Polynomially bounded cohomology and the Novikov Conjecture

P olynomia lly b ounded cohomology a nd the No vik o v Co njecture Cric h ton Ogle Dept. of Mathematics The Ohio State Univ ersit y ogle@math.o hio-state.edu Abstract Using tec hn iques develo p ed for study ing polyn omially boun ded cohomolog y , we sho w that the assem bly map for K t ∗ ( ℓ 1 ( G )) is rationally injectiv e for all ﬁnitely presented discrete groups G . This veriﬁes the ℓ 1 -analogue of the Strong Noviko v Conjecture f or su ch groups. The same metho ds show that the Strong No viko v Conjecture for all ﬁnitely presented groups can b e reduced to proving a certain (conjectural) rigidit y of the topological cy clic c h ain complex C C t ∗ ( H C M m ( F )) where F is a ﬁnitely-generated free group and H C M m ( F ) is the “maximal”Connes-Mosco vici algebra asso ciated to F . Con ten ts 1 In tro duction 2 2 Preliminaries 5 3 Simplici al rapid decay algebras 8 3.1 ℓ 1 -rapid decay alge br as . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 The Connes-Mo scovici algebra o f a discrete group . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Simplicializing the Connes-Moscovici constructio n . . . . . . . . . . . . . . . . . . . . . . 10 4 Detecting the assem bl y map 17 4.1 Spec tr al sequence s in Ho chsc hild and cyclic (co-)homolo gy . . . . . . . . . . . . . . . . . . 17 4.2 The lo cal C her n character asso c ia ted to an in tegral homology cla ss . . . . . . . . . . . . . 19 4.3 Pro of of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.4 Related results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5 App endix 27 2000 Mathematics Subje ct Classiﬁc ation . Primary 58B34 ; Secondary 18G10, 18G30 , 18G35, 18G40, 18G 60, 19K56, 46H25, 46L80, 46L87, 46M20, 55N35, 55 T05, 58B34. Key wor ds and phr a se s . Strong No viko v Conjecture, sim plicial rapid deca y algebras, p olynomially b ounded cohomology . 1 1 In tro d uction The sta r ting p oint for the work pr esented her e is the following Conjecture - No viko v, 1 970 (NC) L et M b e a close d, c omp act, oriente d n -dimensional ma n ifold, L ( M ) its total Hirzebruch L -class, [ M ] its fu n damental homolo gy class, and ι : M → B π . Then for every [ c ] ∈ H ∗ ( B π ; Q ) , the high er signatur es S ig n c ( M ) := hL ( M ) ι ∗ ( c ) , [ M ] i ∈ Q ar e invaria n ts of the oriente d homotopy typ e of M . Let NC( π ) deno te the NC is true for the discrete group π . Ea r ly work of Noviko v and Lustig show ed NC( π ) was true for π = Z n . How ever, further progr ess r equired a reformulation. Beginning with L • ( R ) = the (sy mmetric/quadra tic) L -theory spectr um of a r ing with in volution R , one has the Mishchenk o -Quinn- Ranicki-W all Assembly map A L ∗ ( π ) : H ∗ ( B π ; L • ( Z )) → L ∗ ( Z [ π ]) = π ∗ ( L • ( Z [ π ])) The Mi shc henko-Ranic ki-W all reformulation (MR W - NC) The assembly m ap A L ∗ ( π ) is r ational ly inje ctive . Let MR W-NC( π ) denote this conjecture for the particular g roup π . W all [W1] s how ed that for a ny discrete group π , MR W-NC( π ) implies NC( π ). Subsequently , Mishchenko and Ranicki s howed the conv erse - that NC( π ) implies MR W-NC( π ) - when π is ﬁnitely prese nted 1 . F or a g iven ﬁnitely presented gr oup π , NC( π ) will refer either to the orig inal conjecture p osed by Novik ov, or the equiv alent conjecture MR W-NC( π ). The ﬁrst ma j o r breakthro ugh using this formulation of NC was achiev ed in the mid 1970’s b y Mishchenk o, who show ed NC( π ) is true whenever π is the fundamental group of a clo sed, compact (orien ted) manifold of negative curv ature. A n umber of years later, another ma jor s tep was taken by G. Kasparov. He formulated the so -called Strong Novik ov Conjecture, which for a g iven disc r ete gro up π states SNC( π ) - Kasparo v L et C ∗ ( π ) b e a faithful C ∗ -c ompletion of C [ π ] . Then the assembly map in t op o- lo gic al K -t he ory A ∗ ( π ) : H ∗ ( B π ; K t • ( C )) → K t ∗ ( C ∗ ( π )) is r ational ly inje ctive. It is eas ily seen that SNC( π ) implies NC( π ) using the fact that the top o lo gical Witt groups and top olo g ical K - g roups o f a C ∗ -algebra are na tur ally isomorphic, but this strong er version has other consequences as well, hence its name. Kaspar ov then show ed in [GK] that SNC( π ) is true (for the reduced C ∗ -algebra C ∗ r ( π )) whenever π is a cocompa ct discrete subgr oup o f a virtually connected Lie Gro up. This s tronger version of the conjecture has b ee n veriﬁed for a num b e r of other class es of gr oups; in pa rticular, for word-h yp er b o lic groups [CM], and a menable gr oups [HR]. Inside of C ∗ ( π ) lies the smaller conv olution algebra ℓ 1 ( π ) o f ℓ 1 -functions on π , and one can formulate an ℓ 1 -analog ue of Kaspar ov’s conjecture: 1 If NC( π ) is true for all ﬁnitely presen ted groups, it is true f or all groups, and S. W ein b erger has pointed out that v alidity of MR W-NC( π ) f or all ﬁnitely pr esented groups π implies MR W-NC for all discrete groups, via a sui table direct limit argumen t. It i s not clear if such a statement holds for the Banach al gebra versions of the conjecture stated below. ℓ 1 -NC( π ) Th e assembly map in top olo gic al K - the ory A ∗ ( π ) : H ∗ ( B π ; K t • ( C )) → K t ∗ ( ℓ 1 ( π )) is r ational ly inje ctive. There is a factorizatio n A ∗ ( π ) : H ∗ ( B π ; K t • ( C )) → K t ∗ ( ℓ 1 ( π )) → K t ∗ ( C ∗ ( π )) by which one see s that SNC( π ) implies ℓ 1 -NC( π ); o r equiv alently , that any counterexample to ℓ 1 -NC( π ) would provide a c o unterexample to SNC( π ). Un til recently the class of gr oups for which these last tw o conjectures were known was essent ia lly the same. In [O 1], and independently in [M1], it was sho wn that groups which ar e sy nchronously co mbable in po lynomial time are P - iso c ohomolo gic al , implying e very (complex) group c o homolog y class is represented by a co cycle of polynomia l growth (this is the (PC) condition of [CM]). By [PJ1] and [PJ2], a ny group satisfying this condition a lso sa tisﬁes ℓ 1 -NC( π ). In [JOR2 ] the iso coho mo logical res ults of [O1] were extended, yielding many more gro ups whic h a re P -isoco homolog ic al - and which therefore also sa tis fy ℓ 1 -NC( π ) - for which the or iginal Novik ov Conjecture is no t currently known. How ever, a s is also sho wn in [JOR2 ], there a r e many groups (including even solv able groups) which do not satis fy c o ndition (PC). In fact, for ﬁnitely pr esented g r oups which are not of t yp e F P ∞ , it is e asily seen that there exist cohomology classes which ar e completely unbounded with resp ect to a ﬁxed word-length function on G . So the C - extendability methods o f [CM], or their ℓ 1 -analog ue, clearly do not a pply to such coho mology classes. And so for groups with r a tional homolog y detectable only by such classes, verifying even ℓ 1 -NC( π ) v ia classical co cycle extens io n techniques and cyclic theo ry is pr oblematic. Our main r esult is that these obstructions c a n b e b ypassed, at lea st for the ℓ 1 -algebra . P recisely , we show Theorem A The co njecture ℓ 1 -NC( π ) is true for all ﬁnitely pres ented gro ups π . The pro o f of Theor em A follows b y a deta ile d analy sis o f simplicial r a pid decay alg ebras; simplicial analogues of the ra pid decay algebr as in tro duced by Jolissaint in [P J 1], [PJ2 ]. W e b eg in in section 2 with some preliminaries concerning Hochschild and cyclic homology . W e also explain (using techniques from [O2]) how Theorem A follo ws from verifying the injectivit y of the r estricte d assemb ly map H ∗ ( B π ; Q ) → K t ∗ ( ℓ 1 ( π )) ⊗ Q Although no t essen tial, this reduction simpliﬁes v a rious consider ations later o n. In section 3.1, we construct the ℓ 1 -rapid decay algebra asso c iated to a p-b ounded simplicial group (as deﬁned in [O 1]). In sections 3.2 and 3.3, we give the analog ous extension to p-bounded simplicia l groups of the Connes-Mo scovici algebr a, and relate it to the simplicial ℓ 2 -rapid decay (ak a Jolissaint) alg ebra when the gr oup is degr eewise rapid deca y (e.g ., fr e e ). W e also deﬁne a “maximal”Co nnes-Moscovici alge br a, which (like its ℓ 1 -analog ue) ha s the adv antage of pro ducing a functor ( f .p.g r oups ) 7→ ( F r ´ eche t a l g eb ra s ) from the category of ﬁnitely presented gr oups to the category of F r´ echet algebra s, which can therefore b e naturally ex tended to a s implicia l fra mework. In section 4, we revie w the sp ectral sequences ar ising in Ho chsc hild and cyclic (co- )homology for the simplicial algebras considered in the previous section, b e ginning with the simplicial gro up algebra. The prototypical spe c tral sequences are those for the simplicial group a lgebra C [Γ . ] when Γ . is a free simplicial resolution of π . In s ection 4.2 we in tro duce the k ey ingredient in the proo f of the Theo rem, namely the construction of a lo cal Chern c ha racter asso ciated to an arbitra ry in teg ral homology clas s x ∈ H n ( B π ; Q ). This Cher n character, deﬁned for b o th K t ∗ ( ℓ 1 ( π )) and K t ∗ ( C ∗ ( π )), is taylor-made to detect the image of x under the rationa lized as sembly map, and results from a co mbination of Baum’s retopolo gization Theorem, the Chern character for ﬁne top olo gical alg ebras constructed b y Tillma nn in [T1] (res ults discussed in the ﬁrst app endix), and the “ rigidity” r esults of Go o dwillie [G1]. In section 4.3, we complete the pr o of of Theo rem A by a car eful analy s is of the sp ectral sequence for the to p ologica l cyclic homolog y of the ℓ 1 -rapid decay algebr a c onstructed in the prev ious section. Finally , in section 4 .4, we prove a weak er injectivit y re s ult for the maximal Connes-Mosc ovici a lg ebra H C M m ( G ). The main result of this section, simila r in spirit to the main theorem of [O2], is that a n injectivity result holds for the top olog ical Ho chsc hild homology groups o f a certrain simplicial F r´ echet algebra, and that if the same r esult ho lds upo n passa g e (via the I ma p) from the topolog ic al Ho chsc hild to the topo logical cyclic homolog y groups of this algeba, this would imply the Strong Novik ov Co njecture for all ﬁnitely pr e s ented gr oups. This pap er inco rp orates a num b er of ideas fr om the multiply-revised pre pr int [O0], which was also a source for [O1] and [O2 ]. In some w ays this pap er repres ents the ﬁnal itera tion o f what w as b egun in [O0], at least as far as the ℓ 1 -group alg e bra is concerned. I would lik e to thank Dan Burghelea for his supp ort and encouragement during the evolution of this w or k, as w ell as the v arious (anonymous) referees whose criticisms ov er the y ear s, applied to v ar ious stages o f [O0] (including the current version), have proved to b e quite v aluable. 