Comparison of secondary invariants of algebraic K-theory

COMP ARISON OF SECOND AR Y INV ARIANTS OF A LGEB RAIC K -THEOR Y J. KAAD Departmen t of Mathematical Sciences, Univ ers it y of Cop enhagen Univ ersitets park en 5, DK-2100 Cop enhagen, Denmark Abstract In this pap er w e prov e that the multiplic ativ e character of A. Connes and M. K a roubi and the determinan t in v arian t of L. G. Bro wn, J. W. Helton and R. E. How e agree up to a canonical homomorphism. Contents 1. In t r o du ction 2 2. The determinan t inv arian t of an extension b y L 1 ( H ) 4 2.1. The ˇ Cec h complex of a surjectiv e simplicial map 4 2.2. The determinan t in v ariant 5 3. The multipl icativ e c haracter o f a finitely summable F redholm mo dule 6 3.1. The relative Chern character 6 3.2. The multiplic ativ e c haracter 8 4. Comparing the determinan t in v ariant and the m ultiplicativ e c haracter 9 4.1. F actorization through relativ e con tinuous cyclic homology 9 4.2. Determinan ts and the relativ e logarithm 11 4.3. Pro of of the main result 14 References 17 email: kaad@ math.k u.dk . 1 2 J. KAAD 1. Int r oduction In their pap er, [9], A. Connes and M. Karoubi define a m ultiplicativ e ch aracter on algebraic K -theory M F : K 2 p ( A ) → C / (2 π i ) p Z for eac h o dd unital 2 p -summable F redholm mo dule ( F , H ) o v er the unital C -algebra A . The construction uses the re lativ e K -groups of a un ital Banac h algebra and the re lativ e Chern c ha r a cter with v alues in con tin uous cyclic homology . In a different direction, L.G. Brow n has defined a determinan t inv ariant d ( X,ι ) : K 2 ( B ) → C ∗ for eac h exact seq uence of C -a lgebras X : 0 − − − → L 1 ( H ) i − − − → E π − − − → B − − − → 0 equipped with a unital algebra homomorphism ι : E → L ( H ) suc h that (1) ( ι ◦ i )( T ) = T for all T ∈ L 1 ( H ) The construction uses the F redholm determinan t homomorphism det : G → C ∗ e T 7→ e T r ( T ) Here G ⊆ GL ( L ( H )) denotes the op erators of determinan t class. See [3, 4]. In the case where B is commu tativ e, the determinan t in v arian t is related to the work of J. W. Helton and R. E. How e o n traces of comm utators, [12], via the iden tit y d ( X,ι ) { π ( e S ) , π ( e T ) } = e T r [ S,T ] S, T ∈ E Here { π ( e S ) , π ( e T ) } ∈ K 2 ( B ) denotes the Steinberg sym b ol. Notice also the app earance of the determinan t in v ariant in the pap ers of R. W. Carey and J. D. Pincus. See [5, 6]. The purp ose of this pap er is to sho w that the multipli cativ e c haracter and the determinan t in v arian t coincide up to a canonical homomorphism. In particular, to eac h o dd unital 2 - summable F redholm mo dule ( F , H ) o v er a unital C -algebra A w e a sso ciate a n extens ion o f a certain C - algebra B by the op erato r s of trace class (2) X F : 0 − − − → L 1 ( H ) i − − − → E π − − − → B − − − → 0 an injectiv e homomorphism ι : E → L ( H ) satisfying (1 ) , and a surjectiv e homomorphism R F : A → B The main result of the pap er can then b e expresse d as the commu tativit y of the diagram (3) K 2 ( A ) K 2 ( B ) C / (2 π i ) Z C ∗ ✲ ( R F ) ∗ ❄ M F ❄ d ( X F ,ι ) ✲ exp It follows that the m ultiplicativ e c haracter admits calculation in t he commu tativ e case, and that it factorizes through the second algebraic K -gro up of the Calcin a lg ebra C 1 ( H ) = L ( H ) / L 1 ( H ) . COMP ARISON OF SECONDAR Y INV ARI ANTS OF A LGEBRAIC K -THEOR Y 3 Using the theory of