The Novikov conjecture for algebraic K-theory of the group algebra over the ring of Schatten class operators

The No vik o v conjecture for algebraic K-theory of the group algebra o v er the ring of Sc hatten class op erators Guolia ng Y u ∗ Abstract: In this pap er, we pro v e the algebraic K- theory No vik ov c o njecture for group algebras o ver the ring of Sc hatten class op erators. The main tec hnical to ol in the pro o f is an explicit construction of the Connes-Chern c hara cter. 1 In tro ducti on Let Γ b e a group a nd R b e an H-unital ring. Let R Γ b e the gr o up algebra of the group Γ ov er the ring R . The isomorphism conjecture of F arrell-Jones states that the following assem bly map is an isomorphism: A : H O r Γ n ( E V C Y (Γ) , K ( R ) −∞ ) − → K n ( R Γ) , where V C Y is the family o f virtually cyclic subgroups of Γ, E V C Y (Γ) is the univ ersal Γ-space with isotropy in V C Y , H O r Γ n ( E V C Y (Γ) , K ( R ) −∞ ) is a gene r al- ized Γ-equiv arian t homology theory asso ciated to t he non-connectiv e algebraic K-theory sp ectrum K ( R ) −∞ , and K n ( R Γ) is the algebraic K-theory of R Γ. The isomorphism conjec ture pro vides an algorithm for computing the alge- braic K-theory of R Γ in terms of the algebraic K- theory of R . This conjecture w as intro duced in [FJ1] fo r R = Z and fo r unital rings R in [BFJR ]. When R is H-unital, the isomorphism conjecture follo ws from the unital case b y us- ing the excision theorem in algebraic K-theory [SW]. The algebraic K-theory ∗ The author is partially supp orted by NSF. 1 isomorphism conjecture go es bac k to [H]. There are analogous conjectures in L-theory [Q1] [Q2] and C ∗ -algebra K-theory [BC1]. Imp ortan t cases of the isomorphism conjecture ha ve b een ve r ified in [FJ1] [FJ2] and [BLR]. The algebraic K-theoretic Nov iko v conjecture states that the assem bly map: H n ( B Γ , K ( R ) −∞ ) − → K n ( R Γ) , is rationally inj ectiv e, where B Γ is the classifying space of the group Γ. The algebraic K-theoretic Nov iko v conjecture follo ws from the (r ational) injectivit y part o f the isomorphism conjecture. By a remark able theorem of B¨ okstedt- Hsiang-Madsen [BHM], the algebraic K-theoretic No vik ov conjecture holds for R = Z if the homology g roups of Γ are finitely generated. The main purp ose of this pap er is to prov e t he (ra tional) injectivit y part of t he algebraic K-theory isomorphism conjecture for g roup algebras o ve r the ring of Schatten class op erato rs. As a conseque nce, w e obtain the algebraic K-theory No viko v conjecture for gro up algebras ov er the ring of Sc hatten class op erators. The motiv ation for considering g roup alg ebras ov er the ring of Sc hatten class op erators comes fro m the deep work of Connes -Mosco vici on higher index theory of elliptic op erators and its applications to the No vik ov conjecture [CM]. In Connes-Mosco vici’s higher index theory , the K-theory of the group algebra ov er the r ing of Sc hatten class op erato r s serve s as the re- ceptacle for the higher index of an elliptic op erato r. F or the con v enience o f readers w e recall that, for any p ≥ 1, an op erator T on an infinite dimensional and separable Hilb ert space H is said to b e Sc hatten p -class if tr (( T ∗ T ) p/ 2 ) < ∞ , where tr is the standard trace defined b y tr ( P ) = P n < P e n , e n > fo r an y b ounded op erator P acting on H and an ort ho normal basis { e n } n of H ( tr ( P ) is indep enden t of the c hoice o f the orthonormal basis). Let S p b e the ring of all Sc hatten p -class op erators on an infinite dimensional and separable Hilbert space. W e define the ring S of all Sc hatten class op erators to b e ∪ p ≥ 1 S p . 2 The fo llowing theorem is the main result of t his pap er. Theorem 1.1. L e t S b e the ring of al l Schatten class op er ators on an in fi nite dimensional and sep ar a ble Hilb ert sp ac e. The assembly map A : H O r Γ n ( E V C Y (Γ) , K ( S ) −∞ ) − → K n ( S Γ) is r ational ly in je ctive for any gr oup Γ , wher e S Γ is the gr oup algebr a of the gr o up Γ ov er the ring S . As a consequence, w e obtain the algebraic K-t heoretic No vik ov conjecture for the group algebra S Γ. The main tec hnical to ol in the pro of of Theorem 1.1 is an explicit con- struction of a Connes-Chern c ha racter using an equiv a rian t cyclic simplicial homology theory . As a conseq uence of this explicit construction, w e obtain a lo cal prop ert y of the Connes-Chern c ha r a cter. This lo cal prop erty of the Connes-Chern character pla ys an imp ortant role in the pro of. This pap er is organized as follo ws. In Section 2, we c o llect a few preliminary results whic h will b e used lat er in the pap er. In Section 3, w e reduce our main theorem t o the case of lo we r alg ebraic K-theory . In Section 4, w e in tro duce a cyclic simplicial homolo g y theory to construct a Connes-Chern character. The Connes-Chern c haracter pla ys a crucial r ole in the pro of of the main t heorem. W e use the explic it construction of the Connes -Chern c haracter to prov e an imp ortant lo cal prop erty of the Connes-Chern c haracter for K- theory elemen ts with small propagation. In Section 5, we prov e the main theorem of this pap er. The author wishes to thank Alain Connes, Max Karoubi, Xiang T ang, An- dreas Thom, Shm uel W ein b erger, and Rufus Willett for inspiring discussions and v ery helpful commen ts. In particular, the author w ould lik e to express his gratitude to Guillermo Corti ˜ nas for his detailed commen t s ab o ut the pa p er and sev eral stim ulating discussions. W e w ould lik e to men tion that Guillermo Corti ˜ nas and Giesela T artaglia ha ve giv en a new pro of of Theorem 1.1 in [CT a]. Part of this w ork w as done at Shanghai Cen ter for Mathematical Sci- ences (SCMS) and the author w ishes to t ha nk SCMS for providin g excellen t w orking en vironmen t. 3 2 Preliminaries In this section, w e collect a few concepts and results useful for this pap er. Let R b e a ring and let R + b e the unital ring o btained from R b y adjo ining a unit. The ring R is defined to b e H- unital if T or R + i ( Z , Z ) = 0 for all i . The imp ortance of H-unita lit y is that it guarantees excision in algebraic K-theory [SW]. If R is a Q -alg ebra and R + Q is the unital Q -a lgebra obtained f r om R with the unit adjoined, then R is H-unital if and only if T or R + Q i ( Q , Q ) = 0 [SW]. The fo llowing result follows from [W1] and Theorem 8.2.1 of [CT]. Theorem 2.1. S is H-unital. By Theorem 7.10 in [SW], w e ha ve : Theorem 2.2. If R is H-unital, then R Γ is H-unital for an y gr oup Γ . As a conseque nce, we obtain that S Γ is H-unital. Recall that a ring R is called K n -regular if the natural map: K n ( R ) → K n ( R [ t 1 , · · · , t m ]) , is an isomorphism for eac h m ≥ 1. W e say that R is K -regular if R is K n - regular f o r all n . The fo llowing result is a sp ecial case of Theorem 8.2.5 in [CT]. Theorem 2.3. S is K -r e gular. The fo llowing result follows from the pro of of Prop osition 2.1 4 in [LR]. Prop osition 2.4. If R is a K - r e gular R -algeb r a, then the natur al map: H O r Γ n ( E F I N (Γ) , K ( R ) −∞ ) → H O r Γ n ( E V C Y (Γ) , K ( R ) −∞ ) , is an isom o rphism, whe r e F I N is the family of finite sub gr oups of Γ and E F I N (Γ) is the univ e rsal Γ -sp a c e with isotr opy in F I N . 