2 Preliminaries In this pap er all v ector spaces (b oth algebra ic and topolog ical), algebras, tensor pro ducts, Hom gr oups, are ov er C . Moreov er, ho mology and cohomology will alw ays b e with co eﬃcients in C , unless indicated otherwise by notation. F or a n algebra A , w e will write C H a ∗ ( A ) r esp. C C a ∗ ( A ) for the algebraic Ho chsc hild resp. cyclic complex of A , with H H a ∗ ( A ) r esp. H C a ∗ ( A ) denoting their cor r esp onding ho mology gro ups. In a similar vein, we set C H ∗ a ( A ) := H om ( C H a ∗ ( A ) , C ), C C ∗ a ( A ) := H om ( C C a ∗ ( A ) , C ) with ass o ciated cohomolo gy gro ups denoted b y H H ∗ a ( A ) resp. H C ∗ a ( A ). More g enerally , for a vector space V w e will denote H om ( C H a ∗ ( A ) , V ) resp. H om ( C C a ∗ ( A ) , V ) by C H ∗ a ( A ; V ) resp. C C ∗ a ( A ; V ), a nd the corr esp onding cohomolog y groups by H H ∗ a ( A ; V ) r esp. H C ∗ a ( A ; V ). In the case o f Ho chsc hild ho mology and co homology , this should no t b e confused with the notio n o f the Ho chsc hild (co-)homolo gy of A with co eﬃcien ts in an A -bimo dule M . W e assume familiarity with the traditional constr uctions and pr op erties of these theor ies; a n appropr iate reference is [L]. In the even t the subscript (resp. superscr ipt) “ a ”is omitted, it will b e understo o d we are referring to algebraic Ho chsc hild o r cyclic (co)-homology . If A is a lo c ally conv ex top ologica l a lgebra, then the Ho chsc hild and cyclic complexes for A may b e formed using the pro jectively complete tensor pro duct, re s ulting in top olog ical chain complexes C H t ∗ ( A ) resp. C C t ∗ ( A ). If W is a to p ologica l vector space, one also has C H ∗ t ( A ; W ) := H om cont ( C H t ∗ ( A ) , W ), C C ∗ t ( A ; W ) := H om cont ( C C t ∗ ( A ) , W ). If D ∗ = ( D ∗ , d ∗ ) is a top o logical chain co mplex, the unreduced resp. reduce d homology of D ∗ is g iven by H t n ( D ∗ ) := k e r ( d n ) /im ( d n +1 ) H t n ( D ∗ ) := k e r ( d n ) /im ( d n +1 ) One has similarly deﬁned g r oups in co homolog y: H n t ( H om cont ( D ∗ , W )) := k er ( δ n ) /im ( δ n − 1 ) H n t ( H om cont ( D ∗ , W )) := k er ( δ n ) /im ( δ n − 1 ) Although there is no Universal C o eﬃcient Theorem in the top ologic al setting, there is still a natural homomorphism H n t ( H om cont ( D ∗ , W )) → H om cont ( H t ∗ ( D ∗ ) , W ) , indicated by H n t ( H om cont ( D ∗ , W )) ∋ [ f ] 7→ [ f ] ∗ : H t ∗ ( D ∗ ) → W When A is equipped with a topolo gy , we will write F a ∗ ( A ) r esp. F ∗ a ( A ) for F t ∗ ( A δ ) resp. F ∗ t ( A δ ), where A δ denotes A with the discrete to p o logy and F represents one of the (co -)chain complex functors or asso cia ted (co-)homolog y functors discussed above. F or any choice o f F , there are natural transforma tions F a ∗ ( − ) → F t ∗ ( − ), F ∗ t ( − ) → F ∗ a ( − ). T op ologies o n A are assumed to be contin uous over C . F or a given A , the ﬁne to p o logy on A will b e denoted by A f . F or any other contin uous top olo gy A T on A , the ident ity map on elements determines a contin uous map A f → A T . Beca use the algebra ic tensor pr o duct is co mplete in the ﬁne top olog y , there are iso morphisms of graded vector spaces F a ∗ ( A ) ∼ = F t ∗ ( A f ) ∼ = F t ∗ ( A f ) F ∗ a ( A ) ∼ = F ∗ t ( A f ) ∼ = F ∗ t ( A f ) for F ( − ) = H H ( − ) , H C ( − ). F or the complex group algebra C [ π ], there are w ell-known deco mp o sitions o f C H ∗ ( C [ π ]) and C C ∗ ( C [ π ]) as direct sums of sub co mplexes, indexed on < π > = the set o f conjugacy classe s o f π , which induce corres p o nding decomp ositions in homology: C H ∗ ( C [ π ]) ∼ = M ∈ <π > C H ∗ ( C [ π ]) C C ∗ ( C [ π ]) ∼ = M ∈ <π > C C ∗ ( C [ π ]) H H ∗ ( C [ π ]) ∼ = M ∈ <π > H H ∗ ( C [ π ]) H C ∗ ( C [ π ]) ∼ = M ∈ <π > H C ∗ ( C [ π ]) In cohomolog y , one has similar isomorphisms, with a direct sum of complexes replaced b y a direct pro duct of co complexes. The pro jection onto the s ummand indexe d b y < x > , on both the (co-)c ha in or (co- )homology level, will b e denoted by p . In this paper , w e will b e primarily co ncerned with the pro jection p < 1 > . F or the ℓ 1 -rapid decay algebra H 1 , ∞ L ( π ) asso cia ted to a discrete group with word-length ( π , L ), viewed as a F r ech ´ et a lgebra, o ne has a similar deco mp o sition o n the chain level (cf. [RJ], [JOR1]) C H t ∗ ( H 1 , ∞ ( π )) ∼ = d M ∈ <π > C H t ∗ ( H 1 , ∞ ( π )) C C t ∗ ( H 1 , ∞ ( π )) ∼ = d M ∈ <π > C C t ∗ ( H 1 , ∞ ( π )) This pro duces pro jection maps, also denoted p , for b oth topo logical (co -)chains, as well as top o logical (co-)homolog y , b o th r e duced and unreduced. On the (co-)chain level a nd for the reduced theories, these pro jection ma ps are contin uous in the induced top olo gy . In this pap er , the ca tegory of ﬁnitely presented gro ups will play a ce nt r al ro le. In order to maintain control on the s iz e of this categor y , we ﬁx a countable s et R U := { x α } α ∈I , where I is a countable indexing set. The ob jects of the ca tegory ( f .p.g roups ) are ﬁnitely presented gro ups G = < R, W > on a generating s et R which is r equired to b e a ﬁnite subset of R U , whe r e the presentation is included as pa r t of the data asso ciated to the ob ject. The morphisms of ( f .g .g roup s ) are g roup homomor phisms. Each ob ject in ( f .g .g r oups ) is equipped with a standard word-length function L st . As the gro ups a re ﬁnitely presented, restr ic tio n of generating sets to ﬁnite s ubs ets of the countable univ ers al collection of g enerator s R U results in O bj ( f .p.g r oups ) b eing a countable set. Additionally , the ﬁxed word-length function shows that each H om set H om ( G, G ′ ) is countable as w ell (these facts are used below in the constr uction of the “ma x imal”analo gue of the Connes-Moscovici a lgebra). F or a Bana ch algebr a A , let K t ( A ) denote the top o logical K -theory sp ectrum for A . Let S denote the sphere sp ectrum (i.e., the reduced susp ensio n spe c tr um of S 0 ). Then 1 ∈ K t 0 ( C ) corre ps onds to a map of sp ectra S → K t ( C ); pr e c omp osition of this “unit”with the full a s sembly map yields the restricted assembly map on the lev el of spec tr a: B π + ∧ S → B π + ∧ K t ( C ) → K t ( ℓ 1 ( π )) (1) Lemma 1. The ful l asssembly map B π + ∧ K t ( C ) → K t ( ℓ 1 ( π )) is r ational ly inje ctive on homotopy gr oups iﬀ the r estricte d assembly map is so. Pr o of. In [O2, Cor. 4.5] we pr ov ed this result with C ∗ ( π ) in place of ℓ 1 ( π ). The proo f for the ℓ 1 -group algebra follows by exactly the same argument leading up to [O2, Thm. 4.4] once o ne makes the following observ ations: • If F is a free gr oup, the assembly map B F + ∧ K t ( C ) → K t ( ℓ 1 ( F )) is a rational homotopy equiv ale nce (in fact, it is an integral equiv alence). • Giv en a Banach algebr a A and a closed ideal I ⊆ A , the sho rt-exact sequence I ֌ A ։ A/I induces a homotop y ﬁbra tion sequence of sp ectra: K t ( I ) → K t ( A ) → K t ( A/I ) • An a ugmented free simplicial resolution Γ . + of Γ − 1 = π induces a ﬁltration on the homotopy groups of both B π + ∧ K t ( C ) a nd K t ( ℓ 1 ( π )), with r esp ect to which the a ssembly map is ﬁltra tion-preser ving. The ﬁltra tion of K U ∗ ( B π ) rationally corresp onds to the skeletal ﬁltration, while the ﬁltration of K t ∗ ( ℓ 1 ( π )) is as deﬁned in [O2, (1.5)]. 3 Simplicial rapid d eca y algebras 3.1 ℓ 1 -rapid deca y alge bras Let ℓ 1 ( π ) b e the conv o lution a lgebra of ℓ 1 -functions on π , with standard ℓ 1 -norm. F or a rea l n umber s > 0 , H 1 ,s L ( π ) is the Banach algebra o f functions f : π → C satisfying the inequality ν 1 ,s ( f ) < ∞ , where ν 1 ,s ( f ) = X h ∈ π | f ( h ) | (1 + L ( h )) s (2) F or each s and L , H 1 ,s L ( π ) is the completion of the gr o up a lgebra C [ π ] with resp ect to the semi-no rm ν 1 ,s . Let H 1 , ∞ L ( π ) := \ s H 1 ,s L ( π ) (3) The collection of semi-norms { ν 1 ,n } n ∈ N give H 1 , ∞ L ( π ) the str ucture of a F r´ ec het space. Unlike the ℓ 2 - case dis cussed b elow, the ℓ 1 -rapid decay algebra H 1 , ∞ L ( π ) is always a s uba lgebra o f ℓ 1 ( π ) which is closed in ℓ 1 ( π ) under holomorphic functional calculus [PJ 1]. Moreover, it is functorial with res pe ct to group homomorphisms φ : ( π , L ) → ( π ′ , L ′ ) whic h a re polynomia lly b o unded with respec t to w or d-length (i.e., there is a po lynomial p for which L ′ ( φ ( g )) ≤ p ( L ( g )) for all g ∈ π ). Let ( f .p.g r oups ) b e the categ ory of ﬁnitely-pr esented gr oups deﬁned a b ov e. If G is ﬁnitely presented, any t wo w o rd-length functions on G are linear ly equiv alent, hence linearly equiv alent to the standard word-length function L G st on G . F or this r eason, we will assume for the remainder of the pap er that the word-length function on a ﬁnitely presented group is alwa ys L st . Observing that any homomorphism from a ﬁnitely-genera ted g r oup G is p oly nomially (in fact, linearly) b ounded w ith r esp ect to L G st , we conclude Prop ositi on 1. The asso ciatio n G 7→ H 1 , ∞ L ( G ) deﬁnes a fun ct or H 1 , ∞ L ( − ) : ( f .p.g roup s ) → ( F r ´ echet al g ebra s ) wher e ( F r ´ echet al g eb ras ) denotes the c ate gory of F r´ echet algebr as and c ontinuous F r ´ echet algeb ra s ho- momorphisms. 3.2 The Connes-Mosco vici algeb r a of a discrete group W e r ecall the constr uc tio n of [CM]. As b efor e, π will denote a countable discrete gr oup equipp ed with a word-length function L . By con ven tio n, ℓ 2 ( π ) is the H ilb er t space of complex- v alued ℓ 2 -functions on π , with L ( ℓ 2 ( π )) the space o f b ounded op erato r s on ℓ 2 ( π ). On ℓ 2 ( π ) one has D L : ℓ 2 ( π ) → ℓ 2 ( π ) given on basis elements by D L ( δ g ) = L ( g ) δ g . This op erator in turn deﬁnes an unbounded operator ∂ L : L ( ℓ 2 ( π )) → L ( ℓ 2 ( π )) giv en by ∂ L ( M ) = ad ( D L )( M ) = D L ◦ M − M ◦ D L F ollowing [CM], we deﬁne the Connes-Moscovici alg e br a a sso ciated with the pair ( π, L ) as Deﬁnition 1 . H C M L ( π ) = { a ∈ C ∗ r ( π ) | ∂ k L ( a ) ∈ L ( ℓ 2 ( π )) for al l k } . Note that H C M L ( π ) = ∞ \ i =1 D omain ( ∂ k L ) ! ∩ C ∗ r ( π ). Since ∂ k L ( ab ) = k X i =1  k i  ∂ i L ( a ) ∂ k − i L ( b ) it follows H C M L ( π ) is an algebra . It is also ea sy to s ee that H C M L ( π ) co nt a ins the group a lgebra. Prop ositi on 2. F or al l wor d-length functions L and gr oups π , H C M L ( π ) is a dense sub algebr a of C ∗ r ( π ) close d under holomo rphic functional c alculus. (A detailed proo f of this Pr op osition is giv en in [RJ]). Deﬁne semi-no rms on C [ π ] b y η m ( a ) = m X i =0  m i  k ∂ i L ( a ) k (4) where k − k is the norm on C ∗ r ( π ) and ∂ 0 L = I d . Using standar d prop er ties of k − k we hav e η m ( ab ) ≤ η m ( a ) η m ( b ) (5) η m ( a ) ≤ η n ( a ) when m ≤ n (6) It is observed in [RJ] tha t H C M L ( π ) is complete in the semi-norms { η m } m ≥ 0 . As C [ π ] is dense in H C M L ( π ) in the top olog y induced by these se mi-norms, w e may alter natively describ e H C M L ( π ) as the F r´ echet algebra co mpletio n of C [ π ] with resp ect to { η m } m ≥ 0 . W e consider a related completio n of the gro up alg ebra C [ π ]. F or each real num b er s > 0, H 2 ,s L ( π ) is the Hilber t spa ce of functions f : π → C satisfying the inequalit y ν 2 ,s ( f ) < ∞ , wher e ν 2 ,s ( f ) = ( < f , f > 2 ,s,L ) 1 / 2 , < f , g > 2 ,s,L = X h ∈ π f ( h ) g ( h )(1 + L ( h )) 2 s (7) F or ea ch s and L , H 2 ,s L ( π ) is the completion of the gr o up algebra C [ π ] with resp ect to the semi-norm ν 2 ,s . Let H 2 , ∞ L ( π ) := \ s H 2 ,s L ( π ) (8) The colle c tio n of se mi-norms { ν 2 ,n } n ∈ N give H 2 , ∞ L ( π ) the structur e of a F r´ echet space, as H 2 ,t L ( π ) ⊂ H 2 ,s L ( π ) for 0 < s < t . If ther e exist n umbers C, s such that k f k ≤ C ν s ( f ) for all f ∈ C [ π ] (9) where k f k denotes the reduced C ∗ -norm of f , then H 2 ,t L ( π ) is a subspace o f C ∗ r ( π ) for t > s , and H 2 , ∞ L ( π ) is a F r´ ec het s ubalgebra of C ∗ r ( π ) . This prop erty was ﬁrst shown to hold for