cen tral extensions w e can state the main result in a differen t fashion. In [9, Paragraph 5 ] it is show n that the univ ersal m ultiplicativ e c haracter M U : K 2 ( M 1 ) ∼ = H 2 ( E ( M 1 )) → C / (2 π i ) Z ∼ = C ∗ is induced by a cen tral extension (4) ϕ : 1 − − − → C ∗ − − − → Γ − − − → E ( M 1 ) − − − → 1 On the other hand, the univ ersal determinan t inv aria n t d : K 2 ( E ( C 1 ( H ))) ∼ = H 2 ( E ( C 1 ( H ))) → C ∗ could b e understo o d as the ho momorphism induc ed by the cen tra l extensi on (5) ψ : 1 − − − → C ∗ − − − → e Γ E ( q ) − − − → E ( C 1 ( H )) − − − → 1 where e Γ = GL ( L ( H )) / G 0 with G 0 defined as the normal subgroup consisting of op erators of determinan t one. The main theorem then translates in to the follo wing statemen t: The elemen ts in group cohomology with co efficien ts in C ∗ determined b y the cen tral extensions (4) and (5 ) coincide up to a homomorphism E ( R ) ∗ ( ψ ) = ϕ E ( R ) ∗ : H 2 ( E ( C 1 ( H )) , C ∗ ) → H 2 ( E ( M 1 ) , C ∗ ) W e th us obtain refinemen ts o f results giv en in [27]. Here w e hav e applied the univ ersal co efficien t Theorem together with the p erfectness of the elemen tary matrices. The presen t paper is organized as follows . In the first tw o sections w e recall the construction of the determinan t inv ariant and the mul- tiplicativ e c haracter. Our definition of the determinan t in v ariant differs sligh tly f r o m the one giv en in [3, 4 ]. W e use a description of the relativ e homology groups of the surjectiv e homo- morphism E ( q ) : E ( L ( H )) → E ( C 1 ( H )) in terms of an algebraic analogue of t he ˇ Cec h complex of an op en co v er of a top olo gical space. See [13 , 17]. A com binatorial argumen t can b e giv en, pro ving that the t w o definitions yield the same map. The alternativ e a pproa c h seems to b e essen t ia l f or the working of our main pro of . In the last section we giv e a pro of of the main result. There are tw o main ingredien ts. The first ingredien t is a factorization result for the homomorphism τ 1 : H C 1 ( M 1 ) → C ( x 0 , x 1 ) 7→ − 1 4 T r ( F U [ F U , x 0 ][ F U , x 1 ]) through a relativ e contin uous cyclic homology g r o up. This result is essen tially con tained in the pro of of [9, Theorem 5 . 6 ]. The second ingredien t is a n application of the (well kno wn) description of the F redholm determinan t using a logarithm. T o b e precise, for each smo oth map σ : [0 , 1] → G σ (0) = 1 w e can calculate the F redholm determinan t of the endp oin t σ (1) ∈ G , using the in tegral T r ( Z 1 0 dσ dt · σ − 1 dt ) ∈ C 4 J. KAAD W e use this observ ation in a relativ e setting, in v o king the algebraic ˇ Cec h complex a second time. These considerations lead directly to a pro of of the main result. W e would lik e to end this introduction b y mentionin g that ev en though the determinan t in- v arian t has no ob vious generalization to the higher K - groups, w e are still a ble to calculate the m ultiplicativ e c haracter in this setting, o bta ining higher dimensional analogues of the iden tity (exp ◦M F ) { e T , e S } = exp( T r [ P T P , P S P ]) This will b e the sub ject of a forthcoming pap er. A c kno wledgemen ts: I would lik e to thank Ryszard Nest fo r his con tin uous supp ort and man y helpful commen ts. 2. The determinant inv ariant of an extension by L 1 ( H ) The main sub ject o f this section is the definition of the determinan t in v