4 The ab ov e prop osition implies that the isomorphism conjecture f or the ring S is equiv alen t to the statemen t that the assem bly map: A : H O r Γ n ( E F I N (Γ) , K ( S ) −∞ ) → K n ( S Γ) , is a n isomorphism and Theorem 1.1 is equiv alen t to the statemen t that the ab ov e assem bly ma p is ratio nally injectiv e. By a result in [CT ], we kno w that K n ( S ) is 2-p erio dic and K 0 ( S ) = Z and K 1 ( S ) = 0. This implies tha t the domain of the assem bly map A in Theorem 1.1 is r ationally isomorphic to ⊕ k even H O r Γ n + k ( E F I N (Γ) , Q ) . 3 Reduction to the lo wer algebraic K -theory case In this section, w e prov e the following reduction result. Prop osition 3.1. The or em 1.1 fol lows fr om the fol lowi n g sp e cial c ase of the the or em for low er algebr ai c K-the ory: i.e . the assembly ma p A : H O r Γ n ( E V C Y (Γ) , K ( S ) −∞ ) → K n ( S Γ) is r ational ly inje ctive for n ≤ 0 . Pr o of. By Prop osition 7.2.3, R emark 7.2.6 a nd Theorem 8.2.5 in [CT], w e ha v e K n ( S ) = Z when n is ev en a nd K n ( S ) = 0 when n is o dd. It follo ws that H − 2 ( pt, K ( S ) −∞ ) = Z . By definition, the assem bly map: A : H − 2 ( pt, K ( S ) −∞ ) → K − 2 ( S ) is an isomorphism and it maps t he generator z of H − 2 ( pt, K ( S ) −∞ ) to the Bott elemen t of K − 2 ( S ) (denoted b y b ). F or any p ositiv e in teger k , w e can use the pro duct op eratio n to construct the Bott elem ent b k in K − 2 k ( S ) , whe re the pro duct is defined using a natural (injectiv e) homomorphism S ⊗ S → S 5 induced b y the homomorphism S ( H ) ⊗ S ( H ) → S ( H ⊗ H ) and the isomor- phism S ( H ⊗ H ) ∼ = S ( H ) (here H is an infinite dimensional and separable Hilb ert space, S ( H ) a nd S ( H ⊗ H ) res p ectiv ely denote t he r ing s of Sc hatten class operato rs on H and H ⊗ H , and S ⊗ S is the algebraic tensor product of S with S ). When n = 2 k , w e ha v e the fo llo wing comm utativ e diagr a m: H O r Γ n ( E V C Y (Γ) , K ( S ) −∞ ) A → K n ( S Γ) ↓ × z k ↓ × b k H O r Γ 0 ( E V C Y (Γ) , K ( S ) −∞ ) A → K 0 ( S Γ) , where the ve rt ical pro duct maps are w ell defined with the help of a natura l homomorphism S ⊗ S → S and the K -theory prop erties of S and S Γ b eing H -unital (Theorems 2.1 and 2 .2). By Theorems 8 .2.5 and 6.5.3 in [CT] and Theorem 8.3 (the Bott p erio dicit y) in [Cu2], w e kno w tha t the Bott elemen t b k is a generato r of K − 2 k ( S ) . It f o llo ws that the pro duct map H i ( pt, K ( S ) −∞ ) × z k − → H i − 2 k ( pt, K ( S ) −∞ ) is an isomorphism for ev ery integer i . This implies that the pro duct map H O r Γ i ( E V C Y (Γ) , K ( S ) −∞ ) × z k − → H O r Γ i − 2 k ( E V C Y (Γ) , K ( S ) −∞ ) is an isomorphism b y using t he fa ct that b oth ho mology theories { H O r Γ i ( · , K ( S ) −∞ ) } i ∈ Z and { H O r Γ i − 2 k ( · , K ( S ) −∞ ) } i ∈ Z ha ve a Ma y er-Vietor is sequence and a fiv e lemma argument. When n = 2 k + 1, w e hav e the follo wing comm utativ e diagr a m: H O r Γ n ( E V C Y (Γ) , K ( S ) −∞ ) A → K n ( S Γ) ↓ × z k +1 ↓ × b k +1 H O r Γ − 1 ( E V C Y (Γ) , K ( S ) −∞ ) A → K − 1 ( S Γ) , where the ve rt ical pro duct maps are w ell defined with the help of a natura l homomorphism S ⊗ S → S and the K -theory prop erties of S and S Γ b eing H -unital (Theorems 2.1 and 2.2). By the same arg umen t as in the ev en case, w e kno w tha t the pro duct map H i ( pt, K ( S ) −∞ ) × z k +1 − → H i − (2 k +2) ( pt, K ( S ) −∞ ) 6 is an isomorphism for ev ery in teger i . This fact, together with a standard Ma y er- Vietoris sequenc e and fiv e lemma argumen t, implies that the pro duct map H O r Γ i ( E V C Y (Γ) , K ( S ) −∞ ) × z k +1 − → H O r Γ i − (2 k +2) ( E V C Y (Γ) , K ( S ) −∞ ) is an isomorphism. No w Prop o sition 3.1 follows from t he ab o ve comm utativ e diagrams and the fact that the left ve rtical maps in the dia grams are isomorphisms. 4 Cyclic simplicial ho mology theory and the Connes -Chern c haracter In this section, w e in tro duce an equiv ariant cyclic simplicial homology theory to construct the Connes - Chern c haracter for K n ( S Γ) when n ≤ 0. The Connes - Chern character is a k ey to ol in t he pro of of the main theorem. W e use this explicit construction to prov e an imp ortan t lo cal prop erty of the Connes-Chern c haracter for K-theory ele ments with sm a ll propagation. This lo cal prop erty will b e useful in the pro of of the main theorem. Let X b e a simplicial complex. Let σ b e a simplex of X . Define tw o orderings of its v ertex set to b e equiv alen t if they differ from each other b y an ev en p erm utation. Each of the equiv alence classes is called an orientation of σ . If { v 0 , ..., v k } is the set of all v ertices of σ , w e use the sym b o l [ v 0 , · · · , v k ] to denote the orien ted simplex with the particular ordering ( v 0 , · · · , v k ). A lo cally finite k -c hain on X is a forma l sum X ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) [ v 0 , · · · , v k ] , where (1) the summation is tak en ov er all orderings ( v 0 , · · · , v k ) of all k -simplices { v 0 , · · · , v k } of X and c ( v 0 , ··· ,v k ) ∈ C ; (2) [ v 0 , · · · , v k ] is iden tified with − [ v ′ 0 , · · · , v ′ k ] in the ab ov e sum if ( v 0 , · · · , v k ) and ( v ′ 0 , · · · , v ′ k ) are opp osite orien t a tions of the same simplex; 7 (3) f or any compact subs et K of X , there are at most finitely man y or dered simplices ( v 0 , · · · , v k ) interse cting K suc h that c ( v 0 , ··· ,v k ) 6 = 0. W e remark that in the a b o ve definition the summation is tak en o ver all ( v 0 , · · · , v k ) instead of [ v 0 , · · · , v k ] for the purp ose to hav e consis t ent notations in the definitions o f the Connes-Chern c hara cters for lo w er a lgebraic K -groups of S 1 Γ using simplicial homology groups and lo wer algebraic K - groups of S p Γ using cyclic simplicial homolo gy groups lat er in this section. Let C k ( X ) b e the ab elian group of all lo cally finite k - c hains on X . Let ∂ k : C k ( X ) → C k − 1 ( X ) b e the standard simplicial b oundary map. W e define the lo cally finite simpli- cial homolog y group: H n ( X ) = K er ∂ n /I m ∂ n +1 . If X has a prop er simplicial action of Γ, let C Γ k ( X ) ⊂ C k ( X ) b e the ab elian group consisting of all Γ-in v arian t lo cally finite k -chains on X . Let ∂ Γ k : C Γ k ( X ) → C Γ k − 1 ( X ) b e the r estriction of the standard simplicial b o undary map. W e define the lo cally finite Γ-equiv ariant simplicial ho molo gy group: H Γ n ( X ) = K er ∂ Γ n /I m ∂ Γ n +1 . Without loss of generalit y , w e can assume