ﬁnitely-genera ted free g roups by Haa garup [H], and later for ﬁnitely-g enerated hyperb olic groups by Jolis s aint [PJ1], [PJ2 ] a nd P . de la Harpe [dH]. A group with word-length function ( π , L ) satisfying condition (9 ) is referred to as R apid De c ay , o r RD. In general, H 2 , ∞ L ( π ) con tains H C M L ( π ) whether o r not π is RD [CM], and when it is, the t wo a re equa l [RJ]. W e will need a reﬁned version of this last result. Call a word-length function L nic e if L does not ta ke a ny v alues in the op en interv al (0 , 1). Lemma 2. L et π b e a discr ete gr oup with nic e wor d-length function L . Then for e ach inte ger k ≥ 0 and ϕ ∈ C [ π ] ν 2 ,k ( ϕ ) ≤ 2 k η 2 k ( ϕ ) If in add ition ther e exists s, C > 0 fo r which ( 9) holds ab ove ( π is RD), t hen η k ( ϕ ) ≤ 2 k C ν 2 ,k + s ( ϕ ) Pr o of. The ﬁrst inequa lity follows from ν 2 ,k ( ϕ ) 2 = X g ∈ π | ϕ ( g ) | 2 (1 + L ( g )) 2 k = X g ∈ π | ϕ ( g ) | 2 2 k X i =0  2 k i  L ( g ) i ! = 2 k X i =0  2 k i  X g ∈ π | ϕ ( g ) | 2 L ( g ) i ! ≤ 2 k X i =0  2 k i  X g ∈ π | ϕ ( g ) | 2 L ( g ) 2 i ! ≤ 2 k X i =0  2 k i  k ∂ i L ( ϕ ) k 2 2 ≤ 2 k X i =0 η 2 k ( ϕ ) 2 = 2 k η 2 k ( ϕ ) 2 Now supp o se π is RD a nd that s, C > 0 have been c hos en for w hich (9) holds. Then a s in the pro of of [RJ, Th. 1.3] one ha s k ∂ m L ( ϕ ) k ≤ C ν 2 ,m + s ( ϕ ) and the second inequality ab ov e follows from η k ( ϕ ) = k X i =0  k i  k ∂ i L ( ϕ ) k ≤ C k X i =0  k i  ν 2 ,i + s ( ϕ ) ≤ C k X i =0  k i  ν 2 ,k + s ( ϕ ) = 2 k C ν 2 ,k ( ϕ ) These explicit inequalities also pr ov e the two semi-no rm top ologies are the same when π is RD. Observe that for all ( π , L ), ther e a re inclusions C [ π ] ֒ → H 1 , ∞ L ( π ) ֒ → H C M L ( π ) ֒ → H 2 , ∞ L ( π ) with the ﬁrst tw o inclusions b eing algebra ho momorphisms, and the last tw o being con tinuous F rech ´ et space maps. 3.3 Simplicializing the Connes-Mosco vici construction The co ns truction o f b oth H C M L ( π ) and H 2 , ∞ L ( π ) lacks functoriality with resp ect to g roup sur jections, even those that are p-b ounded. The following propo sition pr ovides the method for extending the Connes - Moscovici constr uction to simplicial or a ugmented simplicial gr oups. T o prop erly state it, we note that if { η ′ i } is a coun table collection of semi-norms on C [ π ], then the completion of C [ π ] in these semi-norms pro duces a F r´ echet spa ce. If e a ch semi-norm is submultiplicativ e , that is, η ′ i ( ab ) ≤ η ′ i ( a ) η ′ i ( b ) ∀ a, b ∈ C [ π ] then the completion is a F r´ echet a lg ebra. In par ticula r, the semi-norms η m satisfy this proper ty by (5). Prop ositi on 3. Supp ose f : π 1 → π 2 is a gr oup homomorphism. L et S i b e a c ount able c ol le ction of submultiplic ative semi-norms on C [ π i ] , i = 1 , 2 . S upp ose also that for e ach η ∈ S 2 , f ∗ η := η ◦ f ∈ S 1 . Then the gr oup algebr a ho m omorphism induc e d by f extends to a c ont inu ous homomorphi sm of F r´ echet algebr as ˆ f : H S 1 ( π 1 ) → H S 2 ( π 2 ) wher e H S i ( π i ) denotes the c ompletion of C [ π i ] with r esp e ct to the c ol le ction of semi-norms S i . Pr o of. As noted, the completions are F r´ echet algebras. So it will suﬃce to prov e that f extends ov er the completions. Let x ∈ H S 1 ( π 1 ) be re presented by the Cauc hy s e quence { x i } and η 2 be any semi- norm in S 2 . Then { x i } converges in the s e mi- norm η 2 ◦ f , whic h is exactly saying that { f ( x i ) } c o nv erges in the semi-norm η 2 . If x is r epresented b y another sequence { x ′ i } , then { x i − x ′ i } co nv erges to zero in the semi-norm η 2 ◦ f , whence { f ( x i − x ′ i ) = f ( x i ) − f ( x ′ i ) } co nv erges to zer o in the semi-norm η 2 . In particula r, if we start with a collection of semi-norms T 1 on C [ π 1 ], we can enlarg e T 1 to a se t S 1 which contains all semi-nor ms of the form f ∗ ( η ) = η ◦ f , η ∈ S 2 . Then the a b ov e prop o sition applies, and f extends. An augmented simplicial g roup with word-length (Γ . + , L . + ) consists of an augmented simplicia l group Γ . + , with L n a word-length function on Γ n for a ll n ≥ − 1. The a ugmented simplicial group is p-b ounded if all fa ce and degeneracy maps , including the augmentation map ε : Γ 0 ։ Γ − 1 , are p o lynomially bo unded with resp ect to the word-length functions { L n } n ≥− 1 . Let Γ( ε ) 0 := ker ( ε ), Γ( ε ) n := ker( ε n = ε ◦ ∂ ( n ) 0 : Γ n → Γ − 1 ). Then Γ( ε ) . is a simplicial subgroup of simplicial group Γ . = { Γ n } n ≥ 0 , and Γ . + is a resolution if π ∗ (Γ( ε ) . ) = 0 , ∗ ≥ 0. Finally , (Γ . + , L . + ) is a type P resolution if i) (Γ n , L n ) is a countably generated free group with N -v alued word-length metric L n induced b y a prop er function on the set of generator s for Γ n for all n ≥ 0 , and ii) Γ( ε ) . , viewed as a simplicial set, admits a simplicial contraction s . ′ = { s ′ n +1 : Γ( ε ) n → Γ( ε ) n +1 } n ≥ 0 which is p o lynomially b ounded in e ach degr e e . Ev ery countable discrete g r oup π admits a type P r esolution; moreover any p-bo unded simplicial group (Γ ., L . ) includes as a simplicial subgroup of a (larger) type P resolution ( e Γ . + , e L . + ) wher e e Γ − 1 = π 0 (Γ . ) [Appendix, O1]. In fact, s tarting with π , the r esolution can a lwa ys be constructed so that the face and dege neracy maps, the aug ment a tion ε a nd the simplicia l co ntraction s . ′ are a ll linearly b ounded. An y augmented simplicia l group Γ . + may b e viewed as a c ov ar iant functor Γ . + : (∆ + ) op → ( g ps ), where ∆ + denotes the augmented simplicial category . F or each m, n ≥ − 1, let S m,n denote the image under Γ . + of the morphism set H om (∆ + ) op ([ m ] , [ n ]); this is the set of all homomor phisms fro m Γ m to Γ n which o ccur as an iterated comp osition of face and degenera c y maps . F or each n ≥ − 1, let { η n,j } j ≥ 0 resp. { ν 2 ,n,j } j ≥ 0 denote the collection of semi-norms asso ciated with the group with w ord-le ng th (Γ n , L n ) as deﬁned in (4) resp. (7). Finally w e set S m = a n ≥− 1 a λ ∈ S m,n { λ ∗ ( η n,j ) } j ≥ 0 m ≥ − 1 (10) F or ea ch m , S m is a collection of submultiplicativ e semi-norms on C [Γ m ]. Deﬁnition 2. F or al l n ≥ − 1 , H C M L . (Γ . + ) n := H S n (Γ n ) , the c ompletion of C [Γ n ] with r esp e ct t o t he c ol le ction of semi-norms S n . W e will write { [ n ] 7→ H C M L . (Γ . + ) n } n ≥− 1 simply a s H C M L . (Γ . + ). Let ( c. compl exes ) denote the c ategory of connec ted chain co mplexes ov er C , with mo rphisms consisting of degre e 0 chain maps. The term aug ment e d simplicial ch a in complex will refer to a cov ariant functor F : (∆ + ) op → ( c. compl exes ), which we will write as ( F . + ) ∗ = { [ n ] 7→ ( F n ) ∗ } n ≥− 1 . F o r each n ≥ − 1, ( F n ) ∗ = (( F n ) k , d n,k ) k ≥ 0 is a connected chain co mplex, with face and degeneracy maps corr e sp onding to morphisms of co mplexes. W e say that ( F . + ) ∗ is of res olution type if the aug ment e d simplicial ab elian group { [ n ] 7→ ( F n ) k } n ≥− 1 is a r esolution for ea ch k ≥ 0. Le t F ∗∗ denote the bicomplex a sso ciated to the simplicial complex ( F . ) ∗ := { [ n ] 7→ ( F n ) ∗ } n ≥ 0 , a nd T ot ( F ∗∗ ) its total complex. Then the augmentation map ε ∗ : ( F . ) ∗ ։ ( F − 1 ) ∗ induces a chain map T ot ( F ∗∗ ) ։ ( F − 1 ) ∗ which is a quasi-isomorphis m (homolog y isomorphism) when ( F . + ) ∗ is of r esolution t yp e. Dually , a coaug mented cosimplicial (connected) c o ch a in complex is o f reso lution type if each of the cos implicial ab elian gr oups { [ n ] 7→ ( F n ) k } n ≥− 1 is a r esolution for each k ≥ 0. In all cases considered b elow, the dual situation aris es via passage to either linear or contin uous duals. In co njunction with the pr o cedure indicated in Deﬁnition 2, type P resolutions of a discr e te gro up π can be used to cons truct co ntin uous augmen ted simplicial reso lutions of H C M L ( π ) in the category of F r´ echet algebras . Theorem 1 . H C M L . (Γ . + ) is an augmente d simplicial F r´ echet algebr a which in de gr e e − 1 is the Connes- Mosc ovici algebr a H C M L − 1 (Γ − 1 ) . When (Γ . + , L . + ) is a typ e P r esolution, the augmente d simplicial c omplexes { [ n ] 7→ C H t ∗ ( H C M L . (Γ . + ) n ) } n ≥− 1 { [ n ] 7→ C C t ∗ ( H C M L . (Γ . + ) n ) } n ≥− 1 ar e of r esolution typ e, as ar e the c osimpl icial c o c omplexes { [ n ] 7→ C H ∗ t ( H C M L . (Γ . + ) n ) } n ≥− 1 { [ n ] 7→ C C ∗ t ( H C M L . (Γ . + ) n ) } n ≥− 1 Pr o of. By Prop. 3 all the fa ce and degenera c y maps of the a ug mented simplicial gro up a lgebra C [Γ . + ] extend to contin uous algebra homomorphisms satisfying the same augmen ted simplicial identies. This makes H C M L . (Γ . + ) an augmented simplicial a lgebra. Moreover, for each m ≥ − 1, the set S m in (10) ab ove is countable. Thus the completio n in each degr ee is with resp ect to a co unt a ble colle c tion of semi-norms, hence F r ´ echet. W e note the condition that the word-length function tak es v a lues in N guarantees it is nice in the sense o f Lemma 2 . Assume no w that (Γ . + , L . + ) is a type P reso lution. In order to show that the augmen ted simplicial (co- )complexes listed ab ov e are of resolution type, it will suﬃce to show that the a ugmented simplicial F r´ echet space H C M L . (Γ . + ) (gotton b y forg etting the m ultiplication) admits a con tinuous simplicial con tra ction { e s n : H C M L . (Γ . + ) n − 1 → H C M L . (Γ . + ) n } n ≥ 0 By assumption, Γ( ε ) . admits a simplicial co ntraction s . ′ = { s ′ n +1 : Γ( ε ) n → Γ( ε ) n +1 } n ≥ 0 which is p- bo unded. Fix a s e ction s (0) : Γ − 1 ֌ Γ 0 , ε 0 ◦ s (0) = identit y , with s (0)(1 ) = 1 a nd s (0 ) minimal with resp ect to word-length. Deﬁne s ( n ) := s ( n ) 0 ◦ s (0) : Γ − 1 ֌ Γ n . Note that ε n ◦ s ( n ) = iden tity ∀ n ≥ 0 , ∂ i ◦ s ( n ) = s ( n − 1) ∀ n ≥ 1 , 0 ≤ i ≤ n , s i ◦ s ( n − 1) = s ( n ) ∀ n ≥ 1 , 0 ≤ i ≤ n − 1 . (11) F or a r bitrary g ∈ Γ n , g ( s ( n )( ε n ( g ))) − 1 ∈ Γ( ε ) n . W e deﬁne e s n +1 by e s n +1 ( g ) = s ′ n +1 ( g ( s ( n )( ε n ( g ))) − 1 ) s ( n + 1)( ε n ( g )) (12) This deﬁnes a map of sets e s n +1 : Γ n → Γ n +1 , which extends uniquely to a map of vector spaces e s n +1 : C [Γ n ] → C [Γ n +1 ] where e s 0 = s (0). By the simplicial identities d n +1 e s n +1 = ( − 1 ) n +1 ( id ) + e s n d n (13) where d m = P m i =0 ∂ i : C [Γ m ] → C [Γ m − 1 ]. It follows that e s ∗ = { e s n } n ≥ 0 is a p-bo unded contraction of the augmented simplicial vector s pace C [Γ . + ]. Our g o al will be to s how that e s ∗ is contin uous with resp ect to the s e mi-norms used to co mplete C [Γ . + ] to for m H C M L . (Γ . + ). By h y po thesis, (Γ n , L n ) is a free group with prop er word-length metric when n ≥ 0 . How ever, it may be inﬁnitely genera ted. F o r a ﬁnitely gener ated free g roup F on mor e than one generator , one has Ha agerup’s inequa lity [H] k ϕ k ≤ 2 ν 2 , 2 ( ϕ ) (14) Observ ation 1. L et ( F, X , L ) re pr esent a fr e e gr oup F on a c ountable gener ating set X , wher e L is a wor d-length metric on F induc e d by a pr op er function on the gener ating s et X [O1]. Then (as in the c ase of ﬁnitely gener ate d fr e e gr oups) the ine quality k x k ≤ 2 ν 2 ( x ) is satisﬁe d for al l x ∈ H 2 , ∞ L ( F ) . Conse quent ly, H 2 , ∞ L ( F ) is a dense, holomorphic al ly clo se d involutive su b algebr a of C ∗ r ( F ) c ont aining C [ F ] , and the c ompletions of C [ F ] with r esp e