co mplex group alg ebra, topo lo gized with the fine top ology), and (ii) the c omp osition of A G,h ∗ ⊗ C with the complexified Chern-Connes-Karoubi- Tillmann character ch C K ∗ : K ∗ ( C [ G ]) ⊗ C → H C ∗ ( C [ G ]) was effectively c omputable (see below). What we did not do in [O1 ] was show that A G,a ∗ ⊗ C and A G,h ∗ ⊗ C a g ree. Since this initial work, there hav e been numerous exte nsions and reformulations of the Baum-Connes a ssemb ly m ap, as well as of the original Baum-Connes conjec ture, which states that the m ap in (0.1) is an isomorphism. These extensions typically are included under the umbrella term “Isomorphism C o njecture”, ( formulated fo r bot h algebraic and top ologic a l K -theory; cf. [D L], [FJ], [LR]). Thanks to [HP], w e now kno w that the differen t formulations of these assembly m aps (e.g., homotopy-theoretic vs. analytical) agree. Abbreviating K K G ∗ ( C ( E G ) , C ) as K G ∗ ( E G ) (read: the equiv ariant K -homology of the prop er G - space E G ), o ur ma in result is Theorem 1 . Ther e is a c ommuting d iagr am K G ∗ ( E G ) A G,DL ∗ / / ch ? ∗   K t ∗ ( C [ G ]) ch C K ∗   H C f in ∗ ( C [ G ]) / / / / H C ∗ ( C [ G ]) wher e A G,DL ∗ is the ho motopic al ly define d assembly map of [DL], f in H C ∗ ( C [ G ]) := ⊕ x ∈ f in ( ) H C ∗ ( C [ G ]) x ∼ = ⊕ x ∈ f in ( ) H ( B G x ; C ) ⊗ H C ∗ ( C ) is the el lipti c summand of H C ∗ ( C [ G ]) [J O R], the lower hor izontal map is the obvio us inclusion, and the Chern cha r acter ch ? ∗ b e c omes a n i somorph ism up on c omplexific a- tion fo r ∗ ≥ 0 . Let β denote a bounding class, ( G, L ) a discrete group equipped with a word-length, and H β ,L ( G ) the rapid deca y algebra a ssoc iated with this data [JOR]. W e write K t ∗ ( H β ,L ( G )) for the Bott-perio dic top ological K -t heory o f the to po lo gical alg ebra H β ,L ( G ). The Baum-Connes assembl y ma p fo r H β ,L ( G ) is defined to be the co m po sition (BC) A G,β ∗ : K G ∗ ( E G ) A G,DL ∗ − → K t ∗ ( C [ G ]) → K t ∗ ( H β ,L ( G )) where the second map is induced b y the natural inclus ion C [ G ] ֒ → H β ,L ( G ). In [JO R], we conjectured that the ima ge of ch ∗ : K t ( H β ,L ( G )) → H C t ∗ ( H β ,L ( G )) lies in the elliptic summand f in H C t ∗ ( H β ,L ( G )) (conjecture β -SrBC). As the inclusion C [ G ] ֒ → H β ,L ( G ) sends f in H C ∗ ( C [ G ]) to f in H C t ∗ ( H β ,L ( G )), naturality of the Chern character ch C K ∗ and Theorem 1 implies Corollary 2 . If A G,β ∗ is r atio nal ly surje ctive, then β -SrBC is tr ue. Since g oing down and then across is rationally injective, we also hav e (compare [O1]) Corollary 3 . The a ssembly ma p A G,DL ∗ ⊗ Q is i nje ctive for al l discr ete gr oups G . W e do not cla im any great originality in this pap er. In fact, Theorem 1, although not officia lly appea ring in prin t b efo re this time, has be e n a “folk-theorem” known to e x p erts for ma n y years. The connection between the Ba um-Connes Conjecture (more precisely a then-h ypo thetical Baum-Co nnes- type Conjecture for C [ G ]) and the stronger Bass Conjecture for C [ G ] discussed in [JOR] was noted by the author in [O2]. 2 There is some ov erlap of this pap er with the results presen ted in [Ji]. A special case of Theore m 1 (for ∗ = 0 and C [ G ] replaced by the ℓ 1 -algebra ℓ 1 ( G )) appea red as the main result of [BCM]. Pro of of Theorem 1 W e use t he notation F f in ∗ ( C [ G ]) to denote the elliptic summand ⊕ x ∈ f in ( ) F ∗ ( C [ G ]) x of F ∗ ( C [ G ]) where F ∗ ( − ) = H H ∗ ( − ) , H N ∗ ( − ) , H C ∗ ( − ) or H P er ( − ). T o maximize co nsistency with [LR], we write S for the (unreduced) susp ension spe ctrum of the zero -sphere S 0 , HN ( R ) re sp. HH ( R ) the Eilenberg- MacLane spe c trum w ho se homotopy groups are the negative cyclic resp. Ho chs child homology groups of the discrete ring