that Γ is a countable g r o up (this is b ecause ev ery group is an inductiv e limit of coun table groups). W e endo w Γ with a prop er left in v a rian t length metric (here prop erness simply means that ev ery ball with finite radius has finitely man y elemen ts). W e remark that suc h a prop er length metric alw a ys exists for an y coun table group. F or eac h d ≥ 0 , the Rips comple x P d (Γ) is the simplicial complex with Γ as its v ertex set and where a finite subset { γ 0 , · · · , γ n } of Γ forms a simplex iff d ( γ i , γ j ) ≤ d for all 0 ≤ i, j ≤ n. It is not diffic ult to sho w that ∪ d ≥ 1 P d (Γ) is a mo del for E F I N (Γ). When Γ is torsion free, ∪ d ≥ 1 P d (Γ) is a univ ersal s pa ce fo r free and 8 prop er action of Γ. In this case, lim d →∞ H Γ n ( P d (Γ)) is the group homology of Γ defined using a standard resolution. T o motiv ate the general construction of the Connes-Chern character, w e shall first consider the special case when Γ is torsion free. Let A (Γ , S ) b e the algebra of all k ernels k : Γ × Γ → S suc h that (1) for eac h k , there exists r ≥ 0 suc h that k ( x, y ) = 0 if d ( x, y ) > r (the smallest suc h r is called the propagation of k ); (2) k is Γ-in v arian t , i.e. k ( g x, g y ) = k ( x, y ) for all g ∈ Γ and ( x, y ) ∈ Γ × Γ; (3) The pro duct in A (Γ , S ) is defined b y: ( k 1 k 2 )( x, y ) = X z ∈ Γ k 1 ( x, z ) k 2 ( z , y ) . W e iden tify S Γ with A (Γ , S ) b y the isomorphism: X g ∈ Γ s g g → k ( x, y ) = s x − 1 y , where s g ∈ S for eac h g ∈ Γ . F or each p ≥ 1, w e can naturally iden tify S p Γ with A (Γ , S p ), where A (Γ , S p ) is defined b y replacing S with S p in the ab o ve definition of A (Γ , S ). F or eac h no n- negativ e even in teger n , w e shall first define the Connes-Chern c haracter c n for a countable torsion free group Γ c n : K 0 ( S 1 Γ) → lim d →∞ ( ⊕ k even , k ≤ n H Γ k ( P d (Γ))) b y: [ ˜ q ] − [ q 0 ] → X k even , k ≤ n X ( x 0 , ··· ,x k ) tr ( q ( x 0 , x 1 ) q ( x 1 , x 2 ) · · · q ( x k , x 0 ))[ x 0 , · · · , x k ] , where P d (Γ) is the Rips complex and ˜ q is an idemp oten t in M m (( S 1 Γ) + ), ˜ q = q + q 0 for some q ∈ M m ( S 1 Γ) and idemp oten t q 0 ∈ M m ( C ), a nd the summation is ta ken ov er all orderings ( x 0 , · · · , x k ) of all n -simplices { x 0 , · · · , x k } of P d (Γ) for some d large enough suc h that q ( x, y ) = 0 if d ( x, y ) > d/ ( n + 1). 9 Prop osition 4.1. L et Γ b e a c o untable torsion fr e e gr oup. F or e ach non- ne gative even in te ger n , the Connes-Che rn ch a r acter c n is a wel l define d homomorphism fr om K 0 ( S 1 Γ) to lim d →∞ ( ⊕ k even, k ≤ n H Γ k ( P d (Γ))) . Pr o of. W e first observ e t ha t c ([ ˜ q ] − [ q 0 ]) is Γ-in v arian t b y using the Γ-inv ariance of q . F or eac h ev en k ≤ n , w e shall pro ve that ∂ Γ k ( X ( x 0 , ··· ,x k ) tr ( q ( x 0 , x 1 ) q ( x 1 , x 2 ) · · · q ( x k , x 0 ))[ x 0 , · · · , x k ]) = 0 . This implies that c ([ ˜ q ] − [ q 0 ]) is a cycle. W e lea v e to the reader t he pro of that the homology class of X k even , k ≤ n X ( x 0 , ··· ,x k ) tr ( q ( x 0 , x 1 ) q ( x 1 , x 2 ) · · · q ( x k , x 0 ))[ x 0 , · · · , x k ] dep ends only on the K-theory class [ ˜ q ] − [ q 0 ]. By the assumption that ˜ q and q 0 are idemp otents, w e ha ve q 2 = q − q 0 q − q q 0 . It follows that ∂ k ( X ( x 0 , ··· ,x k ) tr ( q ( x 0 , x 1 ) q ( x 1 , x 2 ) · · · q ( x k , x 0 ))[ x 0 , · · · , x k ]) = k X i =1 ( − 1) i X ( x 0 , ··· , ˆ x i , ··· ,x k ) tr ( q ( x 0 , x 1 ) q ( x 1 , x 2 ) · · · q 2 ( x i − 1 , x i +1 ) · · · q ( x k , x 0 ))[ x 0 , · · · , ˆ x i , · · · , x k ] + X ( x 1 , ··· ,x k ) tr ( q ( x 1 , x 2 ) · · · q 2 ( x k , x 1 ))[ x 1 , · · · , x k ] = 10 k X i =1 ( − 1) i X ( x 0 , ··· , ˆ x i , ··· ,x k ) tr ( q ( x 0 , x 1 ) q ( x 1 , x 2 ) · · · q ( x i − 1 , x i +1 ) · · · q ( x k , x 0 ))[ x 0 , · · · , ˆ x i , · · · , x k ] + X ( x 1 , ··· ,x k ) tr ( q ( x 1 , x 2 ) · · · q ( x k , x 1 ))[ x 1 , · · · , x k ] − ( k X i =1 ( − 1) i X ( x 0 , ··· , ˆ x i , ··· ,x k ) tr ( q ( x 0 , x 1 ) q ( x 1 , x 2 ) · · · q 0 q ( x i − 1 , x i +1 ) · · · q ( x k , x 0 ))[ x 0 , · · · , ˆ x i , · · · , x k ] + X ( x 1 , ··· ,x k ) tr ( q ( x 1 , x 2 ) · · · q 0 q ( x k , x 1 ))[ x 1 , · · · , x k ] + k X i =1 ( − 1) i X ( x 0 , ··· , ˆ x i , ··· ,x k ) tr ( q ( x 0 , x 1 ) q ( x 1 , x 2 ) · · · q ( x i − 1 , x i +1 ) q 0 · · · q ( x k , x 0 ))[ x 0 , · · · , ˆ x i , · · · , x k ] + X ( x 1 , ··· ,x k ) tr ( q ( x 1 , x 2 ) · · · q ( x k , x 1 ) q 0 )[ x 1 , · · · , x k ]) . Using t he tra ce prop ert y and the definition o f oriented simplices and the as- sumption t ha t k is ev en, we ha ve X ( x 0 , ··· , ˆ x i , ··· ,x k ) tr ( q ( x 0 , x 1 ) q ( x 1 , x 2 ) · · · q ( x i − 1 , x i +1 ) · · · q ( x k , x 0 ))[ x 0 , · · · , ˆ x i , · · · , x k ] = 0 , X ( x 1 , ··· ,x k ) tr ( q ( x 1 , x 2 ) · · · q ( x k , x 1 ))[ x 1 , · · · , x k ] = 0 , X ( x 0 , ··· , ˆ x i , ··· ,x k ) ( tr ( q ( x 0 , x 1 ) q ( x 1 , x 2 ) · · · q 0 q ( x i − 1 , x i +1 ) · · · q ( x k , x 0 ))[ x 0 , · · · , ˆ x i , · · · , x k ] + tr ( q ( x 0 , x 1 ) q ( x 1 , x 2 ) · · · q ( x i − 1 , x i +1 ) q 0 · · · q ( x k , x 0 ))[ x 0 , · · · , ˆ x i , · · · , x k ]) = 0 , X ( x 1 , ··· ,x k ) ( tr ( q ( x 1 , x 2 ) · · · q 0 q ( x k , x 1 ))[ x 1 , · · · , x k ] + tr ( q ( x 1 , x 2 ) · · · q ( x k , x 1 ) q 0 )[ x 1 , · · · , x k ]) = 0 . By the definition of low er algebraic K-theory using the group algebra o ve r the free ab elian g roup Z n , we can similarly define the Connes-Chern character: c n : K i ( S 1 Γ) → lim d →∞ ( ⊕ k + i ev en, k ≤ n H Γ k ( P d (Γ))) for eac h non-negat ive in teger n and i < 0. Next w e extend the construction of the Connes-Chern c haracter c n to the S p case for eac h p ≥ 1 and eac h non-negativ e in teger n when Γ is torsion free: c n : K 0 ( S p Γ) → lim d →∞ ( ⊕ k ev en, k ≤ n H Γ k ( P d (Γ))) . 11 W e need to in t ro duce an equiv arian t cyclic simplicial homology group to define the Connes-Chern c hara cter. Let X b e a simplicial complex. An ordered k -simplex ( v 0 , · · · , v k ) is defined to b e an ordered finite sequence of v ertices in a simplex of X , where v i is allo we d to b e equal to v j for some distinct pair of i and j . Recall that the following p erm utatio n is called a cyclic p erm utation ( v 0 , ..., v k ) → ( v k , v 0 , · · · , v k − 1 ) . W e define tw o ordered sim plices ( v 0 , · · · , v k ) and ( v ′ 0 , · · · , v ′ k ) to b e equiv alen t if one ordered simplex can b e obta ined from the other ordered simplex b y an y n umber of cyclic p ermutations when k is ev en and by an eve n n umber of cyclic p ermutations when k is o dd. Eac h of the equiv alence classes is called a cyclically orien ted simplex. If ( v 0 , ..., v k ) is an ordered simplex of X , w e use the sym b ol [ v 0 , · · · , v k ] λ to denote the corresp onding cyclically oriented simplex. A lo cally finite cyclic k -chain on X is a formal sum X ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) [ v 0 , · · · , v k ] λ , where (1) the summation is ta k en o v er all ordered simplices ( v 0 , · · · , v k ) of X and c ( v 0 , ··· ,v k ) ∈ C ; (2) [ v 0 , · · · , v k ] λ is identified with ( − 1) k [ v k , v 0 , · · · , v k − 1 ] λ in the ab o ve sum; (3) f or any compact subs et K of X , there are at most finitely man y or dered simplices ( v 0 , · · · , v k ) interse cting K suc h that c ( v 0 , ··· ,v k ) 6 = 0. Let C λ k ( X ) b e the abelian g r oup of all lo cally finite cyclic k -chains on X . Let ∂ λ k : C λ k ( X ) → C λ k − 1 ( X ) b e the standard b oundary map. W e define the cyclic simplicial ho mology gro up: H λ n ( X ) = K er ∂ λ n /I m ∂ λ n +1 . If X ha s a simplicial prop er action of a gro up Γ, w e can define C λ, Γ k ( X ) ⊆ C λ k ( X ) to be the subspace of a ll Γ-in v ariant lo cally finite cyclic k -c hains o n X . 