ct to the c ol le ctions of semi-norms { ν 2 ,s } s ≥ 0 and { η m } m ≥ 0 ar e the same. Pr o of. Let X m = { x ∈ X | L ( x ) ≤ m } , and let F m be the s ubgroup of F generated by X m , whe r e L m = L | F m . W e wish to show that H ∞ L ( F ) is contained in C ∗ r ( F ). Let x = P g λ ( g ) g ∈ H 2 , ∞ L ( F ), a nd for each p let x p = P g ∈ F p λ ( g ) g . F or all N > m, n k x m − x n k N ≤ 2 ν 2 , 2 ,N ( x m − x n ) (15) by (14), where the norm on the left is the C ∗ norm in C ∗ r ( F N ), and the norm o n the right is in H 2 , ∞ L N ( F N ). Denote the norm in C ∗ r ( F ) by k − k . Then (15) implies k x m − x n k ≤ 2 ν 2 , 2 ( x m − x n ) Thu s the Cauch y seq uence { x p } co nv erges in C ∗ r ( F ). Since it conv erg es to x in ℓ 2 ( F ), x ∈ C ∗ r ( F ). Returning to the pro of a t hand, w e consider t wo cases. n = 0 W e need to show e s 0 = s (0) : C [Γ − 1 ] → C [Γ 0 ] is con tinuous with resp ect to the semi-norms in S 0 . F or each m , the set S 0 ,m consists o f a sing le elemen t s ( m ) 0 : Γ 0 → Γ m . The pro o f for this case follows from the sequence of inequa lities ( s ( m ) 0 ) ∗ ( η m,j )( s (0)( x )) = η m,j ( s ( m ) 0 ( s (0)( x ))) ≤ (2)2 j ν 2 ,m,j +2 ( s ( m ) 0 ( s (0)( x ))) b y Lemma 2 and Observ ation 1 ≤ C m 2 j +1 ν 2 , 0 ,j +2+ k m ( s (0)( x )) as s ( m ) 0 is p-b o unded = C m 2 j +1 ν 2 , − 1 ,j +2+ k m ( x ) as s (0) preserves word-length ≤ C m 2( j + 2 + k m )2 j +1 η − 1 , 2( j +2+ k m ) ( x ) b y Lemma 2 n > 0 A se mi-norm in S n is of the form f ∗ ( η m,j ) where f ∈ S n,m . Given e s n : Γ n − 1 → Γ n , the comp ositio n f ◦ e s n : Γ n − 1 → Γ m may b e factored as Γ n − 1 s J ◦ ∂ I − → Γ l e s K − → Γ m where s J is an iterated degenera cy , ∂ I an iterated comp osition of face maps, and e s K = e s m ◦ e s m − 1 ◦ · · · ◦ e s l +1 with l < m . This implies f ∗ ( η m,j )( e s n ( x )) = η m,j ( e s K (( s J ◦ ∂ I )( x ))) If m = − 1, then f ◦ e s n : Γ n − 1 → Γ − 1 is the a ugmentation map ε n − 1 , and η − 1 ,j ( f ( e s n ( x ))) = η − 1 ,j ( ε n ( x )) = ε ∗ n − 1 ( η − 1 ,j )( x ) If m ≥ 0, then η m,j ( e s K (( s J ◦ ∂ I )( x ))) ≤ 2 j +1 ν 2 ,m,j +2 ( e s K (( s J ◦ ∂ I )( x ))) b y Lemma 2 and Observ ation 1 ≤ 2 j +1 ( C N ) ν 2 ,l,j +2+ N (( s J ◦ ∂ I )( x )) b y the p-b oundedness of e s K ≤ 2 j +1 C N (2( j + 2 + N )) η l, 2( j +2+ N ) (( s J ◦ ∂ I )( x )) = 2 j +1 C N (2( j + 2 + N ))( s J ◦ ∂ I ) ∗ ( η l, 2( j +2+ N ) )( x ) In other w or ds, s tarting with the semi-norm η = f ∗ ( η m,j ) ∈ S n , there is a co nstant D N and semi-norm η ′ = ( s J ◦ ∂ I ) ∗ ( η l, 2( j +2+ N ) ) ∈ S n − 1 such that η ◦ e s n ≤ D N η ′ W e co nclude that for a ll n ≥ − 1 and se mi- norms η ∈ S n , there is a semi-norm η ′ ∈ S n − 1 and a constant C such that η ◦ e s n ≤ C η ′ , verifying that fo r each n ≥ 0, e s n : C [Γ n − 1 ] → C [Γ n ] extends to a contin uous morphism of F r ´ echet spaces e s n : H C M L . (Γ . + ) n − 1 → H C M L . (Γ . + ) n (16) Because e s ∗ is a simplicial con tra ction on the dense a ug mented simplicial subspac e C [Γ . + ], it follows by contin uity that its extension sa tisﬁes the necessa r y identities to ma ke it a s implicial con tra ction of the augmented simplicia l F r´ echet space H C M L . (Γ . + ), completing the pro o f. One also ha s the ℓ 1 -rapid decay and a lgebraic versions of the prev ious theorem; the pro of in bo th cases is quite straightforw a rd, and is left to the reader. Theorem 2. F or an augmente d simplicial gr oup (Γ . + , L . + ) , let A (Γ . + ) denote either the augmente d simpli- cial gr oup algebr a C [Γ . + ] ( e qu ipp e d de gr e ewise with the ﬁne top olo gy), or the augmente d simplicia l F r´ echet algebr a H 1 , ∞ L . (Γ . + ) (e quipp e d de gr e ewise with the F r ´ echet top olo gy). If (Γ . + , L . + ) is a typ e P r esolution, the augmente d simplici al c omplexes { [ n ] 7→ C H t ∗ ( A (Γ n )) } n ≥− 1 { [ n ] 7→ C C t ∗ ( A (Γ n )) } n ≥− 1 ar e of r esolution typ e, as ar e the c osimpl icial c o c omplexes { [ n ] 7→ C H ∗ t ( A (Γ n )) } n ≥− 1 { [ n ] 7→ C C ∗ t ( A (Γ n )) } n ≥− 1 The deﬁnition of H C M L . (Γ . + ) given above inv olves the minimum amount of adjustmen t necessary to extend the co nstruction of the Connes-Moscovici algebra H C M L ( π ) to a type P resolution of π . It do es not allow for ar bitrary homomorphisms of a ugmented s implicial gr oups Γ . + → Γ ′ . + to be extended to H C M L . (Γ . + ). Over the category ( f .p.g r oups ), this can b e rectiﬁed as follows. F or ea ch ob ject G o f ( f .p.g roup s ), let T ( G ) := [ G ′ ∈ obj ( f .p.g r oups ) H om ( G, G ′ ) be the set of a ll morphisms in ( f .p.g r oups ) or ig inating w ith G . F o r each f ∈ T ( G ) , f : G → G ′ , let ℓ 2 ( f ) denote a copy o f ℓ 2 ( G ′ ) indexed by f . Let ρ f : G → L ( ℓ 2 ( f )) denote the unitar y repres entation induced by the left regular r epresentation of G ′ on ℓ 2 ( f ) = ℓ 2 ( G ′ ) precompos ed with the homomorphism f . Let ℓ 2 ( T ( G )) := ⊕ f ∈ T ( G ) ℓ 2 ( f ) b e the direct sum of the Hilb er t spac e s ℓ 2 ( f ), and let ρ T ( G ) := ⊕ f ∈ T ( G ) ρ f be the direct sum of the unitary representations ρ f . Then C ∗ m ( G ) will denote the completion of C [ G ] in the op erator norm assoc iated to the unitary r epresentation ( ρ T ( G ) , ℓ 2 ( T ( G ))). This C ∗ -algebra lies b etw ee n the maxima l C ∗ -algebra C ∗ ( G ) and the reduced C ∗ -algebra C ∗ r ( G ), in that there is a factorization C ∗ ( G ) → C ∗ m ( G ) → C ∗ r ( G ) for eac h ﬁnitely presen ted group G . Moreov e r , the map C ∗ ( G ) → C ∗ m ( G ) is induced by a natural transformatio n of functors C ∗ ( − ) → C ∗ m ( − ) : ( f .p.g r oups ) → ( C ∗ - al g ebras ). W e can now duplicate the Connes-Moscovici constr uction on this sum. F or eac h f ∈ T ( G ), deﬁne D f L : ℓ 2 ( T ( G )) → ℓ 2 ( T ( G )) o n basis elements b y • D f L ( δ g ) = δ g if δ g ∈ ℓ 2 ( f ′ ), f ′ 6 = f • D f L ( δ g ) = L st ( g ) δ g when δ g ∈ ℓ 2 ( f ). As b efor e , each op e r ator D f L deﬁnes a n unbounded o p erator ∂ f L : L ( ℓ 2 ( T ( G ))) → L ( ℓ 2 ( T ( G ))) giv en by ∂ f L ( M ) = ad ( D f L )( M ) = D f L ◦ M − M ◦ D f L W e ca n now deﬁne the maximal Connes-Moscovici algebr a asso cia ted to G as Deﬁnition 3 . H C M m ( G ) = { a ∈ C ∗ m ( G ) | ( ∂ f L ) k ( a ) ∈ L ( ℓ 2 ( T ( G ))) for al l k ≥ 0 , f ∈ T ( G ) } . This a lgebra admits s emi-norms induced b y the ( ∂ f L ) k : η f n ( a ) := n X i =0  n i  k ( ∂ f L ) i ( a ) k m (17) where k − k m is the C ∗ -norm on C ∗ m ( G ) and ∂ 0 L = I d . Exactly as ab ov e, one has for each f ∈ T ( G ) inequalities η f n ( ab ) ≤ η f n ( a ) η f n ( b ) η f m ( a ) ≤ η f n ( a ) when m ≤ n Remark 1. In Deﬁnition 2 above, we could hav e pro ceeded as follows: let T m = ` n ≥− 1 S m,n denote the set of ma ps in the simplicial group Γ . o riginating with Γ m . F or ea ch T m ∋ f : Γ m → Γ n , let ℓ 2 ( f ) denote a copy of ℓ 2 (Γ n ) indexed by f . W rite ℓ 2 ( T m ) for ⊕ f ∈ T m ℓ 2 ( f ), ρ ( T m ) for ⊕ f ∈ T m ρ ( f ), where ρ ( f ) is deﬁned a s ab ov e . F or each m ≥ 0, C ∗ T . (Γ . ) in degree − 1 is C ∗ r ( π ), π = π 0 (Γ . ), and in dimens io n n ≥ 0 is the C ∗ -algebra co mpletion of C [Γ n ] with respect to the unitary representation ( ρ ( T n ) , ℓ 2 ( T n )). This is a simplicial C ∗ -algebra with π 0 ( C ∗ T . (Γ . )) mapping to C ∗ r ( π ) by a map inducing an isomorphism in topo logical K - theo ry . Similarly , for n = − 1 set H C M T . (Γ . ) = H C M L ( π ), and for n ≥ 0 H C M T . (Γ . ) n := { a ∈ C ∗ T . (Γ . ) n | ( ∂ f L ) k ( a ) ∈ L ( ℓ 2 ( T n )) for all k ≥ 0 , f ∈ T n } The fact that T n is a ﬁnite set of group homomorphisms for each n implies that the natura l map H C M T . (Γ . ) n → H C M L . (Γ . ) n is an isomorphism for eac h n . In the more general case, when the set of homomorphisms is inﬁnite, these tw o construc tio ns are not eq uiv alent , a nd the prop er construction is that given in Def. (3). Prop ositi on 4. The asso ciatio n G 7→ H C M m ( G ) deﬁnes a fun ct or H C M m ( − ) : ( f .p.g roup s ) → ( F r ´ echet al g ebra s ) Pr o of. As w e have seen, us ing the standard word-length function on the ob jects of ( f .p.g r oups ) allows one to enum er ate both the o b jects of ( f .p.g roup s ) and, for any pair of ob jects G, G ′ ∈ obj ( f .p.g roups ), the Hom set H om ( G, G ′ ). Consequently T ( G ) is co untable for each G ∈ obj ( f .p.g r oups ). The functoriality is built in to the deﬁnition of the algebra (compar e P r op. 3 a b ov e). The functoriality of both H C M m ( − ) and C ∗ m ( − ) allows one to treat the inclusion H C M m ( G ) ֒ → C ∗ m ( G ) as the res triction to G ∈ obj ( f .p.g r oups ) of a natura l transformation H C M m ( − ) ֒ → C ∗ m ( − ) If Γ . is a simplicial o b ject in ( f .p.g rou ps ), then this natural transformation induces a contin uous homo- morphism of simplicial algebra s H C M m (Γ . ) ֒ → C ∗ m (Γ . ) F or the following prop o sition, we recall that A f indicates the contin uous top olog ic al algebra A is equipped with the ﬁne top ology . Prop ositi on 5. L et Γ . b e a simpli cial obje ct in ( f .p.g roups ) , with π := π 0 (Γ . ) . Then ther e is an isomor- phism K t ∗ ( π 0 ( H C M m (Γ . ) f )) ∼ = K t ∗ ( C ∗ m ( π )) , ∗ ≥ 1 Mor e over, the natur al map C [ π ] → C ∗ m ( π ) factors as C [ π ] = π 0 ( C [Γ . ]) → π 0 ( H C M m (Γ . ) f ) → C ∗ m ( π ) (18) Pr o of. F or a simplicial algebr a A . , set A j n := ∩ j i =0 k er ( ∂ i : A n → A n − 1 ) fo r 0 ≤ j ≤ n and n ≥ 1. Also, set A 0 0 := ∂ 1 ( A 0 1 ). Then π 0 ( A . ) = A 0 / A 0 0 . As in [O2 , Cor. 1.7 ], one identiﬁes C ∗ m (Γ . ) 0 0 as the norm-closur e of C [Γ . ] 0 0 in C ∗ m (Γ 0 ), with π 0 ( C ∗ m (Γ . )) = C ∗ m ( π ). F ollowing [O2, Le mma 5.2] and applying Baum retop olog ization [App endix], o ne has that the map H C M m (Γ . ) j n → C ∗ m (Γ . ) j n induces an isomorphism on top olog ical K -gr oups K t ∗ (( H C M m (Γ . ) j n ) f ) ∼ = K t ∗ ( C ∗ m (Γ . ) j n ) , ∗ ≥ 0 (19) for all 0 ≤ j ≤ n , n ≥ 0. The isomorphism in (19) induces an isomo rphism 2 K t ∗ ( π 0 ( H C M m (Γ . ) f )) ∼ = K t ∗ ( π 0 ( C ∗ m (Γ . ))) = K t ∗ ( C ∗ m ( π )) , ∗ ≥ 1 T o verify the fac torization of (18) ab ov e, it suﬃces to obser ve the e x istence of a commut a tive diag ram C [Γ 1 ] ∂ i   / / H C M m (Γ 1 ) ∂ i   C [Γ 0 ] / / H C M m (Γ 0 ) which, in turn, follows from the functoriality of both C i ( − ) and H C M m ( − ) on the categ o ry ( f .p.g roup s ) once one knows that fo r ea ch ﬁnitely presented group G , the natural inclus io n C [ G ] ֒ → C ∗ m ( G ) factors as C [ G ] ֒ → H C M m ( G ) → C ∗ m ( G ) 2 see Appendix 4 Detecting the assem bly map 4.1 Sp ectral sequences in Ho chsc hild and cyclic ( co-)homology W e b egin with the obse r v ation (compare[O1 , App endix]) that if Γ . is a free simplicial reso lution of π , the homology and c o homolog y of B π (with co eﬃcients) may be computed as follo ws: let V b e a tr ivial π -mo dule. Set D n := H 1 ( B Γ n − 1 ; Z ), with d n = P n − 1 i =0 ( − 1) i ( ∂ i ) ∗ : D n → D n − 1 , ∂ i : Γ n → Γ n − 1 . Then ( D ∗ , d ∗ ) is a chain complex; its dual (with co eﬃcien ts in V ) is given by D n ( V ) := H 1 ( B Γ n − 1 ; V ) , δ n = P n i =0 ( − 1) i ( ∂ i ) ∗ : D n → D n +1 . F or n ≥ 1 o ne then ha s H n ( B π ; V ) = H n ( D ∗ ⊗ V , d ∗ ⊗ 1) H n ( B π ; V ) = H n ( D ∗ ( V )) (20) Next, cons ider the spectral sequence in cyclic homology E 1 ∗∗ = { E 1 p,q = H C p ( C [Γ q ]) ⇒ H C p + q ( C [Γ . ]) ∼ = H C p + q ( C [ π ]) } p,q ≥ 0 (21) When p > 0 we have H C p ( C [Γ q ]) = H C p ( C [Γ q ]) < 1 > for all q ≥ 0. F o r any free gro up F , the c yclic homology of [ F ] abov e dimension 0 is giv en by H C 2 p +1 ( C [ F ]) = H 1 ( B F ) , p ≥ 0 H C 2 p ( C [ F ]) = C , p ≥ 1 (22) Graphically , then, the E 1 ∗∗ -term in (21) may be represented as . . .   . . .   . . .   . . .   H C 0 ( C [Γ 3 ])   H C 1 ( C [Γ 3 ])   H C 2 ( C [Γ 3 ])   H C 3 ( C [Γ 3 ])   · · · H C 0 ( C [Γ 2 ])   H C 1 ( C [Γ 2 ])   H C 2 ( C [Γ 2 ])   H C 3 ( C [Γ 2 ])   · · · H C 0 ( C [Γ 1 ])   H C 1 ( C [Γ 1 ])   H C 2 ( C [Γ 1 ])   H C 3 ( C [Γ 1 ])   · · · H C 0 ( C [Γ 0 ]) H C 1 ( C [Γ 0 ]) H C 2 ( C [Γ 0 ]) H C 3 ( C [Γ 0 ]) · · · (23) Incorp ora ting the iso mo rphisms of (22) , the diagr am in (23) becomes . . .   . . .   . . . id   . . .   . . . id   . . .   H C 0 ( C [Γ 3 ])   H 1 ( B Γ 3 )   C 0   H 1 ( B Γ 3 )   C 0   H 1 ( B Γ 3 )   · · · H C 0 ( C [Γ 2 ])   H 1 ( B Γ 2 )   C id   H 1 ( B Γ 2 )   C id   H 1 ( B Γ 2 )   · · · H C 0 ( C [Γ 1 ])   H 1 ( B Γ 1 )   C 0   H 1 ( B Γ 1 )   C 0   H 1 ( B Γ 1 )   · · · H C 0 ( C [Γ 0 ]) H 1 ( B Γ 0 ) C H 1 (Γ 0 ) C H 1 (Γ 0 ) · · · (24) Because H C ∗ ( C [ π ]) < 1 > = ⊕ m ≥ 0 H ∗− 2 m ( B π ; C ) is a summand of H C ∗ ( C [ π ]) to whic h the ab ove sp ectral sequence is con verging, the iso morphisms of (20) imply • The spectr al sequence ab ov e, with E 1 ∗∗ -term given as in (23) or (24), co llapses at the E 2 ∗∗ -term; i.e., E 2 ∗∗ = E ∞ ∗∗ . • F or ea ch m ≥ 0, the summand H ∗− 2 m ( B π ; C ) of H C ∗ ( C [ π ]) < 1 > ident iﬁes with E 2 2 m +1 , ∗− 2 m − 1 when ( ∗ − 2 m ) > 0. In par ticula r, the unshifted summand H ∗ ( B π ; C ) of H C ∗ ( C [ π ]) < 1 > app ears a s E 2 1 , ∗− 1 = H ∗ ( D ∗ , d ∗ ). The sa me analys is a pplies to the sp ectral se quence in cyclic cohomolog y . The relev an t p oints ar e • The sp e ctral seq uence E ∗∗ 1 = { E pq 1 := H C p ( C [Γ q ]) ⇒ H C p + q ([Γ . ]) ∼ = H C p + q ( C [ π ]) } p,q ≥ 0 (25) collapses at the E ∗∗ 2 -term. • The summand H ∗− 2 m ( B π ; C ) of H C ∗ ( C [ π ]) < 1 > ident iﬁes with the E 2 m +1 , ∗− 1 2 -term of this sp ectral sequence. Note als o that a na logous sp ectral sequences exist in Hochschild homolo gy and coho mology: E 1 ∗∗ = { E 1 pq := H H p ( C [Γ q ]) ⇒ H H p + q ([Γ . ]) ∼ = H H p + q ( C [ π ]) } p,q ≥ 0 E ∗∗ 1 = { E pq 1 := H H p ( C [Γ q ]) ⇒ H H p + q ([Γ . ]) ∼ = H H p + q ( C [ π ]) } p,q ≥ 0 and that these sp ec tr al s e quences in b o th Hochschild and cyclic (co)homology ex ist for a rbitrar y simplicial groups Γ . Moreover, when Γ . is deg reewise free, (but not nec e ssarily a r e solution of π ), the desc ription of the E 2 ∗∗ -term in cyclic homology given a b ov e in (2 4) still applies. In this cas e, how ever, the s p ec tral sequence need not collapse at the E 2 ∗∗ -term, and will conv erge to H C ∗ ( C [Γ . ]) whic h ma y not equal H C ∗ ( C [ π ]). F rom these considerations, one sees that an arbitrary x ∈ H n ( B π ; C ) admits a unique r epresentativ e as x ∈ E 2 1 ,n − 1 = k er ( H C 1 ( C [Γ n − 1 ]) → H C 1 ( C [Γ n − 2 ])) /im ( H C 1 ( C [Γ n ]) → H C 1 ( C [Γ n − 1 ])) (26) Suppo se now that Γ . is a t yp e P resolution of a ﬁnitely-presented gr oup π . By Theorems 2 a nd 1 ab ove, one has corresp onding sp ectral sequences in top olo gical cyclic homology: E 1 ∗∗ = { E 1 p,q = H C t p ( H 1 , ∞ L q (Γ q )) ⇒ H C t p + q ( H 1 , ∞ L . (Γ . )) ∼ = H C t p + q ( H 1 , ∞ L ( π )) } p,q ≥ 0 (27) E 1 ∗∗ = { E 1 p,q = H C t p ( H C M L (Γ . )) ⇒ H C t p + q ( H C M L . (Γ . )) ∼ = H C t p + q ( H C M L ( π )) } p,q ≥ 0 (28) Given an element 0 6 = x as in (26) ab ove, one can c ho o s e a r epresentativ e x ∈ H 1 ( B Γ n − 1 ; C ) ∼ = H C 1 ( C [Γ n − 1 ]), and a ttempt to show that • its image in H C t 1 ( H 1 , ∞ L n − 1 (Γ n − 1 )) survives to E ∞ 1 ,n − 1 , or even more, that • its image in H C t 1 ( H C M L . (Γ . ) n − 1 ) surv ives to E ∞ 1 ,n − 1 . As we shall see, verifying the former for a ll (rational) homolog y clas ses implies rational injectivity of the assembly map for K t ∗ ( ℓ 1 ( π )), while verifying the latter for all such classes implies rational injectivit y of the assembly map for K t ∗ ( C ∗ r ( π )). How ever, even though these sp ectr al sequences exist (and ar e a prime motiv ation for what follows), working with them direc tly to verify injectivity for homolo gy classes for which injectivit y was not already known is problematic. The ﬁrst complicating factor is the size o f the group Γ . , whic h aﬀects bo th spectra l sequences. An additional iss ue aﬀecting the second sp ectral se q uence is its lack of functoriality with repsect to simplicial g roup ho momorphisms, a s w e ha ve discussed above. In the next section, we introduce a technique which gets around the ﬁrst problem, culminating in the pro of of Theor em A. And, with H C M m (Γ . ) in pla c e o f H C M L . (Γ . ), it yields as well a r eduction of the Strong Novik ov Conejcture to a conjectur e ab o ut the topo logical cyclic homo logy gr oup H C t 1 ( H C M m ( F )), where F is a ﬁnitely- generated free g roup. 4.2 The lo cal C hern character asso ciated to an in tegral homology class Suppo se now that π is ﬁnitely presented. In this ca se we can construct an augmented simplicia l res olution Γ . + of π of t yp e H F 2 ; in o ther words, a p-b o unded re solution degreewis e free ab ov e dimension − 1 , with π 0 (Γ . ) = Γ − 1 = π and Γ i ﬁnitely generated for i = 0 , 1. As is tr ue for any re s olution, H ∗ ( B π ; C ) = H ∗ ( D ∗ ⊗ C , d ∗ ) for ∗ ≥ 1 , where D n = H 1 ( B Γ n − 1 ; Z ), with diﬀer e nt ia l induced b y the alternating sum of the b oundar y ma ps in homo lo gy . This complex can b e deﬁned for any simplicia l group (not just free resolutons); for that reason when we wan t to sp ecify the dep endence on the group Γ . , we will write D ∗ (Γ . ) rather than just D ∗ . As in [O1], w e assume Γ . comes equipp ed with a g raded generating set X . which is closed under deg eneracies, a nd whic h (via the ﬁnite pres entabilit y of π ) w e may arra nge to be ﬁnite in dimensons 0 and 1. In what follo ws , an inte gr al homology class x ∈ H n ( B π ; C ), n ≥ 1, will r efer to an element in the image of the map H n ( B π ; Z ) → H n ( B π ; C ) induced b y the inclusion of coeﬃcients Z ֒ → C . Via the iden tiﬁca tion H ∗ ( B π ; Z ) ∼ = H n ( D ∗ (Γ . ) , d ∗ ), w e ma y c ho os e a preimage ˜ x ∈ Γ n − 1 of x ∈ H n ( D ∗ (Γ . ) , d ∗ ) mapping to a cycle r epresenting x under the pro jection Γ n − 1 ։ H 1 ( B Γ n − 1 ; Z ) → H 1 ( B Γ n − 1 ; C ); from this element w e will co ns truct an a ug mented simplicial subgroup Γ( ˜ x ) . + ⊆ Γ . + . W e begin b y deﬁning a gra ded set S ( ˜ x ) . • F or j = 0 , 1, S ( ˜ x ) j = X j ; • If n ≥ 3, S ( ˜ x ) n − 1 contains a single element s ˜ x corres p o nding to ˜ x ; • S ( ˜ x ) n − 2 contains every ele ment of X n − 2 used in writing Q 0 ≤ i ≤ n − 1 ( ∂ i ( ˜ x )) ( − 1) i as a product of commutators in Γ n − 2 ; • F or all 2 ≤ j < n − 2, S ( ˜ x ) j is the minimal s ubset of X j containing all generator s o ccur ing in the unique reduced word repre s enting ∂ I ( y ), where ∂ I ( − ) denotes an iterated face ma p beginning either in dimension n − 1 or n − 2 and y ∈ S ( ˜ x ) n − 1 ` S ( ˜ x ) n − 2 . Now deﬁne X ( ˜ x ) . to be the closure o f S ( ˜ x ) . under degener acies, and Γ( ˜ x ) . to be the free simplicial group generated by X ( ˜ x ) . There is a ca nonical injection Γ( ˜ x ) . ֒ → Γ . which sends s ˜ x to ˜ x , and on other generator s is the identit y map. In fact, one can deﬁne the simplicial structure on Γ( ˜ x ) . to b e that mak ing this inclusion of g r aded sets a simplicial map. W e extend Γ( ˜ x ) . to a n a ugmented simplicial o b ject by setting Γ( ˜ x ) − 1 = π . The following prop osition lists the essential prop erties of Γ( ˜ x ) . Prop ositi on 6. F or any element ˜ x , • Γ( ˜ x ) i = Γ i for − 1 ≤ i ≤ 1 ; • Γ( ˜ x ) . is fr e e and ﬁnitely gener ate d in e ach de gr e e; • F or al l n ≥ 1 ther e is an inte gr al “fundamental”homolo gy class µ ˜ x ∈ H n ( B Γ( ˜ x ) . ; Z ) . Mor e over, we have µ ˜ x 7→ x un der the c omp ositio n in homolo gy H n ( B Γ( ˜ x ) . ; Z ) → H n ( B Γ( ˜ x ) . ; C ) → H n ( B Γ . ; C ) ∼ = H n ( B π ; C ) If n ≥ 3 , H n ( B Γ( ˜ x ) . ; Z ) ∼ = Z , gener ate d by µ ˜ x . • In the sp e ctr al se quenc es E 1 pq = H C p ( C [Γ( ˜ x ) q ]) ⇒ H C p + q ( C [Γ( ˜ x ) . ]) r esp . E pq 1 = H C p ( C [Γ( ˜ x ) q ]) ⇒ H C p + q ( C [Γ( ˜ x ) . ]) the element µ ˜ x r esp. its dual is r epr esente d by a p ermanent cycle in E ∞ 1 ,n − 1 r esp. E 1 ,n − 1 ∞ which is non-zer o if 0 6 = x ∈ H n ( B π ) . Pr o of. The ﬁrst tw o pr o p erties follow directly from the co nstruction of Γ( ˜ x ) . + . F or the third, we observe that the image o f s e x ∈ Γ( e x ) n − 1 in H 1 ( B Γ( e x ) n − 1 ; Z ) is a cyc le 3 representing a canonically deﬁned class µ e x ∈ H n ( D (Γ( e x ) . ; Z ). Now assume n ≥ 3. The homolog y of D ∗ (Γ( ˜ x ) . ) can b e computed b y the normaliz e d complex D ∗ (Γ( ˜ x ) . ), in which degener ate elements have b een identiﬁed with zer o . In this normalized complex, D m (Γ( ˜ x ) . ) = 0 when m > n , D n (Γ( ˜ x ) . ) ∼ = Z g enerated by the cycle µ ˜ x := [ s ˜ x ], a nd for m < n is the free Z -module with ca no nical basis consisting of the non-degenerate elemen ts of X ( ˜ x ) m . This identiﬁes H n ( D ∗ (Γ( ˜ x ) . ); Z ) with Z on basis element µ ˜ x , which b y construction maps to x under the indicated map. The sa me result holds in coho mology , with an identiﬁcation H n ( D ∗ (Γ( ˜ x ) . ); Z ) ∼ = Z gener a ted by the dual cla ss µ ∗ ˜ x . W e now consider the four th claim. Because Γ( ˜ x ) . is a free simplicial gr oup, the cy clic homology spec tr al sequence co nv erging to H C ∗ ( C [Γ( ˜ x ) . ]) satisﬁes E 2 1 ,n − 1 ∼ = H n ( D ∗ (Γ( ˜ x ) . ) ⊗ C ). By the third prop erty , the inclusion Γ( ˜ x ) . ֒ → Γ . induces a map o f cyclic homology s p ectral sequences which is an injection on the E 2 1 ,n − 1 -term. As Γ . is b oth deg r eewise free and a resolution of π , the image of µ ˜ x in the E 2 1 ,n − 1 -term for H C ∗ ( C [Γ . ]) s urvives to E ∞ 1 ,n − 1 . But this implies it surviv es to E ∞ 1 ,n − 1 in the cyclic homolo g y s p e ctral sequence converging to H C ∗ ( C [Γ( ˜ x ) . ]). The dual a rgument applies in cyclic cohomolog y sp ectr al s equence converging to H C ∗ ( C [Γ( ˜ x ) . ]). In g eneral, Γ( ˜ x ) . will not b e a resolution of π = π 0 (Γ( ˜ x ) . ), let alone a t yp e P res olution, a nd so the simpli- cial pro jections C [Γ( ˜ x ) . ] ։ C [ π ], H 1 , ∞ L . (Γ( ˜ x ) . ) ։ H 1 , ∞ L ( π ) will not induce isomo r phisms in (top o logical) cyclic homo logy . The following theorem allows us to bypass this is sue as far as the c o nstruction of Chern characters ar e concer ned. Theorem 3. Supp ose A . + is an augmente d simplicial F r´ echet algebr a with π 0 ( A . ) = A − 1 , for whic h A − 1 is endowe d with a (c o arser) norm top olo gy (in which it ne e d not b e c omplete). The n for ∗ ≥ 1 ther e is a factorization H C t ∗ ( A . ) p ∗   K t ∗ ( π 0 ( A . )) f ch C K ∗ 7 7 ♥ ♥ ♥ ♥ ♥ ♥ ch C K T ∗ / / H C t ∗ ( π 0 ( A . ) f ) wher e ch C K T ∗ denotes the Connes-Kar oub-Til lmann Chern char acter [c onstru cte d in App endix]. Mor e- over, this factori zation is functorial in A . Pr o of. Deﬁne H P er x ∗ ( A. ) := lim ← − S H C x ∗ +2 n ( A. ) , x = a, t The lifting indicated in the ab ov e diagra m is given b y the