arian t. Our definition relies on conside rations in relativ e homology , whic h will find application throughout the pap er. W e will therefore start by a form ulation of these ideas in a general contex t b efore passing on to t he definition of the determinan t in v ar ia n t. 2.1. The ˇ Cec h complex of a surjectiv e simplicial map. Let X and Y b e simplicial sets and sup p ose that w e hav e a surje ctive simplicial map f : X → Y . W e let Ker ∗ ( f ) denote the k ernel ch ain complex of the asso ciated c hain map f : Z [ X ∗ ] → Z [ Y ∗ ] . F urthermore w e let ˇ C ( f ) denote the bicomplex with ˇ C nm ( f ) = Z [ X n × Y n . . . × Y n X n | {z } m +2 ] The generators are th us ( m + 2) -tuples of elemen ts in X n , ( x 1 , . . . , x m +2 ) , whic h coincide in Y n , f ( x 1 ) = . . . = f ( x m +2 ) . The differen tials are give n b y d : ˇ C nm ( f ) → ˇ C ( n − 1) m ( f ) d ( x 1 , . . . , x m +2 ) = P n i =0 ( − 1) i ( d i ( x 1 ) , . . . , d i ( x m +2 )) and δ : ˇ C nm ( f ) → ˇ C n ( m − 1) ( f ) δ ( x 1 , . . . , x m +2 ) = P m +2 j =1 ( − 1) j +1 ( x 1 , . . . , ˇ x j , . . . , x m +2 ) Here the ˇ · signifies t ha t the term has to b e omitted. W e then hav e a long exact sequen ce of ab elian groups Z [ X n ] Z [ Y n ] ˇ C n 0 ( f ) ˇ C n 1 ( f ) ˇ C n 2 ( f ) . . . ✛ f ✛ ε ✛ δ ✛ δ ✛ δ for eac h n ∈ N 0 . Here the ch ain map ε : ˇ C ∗ 0 ( f ) → Z [ Y ∗ ] is giv en b y ε : ( x 1 , x 2 ) 7→ x 1 − x 2 . In particular, letting Cok er ∗ ( δ f ) denote the cok ernel c hain complex of the chain map δ : ˇ C ∗ 1 ( f ) → ˇ C ∗ 0 ( f ) w e ha ve the result COMP ARISON OF SECONDAR Y INV ARI ANTS OF A LGEBRAIC K -THEOR Y 5 Theorem 2.1. [17] F or e ach n ∈ N 0 we ha v e the isomorphisms H n ( f ) ∼ = H n ( Ker ( f )) ∼ = H n ( Cok er ( δ f )) ∼ = H n ( ˇ C ( f )) Her e H n ( f ) denotes the r elative homolo gy o f the simpli c ial map f : X → Y 2.2. The determinan t in v arian t. Let C 1 ( H ) = L ( H ) / L 1 ( H ) denote the Calkin a lgebra. W e then hav e the quotien t map q : L ( H ) → C 1 ( H ) and an induced surjectiv e group homomorphism E ( q ) : E ( L ( H )) → E ( C 1 ( H )) . Here E ( A ) denotes the elemen ta ry matrices of a unital ring A . W e define the determinan t homomorphism on the ab elian group ˇ C 10 ( E ( q )) b y det : Z [ E ( L ( H )) × E ( C 1 ( H )) E ( L ( H ))] → C ∗ ( g 1 , g 2 ) 7→ det ( g 1 g − 1 2 ) Note that g 1 g − 1 2 is of determinan t class since E ( q )( g 1 ) = E ( q )( g 2 ) . Lemma 2.2. T he determin ant det : Z [ E ( L ( H )) × E ( C 1 ( H )) E ( L ( H ))] → C ∗ induc es a map on ho molo gy det : H 1 ( Cok er ( δ E ( q ) )) → C ∗ Pr o of. This is a matter of c hec king the equalit y det ( g 1 g − 1 2 ) · de t ( g 2 g − 1 3 ) = det ( g 1 g − 1 3 ) for E ( q )( g 1 ) = E ( q )( g 2 ) = E ( q )( g 3 ) and the equalit y det ( g 1 g − 1 2 ) det ( h 1 h − 1 2 ) = det ( g 1 h 1 h − 1 2 g − 1 2 ) for E ( q )( g 1 ) = E ( q )( g 2 ) and E ( q )( h 1 ) = E ( q )( h 2 ) . But they follow easily from w ell kno wn prop erties of the F redholm determinan t, [28, 31].  By Theorem 2.1, the surjectivit y o f E ( q ) : E ( L ( H )) → E ( C 1 ( H )) implies that ε : H 1 ( Cok er ( δ E ( q ) )) → H 1 ( E ( q )) is an isomorphi sm. F urthermore, by [1 8, Prop osition 1 . 2 . 