12 Let ∂ λ, Γ k : C λ, Γ k ( X ) → C λ, Γ k − 1 ( X ) b e the restriction of the standard b o undar y map. W e define the Γ-equiv ariant cyc lic simplicial homology gro up H λ, Γ n ( X ) b y: H λ, Γ n ( X ) = K er ∂ λ, Γ n /I m ∂ λ, Γ n +1 . The fo llo wing result computes the Γ-equiv ariant cyclic simplicial homology group in terms of t he Γ-equiv a rian t simplicial homolog y groups. Prop osition 4.2. L et Γ b e a gr oup. L et X b e a sim plicial c omplex with a pr o p er simplic i a l action of Γ . We have H λ, Γ n ( X ) ∼ = ( ⊕ k ≤ n, k = n mod 2 H Γ k ( X )) . Pr o of. Giv en an ordering ( v 0 , · · · , v k ) of k + 1 num b er of v ertices in X , w e use the same notation ( v 0 , · · · , v k ) to denote the corresp onding ordered k - simplex. Let C or d, Γ k ( X ) b e the ab elian gr o up of all Γ- inv ariant lo cally finite ordered k -ch a ins X ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) ( v 0 , · · · , v k ) , where the sum is tak en ov er all ordered k -simplices of X and c ( v 0 , ··· ,v k ) ∈ C . W e remark that, in the ab ov e defin it ion, a pa ir of v ertices in ( v 0 , · · · , v k ) are allo wed to b e the same. Let C or d, Γ k , 0 ( X ) b e the ab elian subgroup o f C or d, Γ k ( X ) consisting of all Γ- in v arian t lo cally finite ordered k -c hains with the fo llo wing special form: X v c v k+1 z }| { ( v , · · · , v ) , where the sum is taken o ver all ve rt ices of X and c v ∈ C . W e define an ab elian group C or d, Γ k ,r ed ( X ) by: C or d, Γ k ,r ed ( X ) :=    C or d, Γ k ( X ) if k is ev en, C or d, Γ k ( X ) /C or d, Γ k , 0 ( X ) if k is o dd, where C or d, Γ k ( X ) /C or d, Γ k , 0 ( X ) is the quotien t group of C or d, Γ k ( X ) o v er C or d, Γ k , 0 ( X ). 13 The standard b oundary ma p on C or d, Γ k ( X ) induces a b o undary map: ∂ or d, Γ k ,r ed : C or d, Γ k ,r ed ( X ) − → C or d, Γ k − 1 ,r ed ( X ) . W e define a new homology group: H or d, Γ n,r ed ( X ) = K er ∂ or d, Γ n,r ed /I m ∂ or d, Γ n +1 ,r ed . Let χ k b e the natural chain map fro m C or d, Γ k ,r ed ( X ) to C λ, Γ k ( X ) defined by : [ X ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) ( v 0 , · · · , v k )] − → X ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) [ v 0 , · · · , v k ] λ , for every [ P ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) ( v 0 , · · · , v k )] ∈ C or d, Γ k ,r ed ( X ). This map is w ell de- fined b ecause k+1 z }| { [ v , · · · , v ] λ = 0 when k is o dd. By a standard Ma ye r- Vietoris and five lemma argumen t, it is not difficult to prov e that χ induces an isomorphism χ ∗ from H or d, Γ n,r ed ( X ) to H λ, Γ n ( X ). This is b ecause b oth homology theories satisfies the Ma y er-Vietoris seq uences and χ ∗ is a n isomorphism when X = Γ /F as Γ-spaces for some finite subgroup F of Γ. W e define a nat ural chain map φ k ,k : C or d, Γ k ,r ed ( X ) → C Γ k ( X ) b y: [ X ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) ( v 0 , · · · , v k )] − → X ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) [ v 0 , · · · , v k ] for every [ P ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) ( v 0 , · · · , v k )] ∈ C or d, Γ k ,r ed ( X ). This map is w ell de- fined b ecause k+1 z }| { [ v , · · · , v ] = 0 when k is o dd (more generally when k ≥ 1). F or eac h ordered k -simplex ( v 0 , · · · , v k ) of X , w e let ϕ k ,k − 2 (( v 0 , · · · , v k )) :=    ( − 1) j − i + 1 [ v 0 , · · · , ˆ v i , · · · , ˆ v j , · · · , v k ] if ( i, j ) is the smallest pair suc h that i < j , v i = v j , 0 if there exists no pair i < j suc h tha t v i = v j , 14 where the smallest ( i, j ) is ta ken with resp ect to the dictionary order of { ( m, l ) : 0 ≤ m < l ≤ k } giv en b y: ( m, l ) < ( m ′ , l ′ ) iff eithe r (1) m < m ′ , or (2) m = m ′ and l < l ′ . W e define a linear map φ k ,k − 2 : C or d, Γ k ,r ed ( X ) → C Γ k − 2 ( X ) b y: [ X ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) ( v 0 , · · · , v k )] − → X ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) ϕ k ,k − 2 (( v 0 , · · · , v k )) for ev ery [ P ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) ( v 0 , · · · , v k )] ∈ C or d, Γ k ,r ed ( X ). Note that φ k ,k − 2 is well defined. Elemen tary computations show that φ k ,k − 2 is a chain map. Similarly w e can construct a chain map φ k ,l : C or d, Γ k ,r ed ( X ) → C Γ l ( X ) if 0 ≤ l ≤ k and k − l is ev en. Using t he ab ov e c hain maps, we construct a chain map: ψ n = ⊕ l ≤ n, n − l ev en φ n,l : C or d, Γ n,r ed ( X ) → ( ⊕ l ≤ n, n − l ev en C Γ l ( X )) . The c hain map ψ n induces an isomorphism ψ ∗ on the homology gro ups since b oth homology theories satisfy the Ma y er- Vietoris sequence and the c hain map induces a n isomorphism at the homolo gy lev el if X = Γ /F as Γ-spaces for some finite subgroup F o f Γ. Finally Proposition 4.2 follows from the facts that χ ∗ and ψ ∗ are isomor- phisms. F or any p ositiv e ev en in teger n ≥ p , w e now define the Connes-Chern c haracter for a coun table to rsion free group Γ c n : K 0 ( S p Γ) → H λ, Γ n ( P d (Γ)) , where Γ is endo w ed with a prop er length metric. Let ˜ q b e an idemp oten t in M m (( S p Γ) + ) and ˜ q = q + q 0 for some q ∈ M m ( S p Γ) and idempo ten t q 0 ∈ M m ( C ). W e iden tify q with an elemen t in 15 A (Γ , S p ) (no t e that A (Γ , M m ( S p )) is isomorphic to A (Γ , S p )). Let d b e greater than or equal to n + 1 times the propagation of q , i.e. q ( x, y ) = 0 if d ( x, y ) > d/ ( n + 1 ) . F or each p ositiv e ev en integer n ≥ p , the Connes-Chern c hara cter c n of [ ˜ q ] − [ q 0 ] is defined to b e homology class of X ( x 0 , ··· ,x n ) tr ( q ( x 0 , x 1 ) · · · q ( x n , x 0 ))[ x 0 , · · · , x n ] λ ∈ H λ, Γ n ( P d (Γ)) , where the summation is tak en o v er all ordered n -simplices ( x 0 , · · · , x n ) of P d (Γ). W e remark that the c hoice o f n guar an tees that the trace in t he ab ov e definition of the Connes-Chern c haracter is finite. By Prop osition 4.2, the ab ov e Connes-Chern c ha r a cter induces a Connes- Chern character: c n : K 0 ( S p Γ) → lim d →∞ ( ⊕ k even , k ≤ n H Γ k ( P d (Γ))) for any non-negativ e in teger n ≥ p . F or an arbitra ry non-negative even integer n , let n ′ b e a p ositiv e ev en in teger satisfying n ′ ≥ max { n, p } . Let π n ′ ,n b e the natural pro jection fr o m lim d →∞ ( ⊕ k ev en, k ≤ n ′ H Γ k ( P d (Γ))) to lim d →∞ ( ⊕ k ev en, k ≤ n H Γ k ( P d (Γ))). W e de- fine the Connes-Chern ch a racter c n from K 0 ( S p Γ) to lim d →∞ ( ⊕ k ev en, k ≤ n H Γ k ( P d (Γ))) to b e π n ′ ,n ◦ c n ′ . It is not difficult to v erify that the definition of c n is indep enden t of the c hoice of n ′ . Prop osition 4.3. L et Γ b e a c ountable torsion fr e e gr oup. F or any n o n- ne gative even inte ger n , the C o nnes-Chern char acter c n is a wel l define d ho- momorphism fr om K 0 ( S p Γ) to lim d →∞ ( ⊕ k even, k ≤ n H Γ k ( P d (Γ))) . The pro of of the ab o ve proposition is similar to the pro of of Prop osition 4.1 and is therefore omitted. Note that when p = 1, the ab ov e definition of the Connes-Che rn c haracter coincides with the prior definition of the Connes- Chern character. Next we shall construct the Connes-Chern c haracter for a general group Γ. 16 Let Γ f in b e the set of all elemen ts with finite order in Γ . The group Γ acts on Γ f in b y conjugations: γ · x = γ xγ − 1 for all γ ∈ Γ and x ∈ Γ f in . Let X b e a simplicial complex with a prop er simplicial action of Γ. Equip the v ertex set V ( X ) of X with a Γ- in v arian t prop er pseudo metric d V . Let Γ act on Γ f in × X diago nally . Let r ≥ 0. F or eac h g ∈ Γ f in , w e define X g ,r to b e the simplicial sub complex of X c onsisting all simplices { v 0 , · · · , v p } satisfying d V ( v i , g v i ) ≤ r for all 0 ≤ i ≤ p. F or eac h ordered simplex ( v 0 , ..., v k ) of X g ,r , we define the following trans- formation to b e a g -cyclic p erm utation ( v 0 , ..., v k ) → ( g v k , v 0 , · · · , v k − 1 ) . W e define tw o ordered simplices ( v 0 , · · · , v k ) and ( v ′ 0 , · · · , v ′ k ) of X g ,r to b e g - equiv alen t if one ordered simple x can b e obtained f rom the other ordered simplex b y a n y n um b er of g -cyclic p erm utations of ordered simplices in X g ,r when k is ev en and by an ev en n umber o f g - cyclic p erm utations when k is o dd. Each of the equiv alence classes is called a g -cyclically o rien ted simplex. If ( v 0 , ..., v k ) is an ordered simplex of X g ,r , w e use the sym b ol [ v 0 , · · · , v k ] λ,g to denote the corresponding g -cyclically oriented simplex. W e define C λ k ,r ( X ) to b e the ab elian gro up of all lo cally finite k -c hains: X g ∈ Γ f in ( g , X ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) ,g [ v 0 , · · · , v k ] λ,g ) , where (1) t he second summation is ta k en o v er all ordered simplices ( v 0 , · · · , v k ) of X g ,r and c ( v 0 , ··· ,v k ) ,g ∈ C ; (2) [ v 0 , · · · , v k ] λ,g is iden tified with ( − 1) k [ g v k , v 0 , · · · , v k − 1 ] λ,g in the ab o ve sum; (3) f o r eac h g ∈ Γ f in and any compact subset K o f X , there are at most finitely man y ordered simplices ( v 0 , · · · , v k ) in tersecting K suc h that c ( v 0 , ··· ,v k ) ,g 6 = 0. 17 The diago nal action of Γ on Γ f in × X induces a natural Γ-action on C λ k ,r ( X ). Let C λ, Γ k ,r ( X ) ⊆ C λ k ,r ( X ) b e the ab elian group consis ting of all Γ-inv arian t k - c hains in C λ k ,r ( X ). W e ha v e a natura l b oundary map: ∂ λ, Γ k ,r : C λ, Γ k ,r ( X ) − → C λ, Γ k − 1 ,r ( X ) . W e define the follo wing equiv arian t homology theory b y: H λ, Γ n,r ( X ) = K er ∂ λ, Γ n,r /I m ∂ λ, Γ n +1 ,r . When Γ is torsion f ree, Γ f in consists of t he iden tity elemen t and w e hav e H λ, Γ n,r ( X ) = H λ, Γ n ( X ) . F or eac h r ≥ 0, let ˆ X r b e the simplical subspace of Γ f in × X defined b y: ˆ X r = { ( g , x ) ∈ Γ f in × X : x ∈ X g ,r } . The diago nal action of Γ on Γ f in × X induces a natural Γ-a ction on ˆ X r . W e define H Γ n,r ( X ) = H Γ n ( ˆ X r ) . The following result computes our new equiv arian t homology theory o f the Rips complex in terms of the (lo cally finite) eq uiv ariant simplicial homolog y theory . Prop osition 4.4. L et Γ b e a c ountable gr oup with a pr op er length metric. We have lim d →∞ lim r →∞ H λ, Γ n,r ( P d (Γ)) ∼ = lim d →∞ lim r →∞ ( ⊕ k ≤ n, k = n mod 2 H Γ n,r ( P d (Γ))) . Pr o of. Let X b e a simplicial complex with a prop er and co compact action o f Γ. W e define an equiv alence relation ∼ o n t he c hain group C λ, Γ k ,r ( X ) as follo ws. Tw o chains z and z ′ in C λ, Γ k ,r ( X ) a r e said to b e equiv alen t if z = X g ∈ Γ f in ( g , X ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) ,g [ v 0 , · · · , v k ] λ,g ) , 18 z ′ = X g ∈ Γ f in ( g , X ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) ,g [ v ′ 0 , · · · , v ′ k ] λ,g ) , and for eac h 0 ≤ i ≤ k the r e exists an in teger j suc h tha t v ′ i = g j v i . Let C λ, Γ k ,r ( X ) b e t he chain group C λ, Γ k ,r ( X ) / ∼ . W e define ˜ H λ, Γ n,r ( X ) to b e the n-th homology group of C λ, Γ k ,r ( X ). The quotient c hain ma p φ from C λ, Γ k ,r ( X ) to C λ, Γ k ,r ( X ) induces a homomor- phism φ ∗ : H λ, Γ n,r ( X ) → ˜ H λ, Γ n,r ( X ) . W e observ e that t he cocompactness of the Γ action on X implies tha t, for eac h r ≥ 0, t here exists N > 0 suc h that if g ∈ Γ f in and X g ,r is nonempt y , then the order of the group elemen t g is bounded b y N . As a conseq uence, for an y d ≥ 0 a nd r ≥ 0 , there exist d ′ ≥ d and r ′ ≥ r suc h tha t, for any g ∈ Γ f in and any simplex in ( P d (Γ)) g ,r with ve rt ices { v 0 , · · · , v k } , the simplex with v ertices { g i 0 v 0 , · · · , g i k v k : 1 ≤ i j ≤ N , 0 ≤ j ≤ k } is a simplex in ( P d ′ (Γ)) g ,r ′ . This implies tha t φ is a c ha in homotop y e quiv alence from the c hain comp lex lim d →∞ lim r →∞ C λ, Γ k ,r ( P d (Γ)) to the chain complex lim d →∞ lim r →∞ C λ, Γ k ,r ( P d (Γ)) with a homotop y in verse c hain map ψ f rom lim d →∞ lim r →∞ C λ, Γ k ,r ( P d (Γ)) to lim d →∞ lim r →∞ C λ, Γ k ,r ( P d (Γ)) defined b y ψ ([ X g ∈ Γ f in ( g , X ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) ,g [ v 0 , · · · , v k ] λ,g )]) = X g ∈ Γ f in ( g , X ( v 0 , ··· ,v k ) 1 n k +1 g n g X i 0 , ··· ,i k =1 c ( v 0 , ··· ,v k ) ,g [ g i 0 v 0 , · · · , g i k v k ] λ,g ) for eac h [ P g ∈ Γ f in ( g , P ( v 0 , ··· ,v k ) c ( v 0 , ··· ,v k ) ,g [ v 0 , · · · , v k ] λ,g )] in lim d →∞ lim r →∞ C λ, Γ k ,r ( P d (Γ)), where n g is the order of the g r o up elemen t g . It follo ws that the homomorphism φ ∗ is an isomorphism from lim d →∞ lim r →∞ H λ, Γ n,r ( P d (Γ)) to lim d →∞ lim r →∞ ˜ H λ, Γ n,r ( P d (Γ)) . Tw o v ertices v and v ′ of X g ,r are defined to b e equiv alen t if v = g j v ′ for some j . W e denote the equiv alence class o f v by [ v ]. W e define ˜ X g ,r to b e the simplicial complex consisting of simplices { [ v 0 ] , · · · , [ v k ] } for all simplices { v 0 , · · · , v k } in X g ,r . Let ˜ X r = { ( g , x ) : g ∈ Γ f in , x ∈ ˜ X g ,r } . 