following c o mp osition 3 The third deﬁning prop erty of the set S ( ˜ x ) guaran tees that [ s ˜ x ] ∈ D n (Γ( ˜ x ) . ) is a cycle. K t ∗ ( π 0 ( A . )) f ch C K ∗   ✤ ✤ ✤ ✤ ✤ ✤ ✤ K t ∗ ( π 0 ( A . ) f ) ∼ = o o f ch T ∗   H P er t ∗ ( π 0 ( A . ) f ) H P er a ∗ ( π 0 ( A . )) H C t ∗ ( A . ) H C a ∗ ( A . ) o o H P er a ∗ ( A . ) o o ∼ = O O As befo re, B f denotes the top o lo gical algebra B e quipp e d with the ﬁne top o lo gy (done degreewis e if B is simplicial or aug ment e d simplicial). The is omorphism in higher top o logical K -theor y follows by Baum’s retop olog ization theorem [Appendix]. The Chern character indicated b y f ch T ∗ is the lifting of Tillmann’s Chern c ha racter for ﬁne top olo gical alg ebras to H P e r t ∗ ( − ) (which identiﬁes with H P er a ∗ ( − ) when applied to topo logical a lgebras equipped with the ﬁne top ology). The pro jection map p : A . → π 0 ( A . ) induces an isomorphism H P er a ∗ ( A . ) ∼ = − → H P er a ∗ ( π 0 ( A . )) by the rigidity results of [G]. Finally , the bottom map in the low er left a r ises from the natura l tr ansformatio n H C a ∗ ( − ) → H C t ∗ ( − ). Each of the homomorphisms in the diagram is functor ial in A . . Corollary 1. F or e ach x ∈ H n ( B π ; C ) , with x, e x , and Γ( e x ) . as ab ove, ther e exist “lo c al”Connes-Kar oubi Chern char acters f ch C K ∗ : K t ∗ ( H 1 , ∞ L ( π )) → H C t ∗ ( H 1 , ∞ L . (Γ( e x ) . )) f ch C K ∗ : K t ∗ ( π 0 ( H C M m (Γ( e x ) . ) f )) → H C t ∗ ( H C M m (Γ( e x ) . )) f ch C K ∗ : K t ∗ ( H C M L ( π )) ∼ = K t ∗ ( π 0 ( H C M L . (Γ( e x ) . ))) → H C t ∗ ( H C M m (Γ( e x ) . )) 4.3 Pro of of Theorem A Let Γ . be a free simplicial resolution of π . Fix an integral class 0 6 = x ∈ i m ( H n ( B π ; Z ) → H n ( B π ; C )), and let e x, Γ( e x ) . b e as de ﬁned above. By Prop osition 6, ther e exists an in tegr a l fundament a l class µ ˜ x ∈ im ( H n ( B Γ( e x ) . ; Z ) → H n ( B Γ( e x ) . ; C )) ma pping to x ∈ H n ( B π ; C ) under the map induced by Γ( e x ) . ֒ → Γ . ≃ ։ π , with µ ˜ x canonically repre s ented by an elemen t in E 1 1 ,n − 1 = H C 1 ( C [Γ( e x ) n − 1 ]) which s ur vives to a non-zero elemen t of E ∞ 1 ,n − 1 . Moreov er, b eca use Γ( e x ) . is degreewise ﬁnitely genera ted, o ne may choos e a “dual”integral fun da mental class µ ∗ ˜ x ∈ im ( H n ( B Γ( e x ) . ; Z ) → H n ( B Γ( e x ) . ; C ) represented by an elemen t 4 in E 1 ,n − 1 1 = H C 1 ( C [Γ( e x ) n − 1 ]) ∼ = H 1 ( B Γ( e x ) ( n − 1) ; C ) which survives to a non-ze ro element in E 1 ,n − 1 ∞ . Under the pairing H C ∗ ( C [Γ( e x )]) ⊗ H C ∗ ( C [Γ( e x )]) → C , one has < µ ∗ ˜ x , µ ˜ x > 6 = 0 Consider now the c o mmut a tive diag ram (D6) H n ( B Γ . ; Q ) ∼ =   H n ( B Γ( ˜ x ) . ; Q ) o o   H n ( B π ; Q ) A π   / / / / H C a n ( C [Γ( ˜ x ) . ])   K t n ( H 1 , ∞ L ( π )) ⊗ Q f ch ( ˜ x ) C K n / / H C t n ( H 1 , ∞ L . (Γ( ˜ x ) . )) 4 F or n ≥ 3, the choice of represent ative µ ∗ ˜ x ∈ H C 1 ( C [Γ( e x ) ( n − 1) ]) is essentially unique, while for n = 1 , 2 is determined b y the canonical basis for H 1 ( B Γ( e x ) ( n − 1) ; C ) ∼ = H C 1 ( C [Γ( e x ) ( n − 1) ]) coming from the generating set for Γ( e x ) ( n − 1) where the arrow in the middle ﬁts into the comm uting diag ram K t ∗ ( C [ π ] f ) ⊗ Q ch T ∗ ( ( ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ H n ( B π ; Q ) 7 7 ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ / / / / ' ' ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ⊕ m ∈ Z H n − 2 m ( B π ; Q ) / / / / H P er a ( C [ π ]) H C a ∗ ( C [Γ( ˜ x ) . ]) H P er a ∗ ( C [Γ( ˜ x ) . ]) ∼ = O O o o Every ratio nal ho mology class in H n ( B π ; Q ) is a scalar multiple of a n integral class , so it suﬃces to verify injectivit y for integral classes. Beca use the choice of integral class x ∈ H n ( B π ; Q ) is ar bitr ary , this ar gument shows that the restricted rationalize d a ssembly map for K t ∗ ( C [ π ] f ) ⊗ Q is injectiv e; a fact already kno wn [T1 ]. Our ob ject, then, is to show that the image of x is non-zero in H C t n ( H 1 , ∞ L . (Γ( ˜ x ) . )). As a ﬁrst step, we hav e Lemma 3. L et 0 6 = x ∈ H n ( B π ; Q ) b e an inte gr al class, with Γ( e x ) . and 0 6 = µ e x ∈ E 1 ,a 1 ,n − 1 = H C 1 ( C [Γ( e x ) n − 1 ]) deﬁne d as ab ove. Then the imag e of µ e x in E 1 ,t 1 ,n − 1 = H C 1 ( H 1 , ∞ L st Γ( e x ) n − 1 )) (under t he map induc e d by the inclusion of simplicial algebr as C [Γ( e x ) . ] ֒ → H 1 , ∞ L . (Γ( e x ) . ) ) is n on-zer o, and survives to a non-zer o element in E 2 ,t 1 ,n − 1 . Pr o of. As Γ( ˜ x ) n − 1 is free and ﬁnitely gener ated, one ha s by [JOR1] that H C t ∗ ( H 1 , ∞ L st (Γ( ˜ x ) n − 1 )) < 1 > ∼ = H C ∗ ( C [Γ( ˜ x ) n − 1 ]) < 1 > = H ∗ ( B Γ( ˜ x ) n − 1 ; C ) H C t ∗ ( H 1 , ∞ L st (Γ( ˜ x ) n − 1 )) ∼ = H ∗ ( C g / ( g ); C ) = 0 , g 6 = 1 , n ≥ 1 F or a giv en group G , the map induced by pro jections o nt o summands indexed by conjugacy clas ses H C t ∗ ( H 1 , ∞ L st ( G )) → Y H C t ∗ ( H 1 , ∞ L st ( G )) is na tural with resp ect to gro up homomor phisms. When G = F is a ﬁnitely gener ated free group, the summands index ed by < g > 6 = 1 v anish in p ositive dimensions (lo c. cit.), re s ulting in a ma p H C t 1 ( H 1 , ∞ L st ( F )) → H C t 1 ( H 1 , ∞ L st ( F )) → H C t 1 ( H 1 , ∞ L st ( F )) < 1 > = H 1 ( B F ; C ) (29) which is functor ial with r esp ect to homo morphisms of ﬁnitely g enerated free groups. The result is a commuting diag ram H 1 ( B Γ( ˜ x ) n ) d n   H C 1 ( C [Γ( ˜ x ) n ]) / / / / d 1 ,H C 1 ,n   H C t 1 ( H 1 , ∞ L st (Γ( ˜ x ) n )) / / / / d 1 ,H C 1 ,n   H 1 ( B Γ( ˜ x ) n ) d n   H 1 ( B Γ( ˜ x ) n − 1 ) H C 1 ( C [Γ( ˜ x ) n − 1 ]) / / / / H C t 1 ( H 1 , ∞ L st (Γ( ˜ x ) n − 1 )) / / / / H 1 ( B Γ( ˜ x ) n − 1 ) by which we co nclude that µ e x survives to a non-zero c la ss in E 2 ,t 1 ,n − 1 , as claimed. T o complete the pro of, w e need to show µ e x is not in the image of the diﬀeren tial d 2 0 ,n +1 : E 2 ,t 0 ,n +1 → E 2 ,t 1 ,n − 1 . Apriori this is a delica te ma tter , b eca use, a lthough E 1 ,a 0 ,m is dense in E 1 ,t 0 ,m in the F r´ echet top o logy , there is no reason to suppo s e that this remains true at the E 2 -level. In fact, the exact relation b etw een E 2 ,a 0 , ∗ and E 2 ,t 0 , ∗ for the ab ov e complex seems v er y diﬃcult to determine. This motiv ates the consider ation of a n int er mediate complex whic h w e now deﬁne, ﬁrs t in the algebraic setting, then in the topolo gical one. Let Γ . b e a s implicial group, with p n : Γ n → π = π 0 (Γ . ) the canonica l pr o jection to π (this is rea lized by any iteration of fac e maps to Γ 0 , fo llowed by the pro jection to π ). F o r n ≥ 0, we deﬁne the homo geneous comp onent of C C ∗ ( C [Γ n ]) to be C C ∗ ( C [Γ n ]) h := ⊕ ∈ < Γ n > p n ( g )=1 C C ∗ ( C [Γ n ]) This sub co mplex is preserved under face maps as n v ar ies, yielding the simplicial s ub c o mplex C C ∗ ( C [Γ . ]) h := { [ n ] 7→ C C ∗ ( C [Γ n ]) h } n ≥ 0 ⊂ C C ∗ ( C [Γ . ]) If (Γ ., L . ) is a p-b ounded simplicia l g roup with word-length, then the decomp osition of C C t ∗ ( H 1 , ∞ L n (Γ n )) as a top ologic a l direct s um indexed on conjugacy classes (as described in § 2) a llows fo r the deﬁnition of a homog e neous comp onent in an analogous fashion C C t ∗ ( H 1 , ∞ L n (Γ n )) h := b ⊕ ∈ < Γ n > p n ( g )=1 C C t ∗ ( H 1 , ∞ L n (Γ n )) This sub co mplex is preserved under face maps as n v ar ies, yielding the simplicial s ub c o mplex C C t ∗ ( H 1 , ∞ L . (Γ . )) h := { [ n ] 7→ C C t ∗ ( H 1 , ∞ L n (Γ n )) h } n ≥ 0 ⊂ C C t ∗ ( H 1 , ∞ L . (Γ . )) Note that in bo th the algebraic and topolo gical settings, the collectio n o f pro jection maps { p n } n ≥ 0 deﬁne (contin uous) pro jections of simplicia l (top olo gical) chain complexes C C ∗ ( C [Γ . ]) ։ C C ∗ ( C [Γ . ]) h C C t ∗ ( H 1 , ∞ L . (Γ . )) ։ C C t ∗ ( H 1 , ∞ L . (Γ . )) h which a re right in verses to the natur al inclusions going the o ther wa y . Lemma 4. Supp ose Γ . is a fr e e simplicial r esolution of π . Then the asso ciate d chain c omplex of the simplicia l ve ctor sp ac e { [ n ] 7→ H C 0 ( C [Γ n ]) h } n ≥ 0 is acyclic ab ove dimension 0 . If (Γ ., L . ) is a typ e P r esolution of ( π , L ) , then the same is true for the asso ciate d chain c omplex of the simplicial top olo gic al ve ctor sp ac e { [ n ] 7→ H C 0 ( H 1 , ∞ L n (Γ n )) } n ≥ 0 . Pr o of. W e consider the algebr a ic case ﬁr st. F or such Γ . , the simplicial chain complex C C ∗ ( C [Γ . ]) h is of r esolution type , res olving C C ∗ ( C [ π ]) < 1 > . How ever, this latter c ha in complex is also resolved b y the simplicial chain complex { [ n ] 7→ C C ∗ ( C [Γ n ]) < 1 > } n ≥ 0 , which is a simplicial s ubc o mplex of C C ∗ ( C [Γ . ]) h . This implies the total co mplex of the quotient simplicia l complex C C ∗ ( C [Γ . ]) h := { [ n ] 7→ C C ∗ ( C [Γ n ]) h := C C ∗ ( C [Γ n ]) h /C C ∗ ( C [Γ n ]) < 1 > } n ≥ 0 is acyclic. Because Γ . is degreewise free, in the simplicial sp ectra l sequence conv erging to the homology of the total complex of C C ∗ ( C [Γ . ]) h one has E 1 p,q = ( 0 p > 0 H C 0 ( C [Γ q ]) h p = 0 where H C 0 ( C [Γ q ]) h = H C 0 ( C [Γ q ]) h /H C 0 ( C [Γ q ]) < 1 > . But the complex { H C 0 ( C [Γ q ]) < 1 > ∼ = C } q ≥ 0 is acyclic ab ove dimension 0, with H 0 = C , implying the stated res ult for the algebraic case. When Γ . is a t yp e P resolution o f π , the same argument w o rks for the simplicial ℓ 1 -rapid deca y algebr a H 1 , ∞ L . (Γ . ). Namely , both the simplicia l topolo g ical c omplex C C t ∗ ( H 1 , ∞ L . (Γ . )) h as well a s its simplicial sub c omplex C C t ∗ ( H 1 , ∞ L . (Γ . )) < 1 > := { [ n ] 7→ C C t ∗ ( H 1 , ∞ L n (Γ n )) < 1 > } n ≥ 0 are res olutions of C C t ∗ ( H 1 , ∞ L ( π )) < 1 > , implying that the quotient simplicial top o logical complex C C t ∗ ( H 1 , ∞ L . (Γ . )) h := { [ n ] 7→ C C t ∗ ( H 1 , ∞ L n (Γ n )) h := C C t ∗ ( H 1 , ∞ L n (Γ n )) h /C C t ∗ ( H 1 , ∞ L n (Γ n )) < 1 > } n ≥ 0 is acy c lic. Aga in, the fact Γ . is degreewis e free implies (by [JOR1], [JO R3]) that in the simplicial sp ectral sequence converging to the homology of the total co mplex one has E 1 p,q = ( 0 p > 0 H C t 0 ( H 1 , ∞ L q (Γ q )) h p = 0 where H C t 0 ( H 1 , ∞ L q (Γ q )) h = H C t 0 ( H 1 , ∞ L q (Γ q )) h /H C t 0 ( H 1 , ∞ L q (Γ q )) < 1 > . But the complex { H C t 0 ( H 1 , ∞ L q (Γ q )) < 1 > ∼ = C } q ≥ 0 is canonica lly isomorphic to its algebraic counterpart, and thus a c yclic ab ov e dimension 0, with H 0 = C . The result follows for the top ologica l case. Theorem 4. F or al l n ≥ 1 and int e gr al classes x ∈ H n ( B π ; Q ) , the class µ ˜ x ∈ E 2 ,t 1 ,n − 1 survives to a non-zer o element in E 3 ,t 1 ,n − 1 . Pr o of. W e in tro duce the method o f pr o of by ﬁrst verifying the statement in the alg ebraic ca se. Of course, for the simplicial group algebra one can use direct knowledge of H C ∗ ( C [ π ]) to conclude that µ ˜ x m us t survive, b ecause it hits the imag e of x ∈ H n ( B π ; Q ) → H n ( B π ; C ), w hich is no n-zero. The ob ject is to ﬁnd a pr o of of surviv ability which do es not rely on this last fact. W e assume Γ . is a free simplicial resolution o f π , with Γ( ˜ x ) . as deﬁned a b ov e. As prev iously noted, there is a pro jection of simplicia l complexes { [ n ] 7→ C C ∗ ( C [Γ( ˜ x ) n ]) } n ≥ 0 ։ C C a ∗∗ := { [ n ] 7→ C C ∗ ( C [Γ( ˜ x ) n ]) h } n ≥ 0 The latter embeds in the simplicia l complex C C a ′ ∗∗ := { [ n ] 7→ C C ∗ ( C [Γ n ]) h } n ≥ 