R , a nd K a ( R ) t he non-co nnective algebraic K -theory sp ectrum of R , with K a ∗ ( R ) represen ting its homot o p y groups. By [LR, diag. 1 .6] there is a commuting diagram (1.1) H G ∗ ( E G ; S )   / / K a ∗ ( Z [ G ]) N T r ∗   H G ∗ ( E G ; HN ( Z ))   ∼ = / / H N f in ∗ ( Z [ G ])   / / / / H N ∗ ( Z [ G ]) h ∗   H G ∗ ( E G ; HH ( Z )) ∼ = / / H H f in ∗ ( Z [ G ]) / / / / H H ∗ ( Z [ G ]) where the top horizontal map is the comp osition H G ∗ ( E G ; S ) → H G ∗ ( E G ; K a ( Z )) A G,DL − → K ∗ ( Z [ G ]) referred to as the the restricted assembl y map for the algegraic K -groups of Z [ G ]. The other tw o horizontal maps are the a ssemb ly m aps for neg a tive c y c lic and Ho chsc hild homology respect ively . The upper lef t-hand map is induced by the map f rom the sphere sp ectrum to the Eilen be rg -MacLane spec- trum HN , which may b e expressed as the co mp osition of spect ra S → K a ( Z ) → HN . By [LR], the comp osition on the left is a rational equiv alence. Let C δ denote the complex n umbers C equipp ed with t he discrete top olog y . T ensoring with C and combined with the inclusion of group alge bras Z [ G ] ֒ → C δ [ G ], (1.1) yields the commuting diagram (1.2) H G ∗ ( E G ; Q ) ⊗ C ∼ =   / / K a ∗ ( C δ [ G ]) ⊗ C N T r ∗   H N f in ∗ ( C [ G ]) / / / / H N ∗ ( C [ G ]) Next, we consider the transforma tion from algebraic to top olog ic a l K -theory , induced by the ma p of group algebras C δ [ G ] → C [ G ] whic h is the identit y on elements . By the results of [CK], [W] and [T], there is a commuting diagram (1.3) K a ∗ ( C δ [ G ]) ⊗ C N T r ∗   / / K t ∗ ( C [ G ]) ⊗ C ch ∗ ( C [ G ])   H N ∗ ( C [ G ]) / / H P e r ∗ ( C [ G ]) 3 where ch ∗ ( C [ G ]) is the Connes-Karoubi Chern ch aracte r fo r the fine top ological algebra C [ G ], and the bot tom map is the transformatio n from negative cyclic to p erio dic cyclic hom ology . W e c an now consider our main diagram (1.4) H G ∗ ( E G ; C ) ⊗ K ∗ ( C ) / /   K a ∗ ( C δ [ G ]) ⊗ C ⊗ K ∗ ( C ) / /   K t ∗ ( C [ G ]) ⊗ C ⊗ K ∗ ( C ) / / ch ∗ ( C [ G ]) ⊗ ch ∗ ( C [ { id } ])   K t ∗ ( C [ G ]) ⊗ C ch ∗ ( C [ G ])   H N f in ∗ ( C [ G ]) ⊗ K ∗ ( C )   / / H N ∗ ( C [ G ]) ⊗ K ∗ ( C ) / / H P er ∗ ( C [ G ]) ⊗ H P er ∗ ( C ) ∼ = / / H P er ∗ ( C [ G ]) H P er f in ∗ ( C [ G ]) ⊗ K ∗ ( C ) ∼ = / / H P er f in ∗ ( C [ G ]) ⊗ H P er ∗ ( C ) ∼ = / / O O O O H P er f in ∗ ( C [ G ]) O O O O The top left square commutes b y (1 .2), and the middle to p square commutes b y (1.3). The upper right square commutes by virtue of the fact that the Connes-Karoubi-Chern character is a homom orphism of g ra ded mo dules, whic h ma ps the K t ∗ ( C )-mo dule K t ∗ ( C [ G ]) to the H P er ∗ ( C )-mo dule H P er ∗ ( C [ G ]), with the m a p of base rings induced b y isomorphism ch ∗ ( C [ { id } ]) : K t ∗ ( C ) ⊗ C ∼ = − → H P er ∗ ( C ). The low er left square com mutes trivially , while the low er righ t commutes by the naturality o f the inclusion H P e r f in ∗ ( C [ G ]) ֒ → H P er ∗ ( C [ G ]) with re spect to the m o du le structure ov er H P er ∗ ( C ). Summarizing, we get a commuting diagram (1.5) H G ∗ ( E G ; C ) ⊗ K ∗ ( C ) / / ∼ =   K t ∗ ( C [ G ]) ⊗ C ch ∗ ( C [ G ])   H P e r f in ∗ ( C [ G ])   / / / / H P e r ∗ ( C [ G ])   H C f in ∗ ( C [ G ]) / / / / H C ∗ ( C [ G ]) where the bottom square is induced b y the tra nsformation H P er ∗ ( − ) → H C ∗ ( − ), which resp e cts the summand decomp osition indexed on conjuga cy cla sses. Restricted the elliptic summand yields the map H P er f in ∗ ( C [ G ]) → H C f in ∗ ( C [ G ]) which is an isom orphism fo r ∗ ≥ 0, implying the result stated in Theorem 1. . References [BC1] P . Baum, A. C onnes, Gemetric K -the ory for Lie gr oups and foliations , Enseign. Mat h. 46 (2000), 3 – 42. [BC2] P . Baum, A . 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