19 By an argumen t similar to the pro of of Propo sition 4.2, w e hav e the fo l- lo wing isomorphism: lim d →∞ lim r →∞ ˜ H λ, Γ n,r ( P d (Γ)) ∼ = lim d →∞ lim r →∞ ( ⊕ k ≤ n, k = n mod 2 H Γ n (( ^ P d (Γ)) r )) . Finally we observ e that the natural homomorphism lim d →∞ lim r →∞ ( ⊕ k ≤ n, k = n mod 2 H Γ n,r ( P d (Γ))) → lim d →∞ lim r →∞ ( ⊕ k ≤ n, k = n mod 2 H Γ n (( ^ P d (Γ)) r )) is an isomorphism. Let Γ b e a coun table gro up with a prop er length metric. L et X b e a simplicial complex with a prop er and co compact action o f Γ. Let ˆ X b e the subspace o f Γ f in × X defined b y: ˆ X = { ( g , x ) ∈ Γ f in × X : g x = x } . The diago nal action of Γ on Γ f in × X induces a na tural Γ- action on ˆ X . Not e that ˆ X is a simplicial complex with a simplicial action of Γ. W e define H Γ k ( X ) = H Γ k ( ˆ X ) . W e remark that H Γ k ( X ) is the equiv a rian t homology theory of Baum- Connes [BC2]. Prop osition 4.5. L et Γ b e a c ountable gr oup with a pr op er length metric. We have lim d →∞ lim r →∞ H Γ n,r ( P d (Γ)) ∼ = lim d →∞ H Γ n ( P d (Γ)) . Pr o of. Let X b e a simplicial complex with a prop er and co compact action o f Γ. F or eac h finite subset F ⊂ Γ f in , let F ′ = { γ f γ − 1 : γ ∈ Γ , f ∈ F } . F or each g ∈ Γ f in and r ≥ 0, w e define ˇ X g ,r to b e the simplicial sub complex of X consisting of simplices with v ertices { g j 0 v 0 , · · · , g j k v k : j i ∈ Z } for all simplices { v 0 , · · · , v k } in X g ,r . 20 W e let ˆ X F = { ( g , x ) ∈ F ′ × X : g x = x } , ˇ X F ,r = { ( g , x ) ∈ F ′ × X : x ∈ ˇ X g ,r } . W e ha v e an inclusion map i : ˆ P d (Γ) F → ˇ ( P d (Γ)) F ,r . The map i induces a homomorphism i ∗ : lim d →∞ H Γ n ( P d (Γ)) → lim d →∞ lim r →∞ H Γ n,r ( P d (Γ)) . By the definition of ˇ ( P d (Γ)) F ,r , for each d ≥ 0 and r ≥ 0, there exists c > 0 suc h that, for ev ery p oint ( g , x ) in ˇ ( P d (Γ)) F ,r , x is within distance c fro m a fixed p oint of g . It follows that, for each d ≥ 0 and r ≥ 0, t here exis t d ′ ≥ d and a con tin uous map ψ : ˇ ( P d (Γ)) F ,r → ˆ ( P d ′ (Γ)) F suc h that if w e write ψ ( g , x ) = ( g , ψ ′ ( x )), then w e ha v e sup { d ( ψ ′ ( x ) , x ) : ( g , x ) ∈ ˆ ( P d ′ (Γ)) F } < ∞ , where d is the restriction of the simplicial metric on P d (Γ). The map ψ induces a homomorphism ψ ∗ : lim d →∞ lim r →∞ H Γ n,r ( P d (Γ)) → lim d →∞ H Γ n ( P d (Γ)) . Using linear homotopies, it is not difficult to che ck that i ∗ and ψ ∗ are in v erses to eac h o ther. F or each no n- negativ e in teger n , w e are no w ready to define the Connes- Chern character c n for a g eneral group Γ: c n : K 0 ( S Γ) − → lim d →∞ ( ⊕ k ev en, k ≤ n H Γ n ( P d (Γ))) . F or each p ≥ 1 and ev en in teger n ≥ p , w e shall first define the Connes- Chern character c m : c n : K 0 ( S p Γ) − → lim d →∞ ( ⊕ k ev en, k ≤ n H Γ k ( P d (Γ))) 21 Let ˜ q b e an idemp oten t in M m (( S p Γ) + ) and ˜ q = q + q 0 for some q ∈ M m ( S p Γ) and idemp oten t q 0 ∈ M m ( C ). Let d b e greater than or equal to n + 1 times the propag ation of q , i.e. q ( x, y ) = 0 if d ( x, y ) > d/ ( n + 1) . The Connes-Chern c haracter c n of [ ˜ q ] − [ q 0 ] is defined to b e ho mo lo gy class of X g ∈ Γ f in ( g , X ( x 0 , ··· ,x n ) tr ( q ( x 0 , x 1 ) · · · q ( x n , g − 1 x 0 ))[ x 0 , · · · , x n ] λ,g ) ∈ H λ, Γ n,d ( P d (Γ)) , where the summation P ( x 0 , ··· ,x n ) is take n o v er a ll ordered n -simplices of ( P d (Γ)) g ,d . W e remark that the choice of n gua ran tees that the trace in the ab o ve defini- tion of the Connes-Chern character is finite. By Pro p ositions 4.4 and 4.5, t he ab ov e Connes-Chern c haracter induces a Connes-Chern character: c n : K 0 ( S p Γ) → lim d →∞ ( ⊕ k even , k ≤ n H Γ k ( P d (Γ))) . F or a n arbitrary non-negative in teger n , let n ′ b e a p o sitive ev en in- teger satisfying n ′ ≥ max { n, p } . Let π n ′ ,n b e the na t ur a l pro jection from lim d →∞ ( ⊕ k ev en, k ≤ n ′ H Γ k ( P d (Γ))) to lim d →∞ ( ⊕ k even , k ≤ n H Γ k ( P d (Γ))). W e de- fine the Connes-Chern ch a racter c n from K 0 ( S p Γ) to lim d →∞ ( ⊕ k ev en, k ≤ n H Γ k ( P d (Γ))) to b e π n ′ ,n ◦ c n ′ . It is not difficult to v erify that the definition of c n is indep enden t of the c hoice of n ′ . The pro of of the follow ing prop osition is similar to the pro of of Prop osition 4.1 and is therefore o mitt ed. Prop osition 4.6. L e t Γ b e a c ountable gr oup. F or any non-ne gative inte ger n , the Connes-Chern char acter c n is a wel l define d homomorphism fr om K 0 ( S p Γ) to lim d →∞ ( ⊕ k even, k ≤ n H Γ k ( P d (Γ))) . Using the definition of lo we r algebraic K-theory , for eac h p ≥ 1 and a n y non-negativ e in teger n , we can similarly define c n : K i ( S p Γ) → lim d →∞ ( ⊕ k + i ev en,k ≤ n H Γ k ( P d (Γ))) for eac h i < 0. 22 Finally , with the help of the equality S = ∪ p ≥ 1 S p , w e obta in a Connes- Chern character c n : K i ( S Γ) → lim d →∞ ( ⊕ k + i ev en, k ≤ n H Γ k ( P d (Γ))) for eac h non-negat ive in teger n and i ≤ 0. Notice that when Γ is t o rsion free, Γ f in consists only of the iden tit y elemen t and the ab o ve definition coincides with the prior definition for the torsion fr ee case. In the rest of this section, w e study a lo cal prop ert y of the Connes-Che rn c haracter for K-theory elemen ts with small propagations. This lo cal pro p ert y will play an imp ortant role in the pro of of the main theorem of this pap er. W e shall need a few preparations to explain the concept of propaga tion in a con tin uo us setting. Let X b e a Γ-inv arian t simplicial subspace of P d 0 (Γ) for some d 0 ≥ 0. Endo w P d 0 (Γ) with a metric d suc h that its r estriction to eac h simplex is the standard metric and d ( γ 1 , γ 2 ) ≤ d Γ ( γ 1 , γ 2 ) for all γ 1 and γ 2 in Γ ⊆ P d 0 (Γ), where d Γ is the proper length metric on Γ. Let X b e giv en the simplicial metric of P d 0 (Γ). Let H b e a Hilb ert space with a Γ-action and let φ b e a ∗ -homomorphism from C 0 ( X ) to B ( H ) whic h is cov arian t in the sense that φ ( γ f ) h = ( γ ( φ ( f )) γ − 1 ) h for all γ ∈ Γ, f ∈ C 0 ( X ) and h ∈ H . Such a triple ( C 0 ( X ) , Γ , φ ) is called a cov arian t system. The fo llowing definition is due to John Ro e [R o e]. Definition 4.7. L et H b e a Hilb ert sp ac e and let φ b e a ∗ -homomorphism fr om C 0 ( X ) to B ( H ) , the C ∗ -algebr a of al l b ounde d o p er ators on H . L e t T b e a b ounde d line ar op er ator acting on H . (1) The supp ort of T is define d to b e the c o mplement (in X × X ) of the set of al l p oints ( x, y ) ∈ X × X for wh i c h ther e exists f ∈ C 0 ( X ) and g ∈ C 0 ( X ) sa tisfying φ ( f ) T φ ( g ) = 0 and f ( x ) 6 = 0 and g ( y ) 6 = 0 ; (2) The pr op agation of T is define d to b e: sup { d ( x, y ) : ( x, y ) ∈ Supp( T ) } ; 23 (3) Given p ≥ 1 , T is sa id to b e lo c al ly Schatten p -class if φ ( f ) T and T φ ( f ) ar e Schatten p -class op er ators for e ach f ∈ C c ( X ) , the algebr a of al l c o m p ac tly supp orte d c