0 . W e deﬁne a sub complex D a ∗∗ ⊂ C a ′ ∗∗ by D a pq =          C C a pq if q ≤ n − 1; C C a ′ pq if either q ≥ n + 2 , or p ≥ 1 and q ≥ n + 1; C C a 0( n +1) + d 0 ,a 0 ,n +2 ( C C a ′ 0( n +2) ) + b 1 ( C C a ′ 1( n +1) ) when ( p, q ) = (0 , n + 1 ); C C a pn + d 0 p ( n + 1)( D a p ( n +1) ) for q = n, D a p ( n +1) deﬁned as a b ove . It follows by Le mma 4 that in the sp ectral sequence conv erg ing to H ∗ ( D a ∗∗ ) o ne has E 2 0 ,n +1 = 0; in other words w e have killed that group. But in the sa me sp ectral s equence it is easily seen that the inclusion C a ∗∗ ֌ D a ∗∗ induces an isomor phism on the E 1 1 ,n − 1 -term, as well a s o n im ( d 1 1 ,n : E 1 1 ,n → E 1 1 ,n − 1 ) (note that the ac tua l E 1 1 ,n -terms may diﬀer, but the diﬀerence comes from the contribution of d 0 1( n +1) ( C C a ′ 1( n +1) ), which must v anish under d 1 1 ,n ). Thus the argument that µ ˜ x survives to a non-ze r o element in E 2 1 ,n − 1 - term of the sp ectral sequence for H ∗ ( C a ∗∗ ) implies the same for the corre s p onding spectral seq uence for H ∗ ( D a ∗∗ ). How ever, in this sp ectra l sequence, the v anishing of the E 2 0 ,n +1 -term implies E 2 1 ,n − 1 = E 3 1 ,n − 1 . Thu s µ ˜ x m us t surv ive to a no n- zero element in this spectral sequence , in turn implying that it must als o for E 3 1 ,n − 1 -term of the spectra l sequence for H ∗ ( C a ∗∗ ) from whic h it came. W e now extend this argument to the ℓ 1 -rapid decay case, following ess entially the same pro cedure. F or this cas e we a ssume Γ . is a t yp e P resolution of π . The ﬁrst step, as b efor e , is to map b y the pro jecton { [ n ] 7→ C C t ∗ ( H 1 , ∞ L n (Γ( ˜ x ) n )) } n ≥ 0 ։ C C t ∗∗ := { [ n ] 7→ C C t ∗ ( H 1 , ∞ L n (Γ( ˜ x ) n )) h } n ≥ 0 The la tter embeds in the simplicial complex C t ′ ∗∗ := { [ n ] 7→ C C t ∗ ( H 1 , ∞ L n (Γ n )) h } n ≥ 0 . As in the algebraic case, we deﬁne a subcomplex D t ∗∗ ⊂ C C a ′ ∗∗ by D t pq =          C C t pq if q ≤ n − 1; C C t ′ pq if either q ≥ n + 2 , or p ≥ 1 and q ≥ n + 1; C C t 0( n +1) + d 0 ,a 0 ,n +2 ( C C t ′ 0( n +2) ) + b 1 ( C C t ′ 1( n +1) ) when ( p, q ) = (0 , n + 1 ); C C t pn + d 0 p ( n + 1)( D t p ( n +1) ) for q = n, D t p ( n +1) deﬁned as a b ove . and argue exactly as befor e. Namely , in the spectra l sequence converging to H ∗ ( D t ∗∗ ) o ne has E 2 0 ,n +1 = 0. Moreov e r , the inclusion C t ∗∗ ֌ D t ∗∗ induces an isomo rphism on the E 1 1 ,n − 1 -term, as w ell as d 1 1 ,n ( E 1 1 ,n ). By Lemma 3, µ ˜ x survives to a no n-zero element in E 2 1 ,n − 1 -term of the sp ectral sequence for H ∗ ( C t ∗∗ ), which implies the same for the co rresp onding spectr a l seq uence conv erging to H ∗ ( D a ∗∗ ). How ever, a s in the algebraic case, the v anishing of the E 2 0 ,n +1 -term for this latter s p ectral sequence implies E 2 1 ,n − 1 = E 3 1 ,n − 1 . Thu s µ ˜ x survives to a no n-zero elemen t in this sp ectra l sequence, implying tha t it m ust also for E 3 1 ,n − 1 - term o f the sp ectr al sequence for H ∗ ( C t ∗∗ ). With this the pro o f of Theore m A is complete. 4.4 Related results Recall that for an y free simplicial gr oup Γ . ′ we have an equality H ∗ ( B Γ . ′ ; Z ) = H ∗ ( D ∗ (Γ . ′ , Z )), with the cohomolog y class µ ∗ ˜ x represented b y an integral co cycle in the (co)complex D ∗ (Γ . ′ , Z ). Fix a minimal simplicial abelia n group mo del 5 K ( Z , n − 1) . for the Eilenber g-MacLa ne space K ( Z , n − 1). If Γ . ′ is not only degreewis e free but als o ﬁnitely generated, then using the co complex D ∗ (Γ . ′ , Z ) as in [O1, Appe ndix ] we may realize a ny integral class [ c ] ∈ H n ( B Γ . ′ ; Z ) via a simplicial g roup homomorphis m φ c : Γ . ′ → K ( Z , n − 1) . which, by ﬁnite generation, is a (linear ly b ounded) homomorphism of simplicial ob jects in ( f .p.g rou p s ). The fundamen tal class ι n ∈ H n ( B K ( Z , n − 1 ) . ; C ) ∼ = C is represented by the genera tor o f H 1 ( B Z ; C ) = H H 1 ( C [ K ( Z , n − 1) n − 1 ]) < 1 > ֌ H H 1 ( C [ K ( Z , n − 1) n − 1 ]) = E 1 ,n − 1 1 ,H H which repr e sents a p er manent cycle in the Ho chschild cohomolo gy spectr al sequence converging to H H ∗ ( C [ K ( Z , n − 1 ) . ]). Tha t it is non- zero follows fro m the fact it pa ir s non-triv ially with the canonical dual generator ι n ∈ H 1 ( B Z ; C ) = H H 1 ( C [ K ( Z , n − 1) n − 1 ]) < 1 > ֌ H H 1 ( C [ K ( Z , n − 1) n − 1 ]) = E 1 ,H H 1 ,n − 1 representing a p erma nent non- z e ro cycle in the Ho chsc hild homolo gy sp ectral s equence converging to H H ∗ ( C [ K ( Z , n − 1) . ]). In particula r, for any integral cla ss x ∈ H n ( B π ; Z ), the complexiﬁed coho mology class µ ∗ ˜ x ∈ H n ( B Γ( ˜ x ) . ; C ) admits a representation by a simplicial group homomorphism φ ˜ x : Γ( ˜ x ) . → K ( Z , n − 1 ) . with φ ∗ e x ( ι n ) = µ ∗ e x . In order to deter mine the eﬀect of this simplicial homomo rphism in alg ebraic and top ologica l Hochschild coho mo logy , we will need Lemma 5. Fix n ≥ 2 . Set A . = C [ K ( Z , n − 1 ) . ] , B . = H 1 , ∞ L . ( K ( Z , n − 1 ) . ) , C. = H C M m ( K ( Z , n − 1 ) . ) . Each of these algeb ra s is augmente d over C ; denote by b A ., b B ., b C . the r esp e ctive c ompletions with r esp e ct t o the p owers of their augmentation ide al. The p assage to c ompletions yields a c ommuting diagr am of simplicial we ak e quivalenc es and isomorphims A . / / / / ≃   B . ≃   C . ≃   b A . b B . b C . Pr o of. As K ( Z , n − 1) . is a ﬁnitely gener ated free abelian group in e a ch degr ee, it is degreewis e of po lynomial growth. F o r any group with word length ( G, L st ) of polyno mia l growth, one has an eq ua lity H 1 , ∞ L st ( G ) = H C M m ( G ), resulting in the equalities B . = C . , b B . = b C . The e q uiv alence A . → b A . follo ws b y Curtis conv er g ence, a s the augment a tion idea l is 0- r educed. If G is a ﬁnitely gener ated free abe lia n group of r a nk N , I [ G ] := k er ( C [ G ] → C ) , I ( G ) := k er ( H 1 , ∞ L st ( G ) → C ), then one ha s shor t-exact sequences I [ G ] 2 ֌ I [ G ] ։ C N I ( G ) 2 ֌ I ( G ) ։ C N The ﬁrs t sequence is purely alg e braic, and can b e viewed in either the discre te or ﬁne to p ology . The exactness of the second seq uenc e I ( G ) 2 ֌ I ( G ) ։ C N implies I ( G ) 2 is o f ﬁnite co dimension in I ( G ), hence closed in the F r ´ echet to p o logy . Moreover, the ﬁnite dimensionality of C N implies that the quotient top ology on I ( G ) /I ( G ) 2 induced b y the F r ´ echet top olo gy on I ( G ) via the pro jection is the same as the ﬁne to po logy . The result is that not only is the pro cess of comp etion B . 7→ b B . well-deﬁned in the F r´ echet top ology (so tha t C ur tis co nv ergence applies here as well), but that o ne has an equality of simplicial top ologica l algebras b A . = b B . , where b o th ar e topo logized deg reewise by the ﬁne topolog y . 5 Here w e take K ( Z , n − 1) . to be the simplicial free abeli an group which is tri vi al i n dimensions j < ( n − 1), i s Z in dimension ( n − 1), and in dimension m > n is the free abelain gr oup on generating set indexed b y the iterated degeneracies from dimension n to di m ension m . Corollary 2. The maps A . ֌ B . = C . in t he pr evious L emma induc e isomorp hisms in b oth top olo gic al Ho chschild and top olo gic al cyclic (c o)homolo gy, wher e A . is top olo gize d de gr e ewise by the ﬁn e top olo gy, B . and C . de gr e ewise by the F r ´ echet top olo gy. Prop ositi on 7. L et µ ˜ x b e the inte gr al class deﬁne d as ab ove. Then it maps to a p ermanent cycle in E 1 ,H H ,t 1 ,n − 1 := H H t 1 ( H C M m (Γ( ˜ x ) n − 1 )) which survives to a non-zer o element E ∞ ,H H ,t 1 ,n − 1 , the E ∞ 1 ,n − 1 -term of t he Ho chschild homolo gy sp e ctr al se quenc e c onver ging to H H t ∗ ( H C M m (Γ( ˜ x ) . )) . Pr o of. The elemen t µ ˜ x comes from an in tegral per manent cycle in E 1 ,H H ,a 1 ,n − 1 = H H 1 ( C [Γ( ˜ x ) n − 1 ]) = H 1 ( B Γ( ˜ x ) n − 1 ; C ) = H H 1 ( C [Γ( ˜ x ) n − 1 ]) < 1 > , and so is cano nically repr esented b y a p ermanent cycle in the Ho chsc hild homology sp ectral seque nce co nv erging to H H t ∗ ( H C M m (Γ( ˜ x ) . )). Now let µ ∗ ˜ x be an integral dual clas s for µ ˜ x with r epresentativ e φ ˜ x : Γ ( ˜ x ) . → K ( Z , n − 1) . This homomorphism induces a contin uous homomorphism o f simplicial F r´ echet algebr as H C M m (Γ( ˜ x ) . ) → H C M m ( K ( Z , n − 1) . ) which, on the lev el of E 1 -terms, sends µ ′ ˜ x ∈ H H t 1 ( H C M m (Γ( ˜ x ) n − 1 )) to a non-zero scala r m ultiple of the fundamen ta l class in ι n ∈ H H t 1 ( H C M m ( K ( Z , n − 1) n − 1 )). By Lemma 5 and Cor. 2, the inclusio n of simplicial (topologica l) algebras C [ K ( Z , n − 1 ) . ] ֒ → H C M m ( K ( Z , n − 1) . ) induces an isomor phism H H ∗ ( C [ K ( Z , n − 1) . ]) ∼ = − → H H t ∗ ( H C M m ( K ( Z , n − 1) . )) implying ι n survives to a non-zero element in the E ∞ 1 ,n − 1 -term o f the sp ectral sequence conv erg ing to the group on the right (as it do e s so on the left). This, in turn, v eriﬁes the same pro p erty for the cy cle µ ′ ˜ x . The lifting pr ovided by the lo cal Chern character f ch ( e x ) C K ∗ allows for a diﬀerent formulation o f the ab ove results alo ng the lines o f [O2 ]. Theorem 5. L et 0 6 = x ∈ H n ( B π ; Q ) , n ≥ 1 b e an inte gr al class, with ˜ x , Γ( ˜ x ) . , and fundamental class [ µ ˜ x ] ∈ H n (Γ( ˜ x ) . ; Q ) as deﬁne d ab ove.Th en • F or al l ﬁnitely pr esente d π , 0 6 = x ∈ H n ( B π ; Q ) , n ≥ 1 , and choic e of inte gr al fundamental class [ µ ˜ x ] , the image of [ µ ˜ x ] under the c omp osition Φ n ( ˜ x ) : H n (Γ( ˜ x ) . ; Q ) ֌ H H n ( C [Γ( ˜ x ) . ]) → H H t n ( H C M m (Γ( ˜ x ) . )) is non-zer o. • If the image of [ µ ˜ x ] r emains non-zer o under t he c omp osition I t n ◦ Φ n ( ˜ x ) : H n (Γ( ˜ x ) . ; Q ) → H H t n ( H C M m (Γ( ˜ x ) . )) → H C t n ( H C M m (Γ( ˜ x ) . )) for al l ﬁnitely pr esente d π and 0 6 = x ∈ H n ( B π ; Q ) , n ≥ 1 , then S NC( π ) is t rue for al l ﬁnitely pr esente d gr oups π . 