1 ] w e can identify the ab elian groups K 2 ( C 1 ( H )) a nd H 2 ( E ( C 1 ( H ))) . W e can thus define the univ ersal determinan t in v ariant as t he comp osition d = det ◦ ∂ : K 2 ( C 1 ( H )) ∼ = H 2 ( E ( C 1 ( H ))) → C ∗ Here ∂ : H 2 ( E ( C 1 ( H ))) → H 1 ( E ( q )) is the b oundary map a sso ciated with the group homomor- phism E ( q ) : E ( L ( H )) → E ( C 1 ( H )) . No w, suppo se that w e hav e an exact sequence of C -algebras X : 0 − − − → L 1 ( H ) i − − − → E π − − − → B − − − → 1 equipped with a unital algebra homomorphism ι : E → L ( H ) suc h that ( ι ◦ i )( T ) = T for all T ∈ L 1 ( H ) W e then ha v e an induced homomorphism ι : B → C 1 ( H ) whic h b y functorialit y of algebraic K -theory yields a homomorphism ι ∗ : K 2 ( B ) → K 2 ( C 1 ( H )) 6 J. KAAD W e define the determinan t in v ar ia nt of the pair ( X , ι ) as the comp osition d ( X,ι ) = d ◦ ι ∗ : K 2 ( B ) → C ∗ 3. The mul tiplica tive cha ra cter of a finitel y summable Fredholm module In this section w e recall the construction of the m ultiplicativ e c ha ra cter asso ciated with an o dd, finitely summable F redholm mo dule. T o this end w e will describe the relativ e Chern c ha r a cter and the relativ e K -groups in detail. The references for this section are [9, 16] and [2 9]. Note that the cyclic homology and Ho c hsc hild homolog y groups encounte red in this section, and throughout the pap er a r e the non-Hausdorff ve rsions of c on tinuous cyclic homology a nd c ontinuous Ho c hsc hild homology groups. 3.1. The relativ e Chern c haracter. Let A b e a unital Banac h algebra. W e let ∆ n = { t ∈ [0 , 1] n | n X i =1 t i ≤ 1 } denote the standard n -simplex. The vertic es will b e denoted by 0 , . . . , n ∈ ∆ n . Let R ( A ) denote the simplicial set with R ( A ) n = { σ ∈ GL ( C ∞ (∆ n , A )) | σ ( 0 ) = 1 } in degree n ∈ N 0 and with face op erators and degeneracy o p erators defined by d i ( σ )( t 1 , . . . , t n − 1 ) =  σ (1 − P n − 1 j =1 t j , t 1 , . . . , t n − 1 ) · σ ( 1 ) − 1 for j = 0 σ ( t 1 , . . . , t i − 1 , 0 , t i , . . . , t n − 1 ) for j ∈ { 1 , . . . , n } s j ( σ )( t 1 , . . . , t n +1 ) =  σ ( t 2 , . . . , t n +1 ) for j = 0 σ ( t 1 , . . . , t i − 1 , t i + t i +1 , . . . , t n +1 ) for j ∈ { 1 , . . . , n } Remark the extra f a ctor σ ( 1 ) − 1 in the express ion f or d 0 . Let | R ( A ) | denote the geometric realization of the simplicial set R ( A ) . The fundamen tal group is then given by π 1 ( | R ( A ) | ) = R ( A ) 1 / ∼ where ∼ denotes the equiv alence relation given by smo oth homotopies with fixed endp oin ts. See [22]. Since the commutator subgroup is p erfect and normal, w e can apply the plus construction of D. Quillen, obtaining the p oin ted C W -complex | R ( A ) | + . See [24]. Definition 3.1. [16] By the relativ e K - theory of A we wil l understand the homotopy gr oups of | R ( A ) | + thus b y de fi nition K rel n ( A ) = π n ( | R ( A ) | + ) for e ach n ∈ N 0 . COMP ARISON OF SECONDAR Y INV ARI ANTS OF A LGEBRAIC K -THEOR Y 7 By [16] the relative K -g ro ups fit in a long exact sequence (6) . . . K top n +1 ( A ) K rel n ( A ) K n ( A ) . . . K n − 1 ( A ) K rel n − 1 ( A ) K top n ( A ) ✲ i ✲ v ✲ θ ❄ i ✛ i ✛ θ ✛ v whic h terminates at K top 1 ( A ) . Here θ : K rel n ( A ) → K n ( A ) is induced b y the simplicial map θ : R n ( A ) → GL ( A ) n θ ( σ ) = ( σ ( 0 ) σ ( 1 ) − 1 , . . . , σ ( n − 1 ) σ ( n ) − 1 ) F urthermore, in [9] it is show n that the long exact sequence (6) is related to t he con tin uous S B I -sequence in homology by means of Chern c haracters . . . K top n +1 ( A ) K rel n ( A ) K n ( A ) K top n ( A ) . . . . . . H C n +1 ( A ) H C n − 1 ( A ) H H n ( A ) H C n ( A ) . . . ✲ i ✲ v ❄ c h top n +1 ✲ θ ❄ ( − 1) n ( n − 1)! c h rel n ✲ i ❄ 1 n ! D n ✲ v ❄ c h top n ✲ I ✲ S ✲ 1 n B ✲ I ✲ S W e will giv e a precise description of the r elative Chern c haracter c h rel n : K rel n ( A ) → H C n − 1 ( A ) as the comp osition of three homomorphisms. The first one is the Hurewicz homomorphism h n : K rel n ( A ) → H n ( | R ( A ) | + ) ∼ = H n ( R ( A )) The second one is the loga rithm L : H n ( R ( A )) → lim m →∞ H C n − 1 ( M m ( A )) whic h is induced b y (7) L : σ 7→ 1 n X s ∈ Σ n sgn ( s ) Z ∆ n ∂ σ ∂ t s (1) · σ − 1 ⊗ . . . ⊗ ∂ σ ∂ t s ( n ) · σ − 1 dt 1 ∧ . . . ∧ dt n for eac h smo o th map σ : ∆ n → GL m ( A ) . The last one is the generalized trace on con tin uous cyclic homolo g y TR : lim m →∞ H C n − 1 ( M m ( A )) → H C n − 1 ( A ) Definition 3.2. [9, 16] By the relativ e Chern c hara cter c h rel n : K rel n ( A ) → H C n − 1 ( A ) we wi l l understand the c omp osition (8) c h rel n = TR ◦ L ◦ h n of the Hur ewicz homomorphism , the lo garithm and the gener alize d tr ac e . 8 J. KAAD 3.2. The m ultiplicativ e c haracter. Let H be a separable Hilb ert space o f infinite dimension. F or eac h p ∈ N let M 2 p − 1 denote the unital C -subalgebra of L ( H ⊕ H ) consisting of op erators of the f orm  x 11 x 12 x 21 x 22  ∈ L ( H ⊕ H ) with x 12 , x 21 ∈ L 2 p ( H ) in the 2 p th Sc hatten ideal. The C -algebra M 2 p − 1 b ecomes a unital Banac h algebra when eq uipp ed with the norm k x k = k x k ∞ + k [ F U , x ] k 2 p F U =  1 0 0 − 1  Here the norms k · k ∞ and k · k 2 p are the op erator norm and the norm on the 2 p th Sc hatten ideal respectiv ely . F or details on the Sc hatten ideals w e refer to [28]. The contin uous linear map τ 2 p − 1 : C λ 2 p − 1 ( M 2 p − 1 ) → C ( x 0 , . . . , x 2 p − 1 ) 7→ − 1 2 2 p ( p − 1)! T r ( F U [ F U , x 0 ] · . . . · [ F U , x 2 p − 1 ]) determines a con tinuous cyclic co cycle and consequen tly a homomorphism τ 2 p − 1 : H C 2 p − 1 ( M 2 p − 1 ) → C F or details w e refer to [8]. Pre-comp osition with the relative Chern characte r th us yields a homomorphism τ 2 p − 1 ◦ c h rel 2 p : K rel 2 p ( M 2 p − 1 ) → C In [9] it is shown that K 2 p ( M 2 p − 1 ) is the cok ernel of v : K top 2 p +1 ( M 2 p − 1 ) → K rel 2 p ( M 2 p − 1 ) from the exact seq uence (6). F urthermore it is sho wn that the image of the homomorphism τ 2 p − 1 ◦ c h rel 2 p ◦ v : K top 2 p +1 ( M 2 p − 1 ) → C equals the additiv e subgroup (2 π i ) p Z of C . By consequen ce w e g et a homomorphism M U : K 2 p ( M 2 p − 1 ) ∼ = Cok er ( v ) → C / (2 π i ) p Z This is the o dd unive rsal multipli cativ e c haracter. No w, to eac h o dd unital 2 p - summable F redholm ( F , H ) o ver a unital C -algebra A w e can asso ciate a unital algebra homomorphism ρ F : A → M 2 p − 1 whic h b y functorialit y of algebraic K -theory yields a homomorphism ( ρ F ) ∗ : K 2 p ( A ) → K 2 p ( M 2 p − 1 ) The m ultiplicativ e ch aracter of t he F r edholm mo dule ( F , H ) o v er A is then defined as the comp osition M F = M ◦ ( ρ F ) ∗ : K 2 p ( A ) → C / (2 π i ) p Z COMP ARISON OF SECONDAR Y INV ARI ANTS OF A LGEBRAIC K -THEOR Y 9 4. Comp aring the determinant inv ariant and the mul t iplica tive chara cter 4.1. F actorization through r elativ e contin uous c yclic homology. In this section w e sho w that the homomorphism τ 1 : H C 1 ( M 1 ) → C factorizes thro ug h a relativ e con tin uous cyclic homology group, asso ciated with an extension of M 1 b y the op erators of trace class. This result can b e found in condensed form in the pro of of [9, Theorem 5 . 