ontinuous functions on X . Definition 4.8. We defi n e the c ovariant system ( C 0 ( X ) , Γ , φ ) to b e admissible if (1) the Γ -a ction on X is pr op er and c o c omp act; (2) φ is no nde g ener a te in the sens e that φ ( C 0 ( X )) H is dense in H ; (3) φ ( f ) is non c omp act for any nonzer o function f ∈ C 0 ( X ) ; (4) for e ach x ∈ X , the action of the stabilizer g r oup Γ x on H is r e gular in the sense that it is i s o morphic to the action of Γ x on l 2 (Γ x ) ⊗ W for some infinite d imensional Hilb ert sp ac e W , wher e the Γ x action on l 2 (Γ x ) is r e gular and the Γ x action on W is t ri v ial. W e remark that condition (4) in t he ab o ve definition is unnecessary if Γ acts on X freely . In particular, if M is a compact manifold and Γ = π 1 ( M ), then ( C 0 ( f M ) , Γ , φ ) is an admissible cov ariant system, where f M is the univ ersal co v er of M and φ ( f ) ξ = f ξ for eac h f ∈ C 0 ( f M ) and all ξ ∈ L 2 ( f M ) . In general, for eac h lo cally compact metric space with a prop er and co compact isometric action of Γ, there exists an admissible co v arian t system ( C 0 ( X ) , Γ , φ ). Definition 4.9. F o r any p ≥ 1 , let ( C 0 ( X ) , Γ , φ ) b e an admissible c ovaria n t system. We define C p (Γ , X, H ) to b e the ring of Γ -invariant lo c al ly Schatten p -class op er ators acting on H with finite pr op a g ation. The pro of of the following useful result is straightforw ard and is therefore omitted. Prop osition 4.10. L et Γ b e a c ountable gr oup. L et X b e a s implicial c omplex with a s i m plicial pr op er and c o c omp act action of Γ . I f ( C 0 ( X ) , Γ , φ ) is an admissible c ovariant system, then the ring C p (Γ , X, H ) is isomorphic to the ring S p Γ . 24 F or eac h r > 0, let X r b e a Γ-in v arian t discrete subset of X suc h that (1) X r has bo unded geometry , i.e. for eac h R > 0, there exis t s N > 0 suc h that any ball in X r with radius R has at most N elemen ts; (2) X r is r -dense in X , i.e. d ( x, X r ) < r f or ev ery x ∈ X ; (3) X r is uniformly discrete, i.e. there exists k r > 0 suc h that d ( z , z ′ ) ≥ k r for all distinct pairs of elemen ts z and z ′ in X r . Let { U z } z ∈ X r b e a Γ-equiv ar ia n t disjoin t Borel cov er of X suc h that z ∈ U z and diameter ( U z ) < r for all z . Let χ z b e the c ha racteristic function of U z . Extend the ∗ -represe ntation φ to the a lg ebra of all b ounded Borel functions. If k ∈ C p (Γ , X, H ), let k ( x, y ) = φ ( χ x ) k φ ( χ y ) for all x and y in X r . F or an y r > 0, let U ′ z b e the 10 r -neighborho o d of U z for eac h z ∈ X r , i.e. U ′ z = { x ∈ X : d ( x, U z ) < 10 r } . Let O r ( X ) = { U ′ z } z ∈ X r . Note that O r ( X ) is an op en cov er of X . Let N ( O r ( X )) b e the nerv e space of the op en co v er O r ( X ). W e equip the v ertex set V of the s implicial complex N ( O r ( X )) with the pseudo metric d V defined by : d V ( W , W ′ ) = sup { d ( x, y ) : x ∈ W , y ∈ W ′ } for any pair of v ertices W and W ′ in N ( O r ( X )). F or each non-nega t ive ev en in teger n ≥ p , w e define the Connes-Chern c haracter c n : K 0 ( C p (Γ , X, H )) → ⊕ k even ,k ≤ n H Γ k ,r ( N ( O r ( X ))) as follows. Let ˜ q b e an idemp ot en t in M m ( C p (Γ , X, H )) + ) and ˜ q = q + q 0 for some q ∈ M m ( C p (Γ , X, H )) and idemp oten t q 0 ∈ M m ( C ). Let r b e the propaga t ion of q . Let n b e an ev en in teger satisfying n ≥ p . The Conne s-Chern c haracter of [ ˜ q ] − [ q 0 ] is defined to b e homology class of X g ∈ Γ f in ( g , X ( x 0 , ··· ,x n ) tr ( q ( x 0 , x 1 ) · · · q ( x n , g − 1 x 0 ))[ x 0 , · · · , x n ] λ,g ) in the homology gro up H λ, Γ n, ( n +1) r ( N ( O ( n +1) r ( X ))), where, for each g and r , ( x 0 , · · · , x n ) denotes the ordered simplex ( U ′ x 0 , · · · , U ′ x n ) in the space 25 ( N ( O ( n +1) r ( X ))) g ,r , [ x 0 , · · · , x n ] λ,g denotes the g -cyclically oriented simplex [ U ′ x 0 , · · · , U ′ x n ] λ,g in ( N ( O ( n +1) r ( X ))) g ,r , and the summation P ( x 0 , ··· ,x n ) in the ab ov e form ula is tak en ov er all ordered simplices in ( N ( O ( n +1) r ( X ))) g ,r . The followin g propo sition f ollo ws f rom the ab o ve definition and the pro of of Prop osition 4.1. Prop osition 4.11. L et Γ b e a c o untable gr oup. L et X b e a simplicial c om- plex with a simp l i c ial pr op er and c o c omp act action of Γ . F or e ach r > 0 , let n b e an even inte ger satisfying n ≥ p . I f ( C 0 ( X ) , Γ , φ ) is an admissi- ble c ovariant system, then the Conn es-Chern char acter c n of an elem ent in K 0 ( C p (Γ , X, H )) with pr op agation less than o r e qual to r > 0 is a hom olo g y class in H λ, Γ n,r ( N ( O ( n +1) r ( X ))) . W e iden tify K 0 ( S p Γ) with lim d →∞ lim X K 0 ( C p (Γ , X, H )) using Prop o sition 4.10, where the direct limit lim X is take n o ve r the directed system of all Γ- in v arian t, Γ-compact subsets of P d (Γ). W e also iden tif y lim d →∞ lim X H λ, Γ n,r ( N ( O r ( X )))) with lim d →∞ ( ⊕ k ev en,k ≤ n H Γ k ( P d (Γ))) using Prop ositions 4.4 and 4.5, where the direct limit lim X is again tak en o v er the di- rected system of all Γ-inv arian t, Γ-compact subsets of P d (Γ). Using the pro jec- tion π n ′ ,n from lim d →∞ ( ⊕ k even ,k ≤ n ′ H Γ k ( P d (Γ))) to lim d →∞ ( ⊕ k ev en,k ≤ n H Γ k ( P d (Γ))) for n ′ = max { n, p } , the ab ov e Connes-Chern c hara cter induces a Connes- Chern character: c n = π n ′ ,n ◦ c n ′ : K 0 ( S p Γ) → lim d →∞ ( ⊕ k ev en, k ≤ n H Γ k ( P d (Γ))) for any non-negativ e ev en in teger n . This construction giv es back the Connes- Chern character in Prop o sition 4.6. In the following corollary , w e demonstrate a lo cal prop erty of the Connes- Chern character. This lo cal prop erty of the Connes-Che rn c hara cter play s an imp ortant role in the pro of of Theorem 1.1. Corollary 4.12. L et Γ b e a c ountable gr oup. L et X b e a simplic i al c omplex with a sim plicial pr op er an d c o c omp act action of Γ . L et ˜ q = q + q 0 b e an eleme nt in M m ( C p (Γ , X, H ) + ) such t h at q ∈ M m ( C p (Γ , X, H )) , q 0 ∈ M m ( C ) , ˜ q an d q 0 ar e idemp otents. F or any non-n e gative inte ger n , when the pr op agation of q 26 is sufficiently smal l, c n ([ ˜ q ] − [ q 0 ]) c an b e r epr esente d by a homolo gy class in ⊕ k even, k ≤ n H Γ k ( X ) . Mor e gener al ly for e a c h i ≤ 0 and p ≥ 1 , the Connes- Chern char acter c n of an element in K i ( C p (Γ , X, H )) with sufficien tly sma l l pr o p agation c an b e r epr e s e nte d by a homolo gy class in ⊕ k even, k ≤ n H Γ k + i ( X ) . Pr o of. Let ˜ q = q + q 0 b e an elemen t in M m ( C p (Γ , X, H ) + ) such that q ∈ M m ( C p (Γ , X, H )), q 0 ∈ M m ( C ), ˜ q and q 0 are idemp otents. If n ≥ p and the propa g ation of q is less than or equal to r > 0 , then b y Prop osition 4.11 w e kno w tha t c n ([ ˜ q ] − [ q 0 ]) can b e represen ted by a homology class in H λ, Γ n,r ( N ( O ( n +1) r ( X ))). Let r 0 = inf { d ( g U ′ z , U ′ z ) : z ∈ X ǫ , g ∈ Γ , g U ′ z 6 = U ′ z } , where { U ′ z } z ∈ X r is the op en co v er O r ( X ) in the definition of the Connes-Chern c haracter. Assume that r is a sufficien tly small p ositiv e n umber for the rest of this pro of. W e ha v e r 0 > 0. W e choose { U z } z ∈ X r suc h that { U ′ z } z ∈ X r is a go o d co v er. If r < r 0 , then w e ha v e the following: [1] b y the definition of H λ, Γ n,r ( N ( O r ( X ))), the homolo g y group H λ, Γ n,r ( N ( O r ( X ))) is equal to H λ, Γ n ( \ ( N ( O r ( X )))); [2] b y Prop osition 4.2, the homolog y g roup H λ, Γ n ( \ ( N ( O r ( X )))) can b e iden tified with ⊕ k ev en,k ≤ n H Γ k ( \ ( N ( O r ( X ))); [3] by the c hoice of { U ′ z } z ∈ X r , the homology group H Γ k ( \ ( N ( O r ( X )))) is equal to H Γ k ( X ) f o r eac h k . When n ≥ p , Coro llary 4.12 follo ws from the ab o ve statemen t s and the definition of the low er algebraic K-groups. T he case of a n arbitrary no n- negativ e in teger n can b e reduced to this sp ecial case by considering n ′ = max { n, p } and using the identit y c n = π n ′ ,n ◦ c n ′ , where π n ′ ,n is the pro j ection from ⊕ k ev en, k ≤ n ′ H Γ k + i ( X ) to ⊕ k even , k ≤ n H Γ k + i ( X ). 27 5 Pro o f of the main result In this section, w e giv e a pro of of Theorem 1 .1 . By Prop osition 2.4, Theorem 1.1 fo llows from the f o llo wing result. Theorem 5.1. L e t S b e the ring of al l Schatten class op er ators on an in fi nite dimensional and sep ar a ble Hilb ert sp ac e. The assembly map A : H O r Γ n ( E F I N (Γ) , K ( S ) −∞ ) → K n ( S Γ) is r ational ly inje ctive for any gr oup Γ . Pr o of. Without loss of generalit y , w e can assume that Γ is coun table (this is b ecause ev ery group is an inductiv e limit of countable groups). W e recall that E F I N (Γ) can be ide ntified with ∪ d ≥ 1 P d (Γ). F or eac h i ≤ 0 and non-negativ e in teger n , comp osing the assem bly map A : H O r Γ i ( E F I N (Γ) , K ( S ) −∞ ) → K i ( S Γ) with the Connes-Che rn c har a cter c n : K i ( S Γ) − → lim d →∞ ( ⊕ k ev en, k ≤ n H Γ k + i ( P d (Γ))) , w e obtain a homomor phism ψ i,n : H O r Γ i ( E F I N (Γ) , K ( S ) −∞ ) → lim d →∞ ( ⊕ k even ,k ≤ n H Γ k + i ( P d (Γ))) . By using the fact that H O r Γ i ( E F I N (Γ) , K ( S ) −∞ ) is lim d →∞ H O r Γ i ( P d (Γ) , K ( S ) −∞ ), we obtain a ho momorphism ψ i : H O r Γ i ( E F I N (Γ) , K ( S ) −∞ ) → lim d →∞ ( ⊕ k even H Γ k + i ( P d (Γ))) suc h tha t ψ i coincides with ψ i,n on the image o f the natural map fr o m H O r Γ i ( P d (Γ) , K ( S ) −∞ ) to H O r Γ i ( E F I N (Γ) , K ( S ) −∞ ) when n + i is greater t ha n or equal to the dimension of P d (Γ). Such a map ψ i is unique and is indep enden t of the c hoice o f n . F or eac h simplicial Γ-inv arian t and Γ- co compact subspace X of E F I N (Γ), let C (Γ , X, H ) = ∪ p ≥ 1 C p (Γ , X, H ) , where C p (Γ , X, H ) is as in definition 4.9. 28 By the definition of the assem bly map in [BFJR] and the fact that t his a ssem bly map coincides with the classic assem bly map (Corollary 6.3 in [BFJR]), K- theory elemen ts in the image of the assem bly map A : H O r Γ i ( X , K ( S ) −∞ ) → K i ( C (Γ , X , H )) can b e represen ted b y elemen ts with arbitrarily small propagation f or i ≤ 0. This, together with Coro lla ry 4.12, implies that there exists a map (still denoted by ψ i ) ψ i : H O r Γ i ( X , K ( S ) −∞ ) → ( ⊕ k even H Γ k + i ( X )) for eac h non-p ositiv e integer i suc h that the following diagram commu t es: H O r Γ i ( X , K ( S ) −∞ ) ψ i − → ( ⊕ k even H Γ k + i ( X )) ↓ j ∗ ↓ j ′ ∗ H O r Γ i ( E F I N (Γ) , K ( S ) −∞ ) ψ i − → lim d →∞ ( ⊕ k ev en H Γ k + i ( P d (Γ))) , where j ∗ and j ′ ∗ are resp ectiv ely induced b y the inclusion maps. If X = Γ /F as Γ-spaces for some finite subgroup F of Γ, then it is straigh t- forw ard to v erify that ψ i is an isomorphism after tensoring with C . In fact, b oth sides are naturally isomorphic to the group R ( F ) ⊗ C , where R ( F ) is t he represen tation r ing of F view ed as an additiv e gro up. Recall that, b y Prop o- sition 7.2.3, R emark 7.2.6 and Theorem 8.2.5 in [CT], w e hav e K n ( S ) = Z when n is ev en and K n ( S ) = 0 when n is o dd. As a consequence , the homol- ogy theory H O r Γ i ( X , K ( S ) −∞ ) is 2-p erio dic. Note that the homolog y t heory ⊕ k ev en H Γ k + i ( X ) is 2- p erio dic b y definition. By the pro of of the Ma yer-Vietoris sequence using the mapping cone and the definition o f the Connes-Chern c har- acter, w e kno w the homomorphisms ψ i comm ute with the M ay er-Vietoris s e- quences (up to scalars) Using the abov e results, the fact that b oth homology the o ries satisfy the Ma y er- Vietoris sequence and a fiv e lemma argumen t, w e can pro v e that the map ψ i : H O r Γ i ( X , K ( S ) −∞ ) → ( ⊕ k even H Γ k + i ( X )) is an isomor phism a f ter tensoring with C for i ≤ 0. This implies that the assem bly map A is r ationally injectiv e fo r i ≤ 0. No w Theorem 5.1 follo ws from Prop osition 3.1. 29 W e commen t that the algebraic K-theory isomorphism conjecture for the ring S Γ can b e view ed as an algebraic coun terpart of the Baum-Connes con- jecture for the K-theory o f the reduced group C ∗ -algebra of Γ [BC1]. The F arrell-Jones isomorphism conjecture and the Baum-Connes conjecture imply the following conjecture. Conjecture 5.1. L et K b e the C ∗ -algebr a of al l c omp act op er ators on an infinite dimensio n al and sep ar able Hilb ert sp ac e , let C ∗ r (Γ) b e the r e duc e d gr oup C ∗ -algebr a of Γ . The natur al homom orphism i ∗ : K n ( S Γ) → K n ( C ∗ r (Γ) ⊗ K ) is a n isomorp hism, wher e C ∗ r (Γ) ⊗ K is the C ∗ -algebr aic tensor pr o duct of C ∗ r (Γ) with K and i is the in clusion map fr om S Γ to C ∗ r (Γ) ⊗ K . W e remark that , b y a theorem of Sus lin- W o dzic ki [SW], the algebraic K- theory K n ( C ∗ r (Γ) ⊗ K ) is isomorphic to the topo logical K- t heory K top n ( C ∗ r (Γ)). Theorem 1.1 implies that the No vik ov higher signature conjecture follow s from the (ratio na l) injectivit y of i ∗ in the ab o ve conjecture. Finally we sp eculate that the (alg ebraic) biv arian t K-theory of Cuntz , Cun tz-Thom and Corti˜ nas-Thom should b e useful in studying the algebraic K-theory isomorphism conjecture for S Γ [Cu1] [Cu2] [CuT] [CT1]. References [BC1] P . Baum , A. Connes , K-the o ry for discr ete gr oups, Op erator Alge- bras and Applications. (D. Ev a ns and M. T akes aki, editors), Cam bridge Univ ersit y Press (1989), 1–20 . [BC2] P . Baum , A. Connes , Chern char acter for dis c r ete gr oups, A fˆ ete of top ology , 163–23 2, Academic Press, Boston, MA, 198 8 . [BFJR] A. Bar tels , T. F arrell , L . Jones , H. Reich . On the isomorphism c o n je ctur e in algebr aic K-the ory. T op ology 43 (2004), no. 1 , 157–21 3 . [BLR] A. Bar tels , W. L ¨ uck , H. R eich . The K-the or etic F arr el l-Jones c on- je c tur e for hyp erb olic gr oups. In ven t. 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