5 App endix 1: K -theory of ﬁne top ological algebras and the Chern c haracter ch C K T ∗ . W e review the deﬁnition o f topo lo gical K -theory for ﬁne topolog ical algebras, and prov e the retop o lo- gization theorem of Baum a s it is needed in o ur context. W e then show ho w it a pplies to yield the lifting of the Connes-Karo ubi Chern c harac ter us ed in s ection 4 o f this paper . Let V b e a vector spac e o ver C . The ﬁne top ology on V is by deﬁnition the inductive limit to p o logy on V with re s p e ct to the family of a ll one- dimensional (complex) subspaces, where each such subspa ce is taken with the standard top olog y . In o ther words, a subspace U ⊂ V is c losed iﬀ the intersection U ∩ W is closed in W for ev ery ﬁnite-dimensional subspace W of V . This deﬁnition is due to Grothendieck. If A is an algebra over C , then A f will denote A equipp ed with the ﬁne topolog y . If A is a top ologica l algebra ov er C , w e let S q ( A ) denote the a lgebra of C ∞ functions 6 from the standar d q -simplex to A . Then { [ q ] 7→ S q A } q ≥ 0 is a simplicial algebra , which we equip degreewise with the discrete top ology . The topolo gical K -theory space K ( A ) o f A is deﬁned as K ( A ) = | [ q ] 7→ B GL ( S q A ) | + (A.1) where “ + ” denotes Quillen’s plus constructio n (compare [T , Def. 1.1]). It is often the case that the top ologica l mo no id GL ( A ) is open in M ( A ) (for example, when A is a Banach algebra ). In this ca s e, the space in (A.1) is homotopy eq uiv alent to | [ q ] 7→ B GL ( S q A ) | . If GL ( A ) is a topolo gical group, then | [ q ] 7→ B GL ( S q A ) | is homo topy equiv a lent to B GL ( A ), in which case the spa ce in (A.1) is ho mo topy equiv alent to B GL ( A ). Ag a in, Banach algebras are a go o d exa mple of when these e q uiv alences hold. The higher top o lo gical K -gr oups of A are deﬁned by K t ∗ ( A ) = π ∗ ( K ( A )) for ∗ > 0. W e wr ite GL. ( A ) for the simplicial group { [ q ] 7→ GL ( S q A ) } q ≥ 0 . W e will w a nt to know the eﬀect of retop olo g ization on to p o logical K -theor y . In other words, if A is an algebra equipped with tw o (pos sibly distinct) top ologies, denoted by A T 1 and A T 2 with T 1 ﬁner than T 2 , then the identit y map on A induces a contin uous map A T 1 → A T 2 and so an induced map on top ologica l K -g roups K t ∗ ( A T 1 ) → K t ∗ ( A T 2 ) Theorem 6. (Ba u m) If A is a (not ne c essarily c omplete) norme d algebr a, T 1 , T 2 two top olo gies on A for which the id ent ity map on obje cts induc es c ontinuous maps A f → A T 1 → A T 2 → A (wher e A by itself me ans t hat A is t aken with the norm top olo gy), and GL ( A ) is op en in M ( A ) , then the map A T 1 → A T 2 induc es a we ak e quivalenc e GL ( A T 1 ) ≃ →→ GL ( A T 2 ) Pr o of. Let T denote a top olo gy ﬁner than the norm top olo gy . Set S ( A ) = M ( A ) − GL ( A ). T hen S ( A ) is closed in M ( A ) in the norm top olo g y . Let K be a ﬁnite simplicia l co mplex , and f : K → GL ( A ) a con tinuous map. As S ( A ) is closed, d = d ( f ( K ) , S ( A )) > 0 . Fix an ǫ with 0 < ǫ < d 4 . Via barycentric sub division, we can as sume the tr ia ngulation s a tisﬁes the pro pe rty that the imag e of any simplex in K under f lies within a ball of ra dius ǫ . Supp ose giv en an m - simplex ∆ m i of K , with vertices x 0 ,i , . . . , x m,i . Set g j,i = f ( x j,i ). Deﬁne P ( f ) : ∆ m i → GL ( A ) in terms of ba rycentric coor dinates by P ( f )( P m j =0 α j x j,i ) = P m j =0 α j g j,i . This map is contin uous in the ﬁne top ology , and the construction is 6 the C ∞ condition on the maps is neede d for the construction of the Chern c haracter. The topological K - groups r emain unc hanged i f w e replace C ∞ maps with con tinuous maps compatible on in ters ections of simplices. F o r x ∈ ∆ m i with barycentric co or dinates as a bove, and x k,i a vertex o f δ m i we hav e d ( P ( f )( x ) , S ( A )) > d ( P ( f )( x k,i ) , S ( A )) − d ( P ( f )( x ) , P ( f )( x k,i )) > d − ǫ > 0 Also d ( P ( f )( x ) , f ( x )) ≤ d ( P ( f )( m X j =0 α j x j,i ) , f ( x k,i )) + d ( f ( x k,i ) , f ( x )) < m X j =0 α j d ( P ( f )( x j,i ) , f ( x k,i )) + ǫ < 2 ǫ So this piece-w is e linear extensio n deﬁnes a c o ntin uous map P ( f ) : K → GL ( A T ), a nd the same distanc e argument implies that the linear homotopy on K × [0 , 1] giv en by h ( x, t ) = tP ( f )( x ) + (1 − t ) f ( x ) yields a co ntin uo us homotopy h : K × [0 , 1 ] → GL ( A ). Thus the induced map o n homotopy gr o ups π ∗ ( GL ( A T )) → π ∗ ( GL ( A )) is surjectiv e. In addition, we note that P ( P ( f )) = P ( f ). Ther efore, if K ′ ⊂ K and g = f | K ′ , then P ( g ) = P ( f ) | K ′ . Th us homotopies may b e lifted in a wa y compatible with the lifting of their restriction a t the tw o ends, implying the injectivit y of π ∗ ( GL ( A T )) → π ∗ ( GL ( A )). Remark 2. This r esult appl ies, for example, when the algebr a A is holomorphic al ly close d in a Banach algebr a, for in this c ase GL ( A ) is op en in M ( A ) in the (induc e d) norm top olo gy. In p articular, the r esult applies to the r api d de c ay su b algebr a H 1 , ∞ L ( π ) of ℓ 1 ( π ) . A technical p oint a rises here, for in o rder to apply Baum’s theorem as it stands to top olog ic al K -theor y , we would need the natural inclusion GL. ( A ) = { GL ( S q A ) } q ≥ 0 ֌ S.GL ( A ) = { S q GL ( A ) } q ≥ 0 (A.3) to be an equiv alence. When the map g 7→ g − 1 is contin uous in the top ology on GL ( A ), the ma p in (A.3) is an isomorphism of simplicial sets. The following v ariatio n provides the version o f Baum’s theorem that applies to our situation; the pro of is essen tially the same. Theorem 7. (Baum variation) If A is a norme d alge br a, T 1 , T 2 two t op olo gies on A for which the identity map on obj e cts induc es c ontinu ou s maps A f → A T 1 → A T 2 → A (wher e A by itself me ans t hat A is t aken with the norm top olo gy), and GL ( A ) is op en in M ( A ) , then the map A T 1 → A T 2 induc es a we ak e quivalenc e of simplici al gr oups GL. ( A T 1 ) = { GL ( S q A T 1 ) } q ≥ 0 ≃ − → GL . ( A T 2 ) = { GL ( S q A T 2 ) } q ≥ 0 Pr o of. As b efore, it suﬃces to prov e the theorem when T 2 is the norm topolo gy and T = T 1 any ﬁner top ology . Since GL. ( A ) is a K an complex, an elemen t of π n ( GL. ( A )) is represented by an element f = { f j k } ∈ GL ( S n A ) with ∂ i ( f j k ) = ∗ for all j, k and 0 ≤ i ≤ n . As above, let d = d ( im ( f ) , S ( A )) (mea sured via the embedding GL ( S n A ) ֌ S n GL ( A )) a nd ε = d 4 . The domain o f f j k is S n = ∆ n /∂ (∆ n ), and only ﬁnitely many of the f j k are non- constant. By bar ycentric subdivis ion we may replace f j k : S n → A by e f j k : K → A wher e | K | ∼ = S n , e f j k is the cor resp onding reﬁnement of f j k and for e a ch simplex ∆ i ∈ K , the image e f (∆ i ) = { e f j k (∆ i ) } j,k lies within a ball of radius ε . The rest follows as befor e in the proof of B aum’s theo rem. Namely , deﬁne P ( e f ) as the matrix { P ( e f j k ) } . This P is ag a in compatible on intersections of simplices, is an idemp otent op eratio n, and is contin uous in the ﬁne top ology . The distance condition guarantees that the image of P ( e f ) lies in the n -skeleton of GL. ( A ), and is homotopic to e f via a linear homotopy which also lifts by P . This implies the sur jectivity and injectivity of the induced map on homotopy gr oups. If GL ( A ) is op en in M ( A ), then I d : A T 1 → A T 2 induces an isomorphism on top o logical K 0 -groups, bec ause in this case K 0 is insensitive to the top olog y on A . B aum’s theorem may b e viewed as the analogue of this isomor phism for higher topolog ical K - theory . If A is a locally co nvex top olog ical algebra, and C C t ∗ ( A ) the completion o f the cy clic complex with resp ect to the topolog y on A , then the homology of this complex yields the unreduced top ologica l cyclic homology g r oups H C t ∗ ( A ), while forming the quo tient s k er ( d t ∗ ) /im ( d t ∗ +1 ) yield the reduced top olo gical cyclic homology g roups H C t ∗ ( A ) of A . The Connes-Ka roubi Chern character, cons tr ucted by Ka roubi in [K], is a graded ho momorphism of abelian g roups K t ∗ ( A ) ch C K ∗ − → H C t ∗ ( A ) Tillmann has s hown in [T] that K aroubi’s framework for co nstructing a Cher n character o n the topo logical K - g roups o f F r´ echet algebr a s car ries over to the ﬁne top olo gy . The adv antage to using this top olo gy in a ho mo logical setting is clea r, fo r in the ﬁne top olo gy the alg e br aic tensor pro duct is complete, and im ( d ∗ +1 ) is closed in k er ( d ∗ ). The r e s ult of [T ] then is a Chern character K t ∗ ( A f ) ch T ∗ − → H C ∗ ( A f ) where A f denotes A with the ﬁne to po logy and H C ∗ ( A f ) its algebr aic cyclic homolog y gr oups. Tillmann also show ed that the Chern character ch f ∗ was compa tible with the Connes-Kar oubi Chern character in the sense that the diag ram K t ∗ ( A f ) / / ch f   K t ∗ ( A ) ch C K ∗   H C ∗ ( A f ) / / H C t ∗ ( A ) commutes. Co mbinin g this result with the previous theor em yields Theorem 8. If A is a lo c al ly c onvex (not n e c essarily c omplete) norme d t op olo gic al al gebr a, then the Connes-Kar oubi Chern char acter for A factors as K t ∗ ( A ) ∼ = K t ∗ ( A f ) ch T ∗ − → H C t ∗ ( A f ) → H C t ∗ ( A ) → H C t ∗ ( A ) The co mpo sition of the ﬁrst tw o maps K t ∗ ( A ) → H C t ∗ ( A f ) is denoted ch C K T ∗ . The deﬁnition of the topolo gical K -theory space given here is appropria te for higher K -gro ups s ta rting in dimension 1, but not so for non- p o sitively g raded K -gr oups. This is not a concern for the a pplica tions to this pap er, beca use there is no place where nega tive groups are used, and only one argument for which K 0 ( − ) is needed: this is in the veriﬁcation of the isomorphism K t ∗ ( π 0 ( H C M L . (Γ . ) f )) ∼ = K t ∗ ( C ∗ m ( π )) , ∗ ≥ 1 (30) where Γ . is a degree wise ﬁnitely ge ne r ated free simplicial g roup with π 0 (Γ . ) = π . W e observe that • π 0 ( H C M L . (Γ . ) f ) = H C M L . (Γ . ) f 0 / ( H C M L . (Γ . ) 0 0 ) f , • H C M L . (Γ . ) 0 = H C M L 0 (Γ 0 ) is dens e a nd holomor phically closed in C ∗ m (Γ 0 ) • H C M L . (Γ . ) 0 0 is dense and holomor phically c lo sed in C ∗ m (Γ . ) 0 0 Prop ositi on 8. L et I ֌ A ։ A b e a short-exact se quenc e of Banach algebr as, B ⊂ A a dense su b al- gebr a of A close d under holomorphi c funct ional c alculus, and J ⊂ I an ide al in B which is dense and holomorph ic al ly close d in I 7 . L et B = B /J , top olo gize d with the ﬁne top olo gy. Then ther e is an exact se quenc e K t 1 ( B f ) → K 1 ( B ) → K t 0 ( J f ) → K t 0 ( B f ) → K t 0 ( B ) 7 W e do not assume here that J i s closed i n B in an y topol ogy coarser than the ﬁne top ology . Pr o of. F or a topolo gical algebra D (p ossibly without unit), K t 1 ( D ) as deﬁned ab ov e may b e rea lized a s the cokernel of the map K a 1 ( S 1 ( D )) d 1 → K a 1 ( S 0 ( D )) where d 1 := ( ∂ 1 ) ∗ − ( ∂ 0 ) ∗ . W e also obse rve that if D is dens e and holomor phically closed in a Banach algebra D ′ , then K a 0 ( D ) = K t 0 ( D ) for any topolo gy on D betw ee n the ﬁne top olog y a nd the norm topo logy induced by the inclusion in D ′ . Consider the commuting dia gram K a 1 ( S 1 ( B f )) d 1   / / K a 1 ( S 1 ( B )) d 1   ∂ / / K a 0 ( S 1 ( J f )) d 1 =0   / / K a 0 ( S 1 ( B f )) d 1 =0   / / / / K a 0 ( S 1 ( B )) d 1 =0   K a 1 ( S 0 ( B f ))     / / K a 1 ( S 0 ( B ))     ∂ / / K a 0 ( S 0 ( J f )) ∼ =   / / K a 0 ( S 0 ( B f )) ∼ =   / / / / K a 0 ( S 0 ( B )) ∼ =   K t 1 ( B f ) / / K t 1 ( B ) ∂ / / K t 0 ( J f ) / / K t 0 ( B f ) / / K t 0 ( B ) Here K a 0 ( S i ( B )) denotes the imag e of K a 0 ( S i ( B f )) in K a 0 ( S i ( B )), and similarly with “ a”repla ced by “ t” . The ﬁrst tw o rows arise from the long-exa ct sequence in algebr aic K -theory . A diagra m chase then s hows that the bottom sequence is exac t. Corollary 3. Under the same c onditions as the pr evious pr op osition, t her e is a long-ex act se quenc e · · · → K i ( J f ) → K t i ( B f ) → K i ( B ) → K t i − 1 ( J f ) → · · · → K t 0 ( B f ) → K t 0 ( B ) this long-exact se quenc e induc es an isomorphism K t i ( B ) ∼ = K t i ( A ) , i ≥ 1 Pr o of. The conditions on J and B imply that K ( J f ) ≃ hof ib ( K ( B f ) → K ( B )), y ie lding a long-exa c t sequence of higher topolo gical K -g roups, which spliced together with the se q uence from the preceding prop osition yields the indicated long-exact seq uence terminating in K t 0 ( B ). The compatible inclusions J ֒ → I , B ֒ → A then induce iso morphisms in topologic a l K -theor y , as well as a map o f lo ng-exact sequences o f topo lo gical K -gro ups. The result follows fro m the ﬁve lemma. References [B] D. Burg helea, The cyclic homolo gy of the gr oup rings , Comm. Math 60 (1985), 3 54 – 365. [CM] A. Connes and H. 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