6 ]. Let H b e a separable Hilb ert space and let M 1 b e the Banac h algebra considered in Section 3. Let P b e the pro jection P =  1 0 0 0  ∈ L ( H ⊕ H ) and let q : L ( H ) → L ( H ) / L 1 ( H ) = C 1 ( H ) b e the quotien t map. W e then ha ve a unital algebra homomorphism R : M 1 → C 1 ( H ) R : x 7→ q ( P xP ) Let T 1 ⊆ L ( H ) × M 1 b e the unital C -subalgebra suc h that ( S, x ) ∈ T 1 precisely when S − P xP ∈ L 1 ( H ) . The diagram (9) T 1 M 1 L ( H ) L ( H ) / L 1 ( H ) ✲ π 2 ❄ π 1 ❄ R ✲ q is th us a comm utativ e diagram o f unital C -algebras. Here π 1 and π 2 are the pro jections giv en b y π 1 ( S, x ) = S and π 2 ( S, x ) = x . The C - algebra T 1 b ecomes a unital Banac h algebra when equipp ed with the norm k ( S, x ) k = k P xP − S k 1 + k x k ∞ + k [2 P − 1 , x ] k 2 It fits in the short exact sequence of Banach algebras (10) 0 − − − → L 1 ( H ) i − − − → T 1 π 2 − − − → M 1 − − − → 0 whic h has a con tin uous linear section s : M 1 → T 1 x 7→ ( P xP , x ) The existence of the con tin uous linear right in v erse to π 2 yields a n isomorphism H C ∗ ( π 2 ) ∼ = H C ∗ ( Ker ( π 2 )) b et w een the relative con tinuous cyclic homology of π 2 and the homology of the k ernel complex asso ciated with the chain map ( π 2 ) ∗ : C λ ∗ ( T 1 ) → C λ ∗ ( M 1 ) In particular, eac h ele men t in H C 0 ( π 2 ) can b e represen ted b y an op erat o r of trace class ( S, 0) ∈ T 1 S ∈ L 1 ( H ) 10 J. KAAD Lemma 4.1. T he tr ac e T r : K er ( π 2 ) → C ( S, 0) 7→ T r ( S ) induc es a map on r elative c ontinuous cyclic homolo gy T : H C 0 ( π 2 ) → C Pr o of. Let α : C λ 1 ( M 1 ) → L ( H ) denote the map induced by α : ( x 0 , x 1 ) 7→ [ P x 0 P , P x 1 P ] Let β : C λ 1 ( T 1 ) → L 1 ( H ) b e the map induced b y β : (( S 0 , x 0 ) , ( S 1 , x 1 )) 7→ [ S 0 , S 1 ] − [ P x 0 P , P x 1 P ] W e then ha v e T r ◦ β = 0 F urthermore w e can express the comp osition of the Ho c hsc hild b oundary b : C λ 1 ( T 1 ) → C λ 0 ( T 1 ) = T 1 with t he pro jection π 1 : T 1 → L ( H ) as the sum π 1 ◦ b = β + α ◦ ( π 2 ) ∗ It follo ws that t he map T r ◦ b : Ker 1 ( π 2 ) → C v anishes, which in turn implies the desired result.  W e can no w pro v e that τ 1 : H C 1 ( M 1 ) → C factor izes through the relativ e con tin uous cyclic homology group H C 0 ( π 2 ) . Theorem 4.2. The cha r acter τ 1 : H C 1 ( M 1 ) → C factorizes as τ 1 = − T ◦ ∂ Her e ∂ : H C 1 ( M 1 ) → H C 0 ( π 2 ) is the b ound a ry map asso c iate d with the homomorph ism π 2 : T 1 → M 1 . Pr o of. W e start b y noting that the comp osition − T r ◦ ( b ◦ s ∗ − s ∗ ◦ b ) : C λ 1 ( M 1 ) → C coincides with the map τ 1 : C λ 1 ( M 1 ) → C Here s ∗ : C λ 1 ( M 1 ) → C λ 1 ( T 1 ) denotes the map induced b y t he con t inuous linear righ t inv erse s : M 1 → T 1 of π 2 : T 1 → M 1 . Since the b oundary map ∂ : H C 1 ( M 1 ) → H C 0 ( π 2 ) ∼ = H C 0 ( Ker ( π 2 )) is induced by t he map b ◦ s ∗ : Z λ 1 ( M 1 ) → Ker ( π 2 ) w e get the desired result.  COMP ARISON OF SECONDAR Y INV ARI ANTS OF A LGEBRAIC K -THEOR Y 11 As an immediate consequence of Theorem 4.2 w e get the follo wing factorization result for the comp osition of the r elative Chern c haracter and the homomorphism τ 1 : H C 1 ( M 1 ) → C . Corollary 4.3. The c omp osition τ 1 ◦ ch rel 2 : K rel 2 ( M 1 ) → C factorizes as τ 1 ◦ ch rel 2 = − T ◦ TR ◦ L ◦ ∂ ◦ h 2 Her e the map s TR : lim m →∞ H C 1 ( M m ( π 2 )) → H C 1 ( π 2 ) and L : H 2 ( R ( π 2 )) → lim m →∞ H C 1 ( M m ( π 2 )) ar e r elative ve rs i o ns of the g e ner alize d tr ac e a n d the lo garithm. Th e map ∂ : H 2 ( R ( T 1 )) → H 1 ( R ( π 2 )) is the b oundary map a sso ciate d with the simplicial map R ( π 2 ) : R ( T 1 ) → R ( M 1 ) . 4.2. Det er minan ts and the relative logarithm. In this part of the pap er w e start relating the mu ltiplicativ e c haracter with the F redholm determinan t. The main result of this section is th us Lemma 4.6, where w e show that the comp osition of the relativ e logar ithm L : H 1 ( R ( π 2 )) → lim m →∞ H C 0 ( M m ( π 2 )) with t he exp onential of the trace exp ◦ ( − T ◦ TR ) : lim m →∞ H C 0 ( M m ( π 2 )) → C ∗ is a F redholm determinan t. Definition 4.4. L et G den o te the op er ators of determinant class. By a smo oth map σ : ∆ n → G we wi l l understand an element in the gr oup σ ∈ GL  C ∞ (∆ n , L ( H ))  such that σ − 1 ∈ M ∞  C ∞ (∆ n , L 1 ( H ))  Define the map ˜ L : Z [ R ( T 1 ) 1 × R ( M 1 ) 1 R ( T 1 ) 1 ] = ˇ C 10 ( R ( π 2 )) → C b y t he a ssignmen t ˜ L : ( σ 1 , σ 2 ) 7→ − T r Z 1 0 d ( σ 1 σ − 1 2 ) dt · σ 2 σ − 1 1 dt Here the tra ce T r : M ∞ ( L 1 ( H )) → C is induced by T r : x 7→ P m i =1 T r ( x ii ) for all x ∈ M m  L 1 ( H )  12 J. KAAD Lemma 4.5. T he map ˜ L : Z [ R ( T 1 ) 1 × R ( M 1 ) 1 R ( T 1 ) 1 ] → C p asses to a map on homolo gy ˜ L : H 1 ( Cok er ( δ R ( π 2 ) )) → C making the diagr am (11) H 1 ( R ( π 2 )) H 1 ( Cok er ( δ R ( π 2 ) )) C ❄ − T ◦ TR ◦ L ✛ ε ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ˜ L c ommute. Pr o of. W e show that ˜ L : Z [ R ( T 1 ) 1 × R ( M 1 ) 1 R ( T 1 ) 1 ] → C agrees with the map − T ◦ TR ◦ L ◦ ε : Z [ R ( T 1 ) 1 × R ( M 1 ) 1 R ( T 1 ) 1 ] → C See Section 2.1. Here TR ◦ L : Ker 1 ( R ( π 2 )) → Ker ( π 2 ) is induced b y the restriction of TR ◦ L : Z [ R ( T 1 ) 1 ] → T 1 to t he kern el of R ( π 2 ) : Z [ R ( T 1 ) 1 ] → Z [ R ( M 1 ) 1 ] . This essen tially amoun ts to a pro of of the equalit y ˜ L ( σ 1 , σ 2 ) = T r  Z 1 0 dσ 2 dt · σ − 1 2 dt − Z 1 0 dσ 1 dt · σ − 1 1 dt  for all ( σ 1 , σ 2 ) ∈ R ( T 1 ) 1 × R ( M 1 ) 1 R ( T 1 ) 1 . No w, since the trace T r : M m ( L 1 ( H )) → C is con tinuous and linear we ha v e ˜ L ( σ 1 , σ 2 ) = − T r Z 1 0 d ( σ 1 σ − 1 2 ) dt · σ 2 σ − 1 1 dt = − Z 1 0 T r  d ( σ 1 σ − 1 2 ) dt · σ 2 σ − 1 1  dt Using the Leibnitz rule for the differen tia l op erator d dt w e get d ( σ 1 σ − 1 2 ) dt = dσ 1 dt · σ − 1 2 − σ 1 σ − 1 2 · dσ 2 dt · σ − 1 2 COMP ARISON OF SECONDAR Y INV ARI ANTS OF A LGEBRAIC K -THEOR Y 13 Defining α ∈ M ∞  C ∞ (∆ 1 , L 1 ( H ))  b y α + 1 = σ 2 σ − 1 1 w e hav e − T r  d ( σ 1 σ − 1 2 ) dt · σ 2 σ − 1 1  = − T r  dσ 1 dt · σ − 1 1 − ( α + 1) − 1 dσ 2 dt · σ − 1 2 ( α + 1)  = T r  ( α + 1) − 1 dσ 2 dt · σ − 1 2 α  + T r  ( α + 1) − 1 dσ 2 dt · σ − 1 2 − dσ 1 dt · σ − 1 1  = T r  dσ 2 dt · σ − 1 2 − dσ 1 dt · σ − 1 1  pro ving the desired r esult.  In the next Lemma we will sho w that the comp osition of the map ˜ L : H 1 ( Cok er ( δ R ( π 2 ) )) → C with t he exp onential exp : C → C ∗ can b e expresse d using a F redholm determinan t. By [9, Prop osition 5 . 4 ] the first alg ebraic K -g roup of the Banac h algebra T 1 v anishes. In particular, it fo llo ws from the exact sequence (10) that the groups E ( M 1 ) and GL 0 ( M 1 ) coincide. Here GL 0 ( M 1 ) denotes the connected comp o nen t of the iden tit y . W e therefore ha v e a commutativ e diagram of simplicial maps R n ( T 1 ) E ( T 1 ) n R n ( M 1 ) E ( M 1 ) n ✲ θ ❄ R ( π 2 ) ❄ E ( π 2 ) ✲ θ and b y conseque nce an induced map θ : H 1 ( Cok er ( δ R ( π 2 ) )) → H 1 ( Cok er ( δ E ( π 2 ) )) F urthermore w e let D ∗ : H 1 ( Cok er ( δ E ( π 2 ) )) → H 1 ( Cok er ( δ E ( q ) )) denote the map induced by the comm utativ e diagra m E ( T 1 ) E ( L ( H )) E ( M 1 ) E ( C 1 ( H )) ✲ E ( π 1 ) ❄ E ( π 2 ) ❄ E ( q ) ✲ E ( R ) of group homomorphisms. 14 J. KAAD Lemma 4.6. T he diag r am (12) H 1 ( Cok er ( δ R ( π 2 ) )) H 1 ( Cok er ( δ E ( q ) )) C C ∗ ❄ ˜ L ✲ D ∗ ◦ θ ❄ det ✲ exp is c ommutative. Pr o of. W e show that for eac h generator ( σ 1 , σ 2 ) ∈ R ( T 1 ) 1 × R ( M 1 ) 1 R ( T 1 ) 1 w e ha ve (exp ◦ ˜ L )( σ 1 , σ 2 ) = det  E ( π 1 )( σ 1 ( 1 ) − 1 ) · E ( π 1 )( σ 2 ( 1 ))  Indeed the smooth map σ 1 σ − 1 2 : ∆ 1 → G L ( T 1 ) is of the form σ 1 σ − 1 2 = ( α, 1) with α : ∆ 1 → G b eing a smo oth map in the sense of De finition 4.4. It follow s that (exp ◦ ˜ L )( σ 1 , σ 2 ) = exp  − T r Z 1 0 dα dt · α − 1 dt  But this is precisely t he F redholm determinan t of the endp oint det ( α ( 1 ) − 1 ) = det  E ( π 1 )( σ 2 ( 1 )) · E ( π 1 )( σ 1 ( 1 ) − 1 )  pro ving the desired r esult.  4.3. P r o of of the main result. In this section we com bine the results of Lemma 4.2 and Lemma 4.6, in order to obtain a pro of of our main theorem: the determinan t in v a rian t equ als the m ultiplicativ e c haracter up to a canonical homomorphism on algebraic K -theory . W e start b y making a couple of useful o bserv ations. Let θ : H 2 ( R ( M 1 )) → H 2 ( E ( M 1 )) denote the homomorphism induced by the simplicial map θ : R n ( M 1 ) → E ( M 1 ) n The comp osition θ ◦ h 2 : K rel 2 ( M 1 ) → H 2 ( E ( M 1 )) then corr esp onds to the map θ : K rel 2 ( M 1 ) → K 2 ( M 1 ) under the identifi cation K 2 ( M 1 ) ∼ = H 2 ( E ( M 1 )) . F urthermore, the simplicial map R ( π 2 ) : R ( T 1 ) → R ( M 1 ) is surjectiv e in eac h degree. This follo ws from [1, Prop osition 3 . 4 . 3 ] b y the surjectivit y o f the map M m ( π 2 ) : M m ( C ∞ (∆ n , T 1 )) → M m ( C ∞ (∆ n , M 1 )) COMP ARISON OF SECONDAR Y INV ARI ANTS OF A LGEBRAIC K -THEOR Y 15 and since eac h ele men t σ ∈ R ( M 1 ) n can b e represen ted by a n elemen t σ ∈ GL 0  M m ( C ∞ (∆ n , M 1 ))  By Theorem 2.1 the map ε : H ∗ ( Cok er ( δ R ( π 2 ) )) → H ∗ ( R ( π 2 )) th us b ecomes an isomorphism. W e are no w ready to pro v e the main result of the pap er. Theorem 4.7. The c omp osition o f the universal m ultiplic ative c har acter M U : K 2 ( M 1 ) → C / (2 π i ) Z with the exp onential function exp : C / (2 π i ) Z → C ∗ c oincides with the c omp os ition of the ho momorphism on algebr aic K -the ory R ∗ : K 2 ( M 1 ) → K 2 ( C 1 ( H )) induc e d by the unital algebr a h o momorphism R : M 1 → C 1 ( H ) x 7→ q ( P xP ) and the universal determinant inva riant d : K 2 ( C 1 ( H )) → C ∗ That is exp ◦M U = d ◦ R ∗ Pr o of. F rom the long exact sequence (6) w e see that θ : K rel 2 ( M 1 ) → K 2 ( M 1 ) implemen ts the isomorphism Cok er ( v ) ∼ = K 2 ( M 1 ) By construction of the multiplic ativ e c haracter it is therefore enough to pro v e the equalit y exp ◦ τ 1 ◦ ch rel 2 = d ◦ R ∗ ◦ θ See Section 3. W e ev olv e on the left hand side. The iden tifications H 1 ( R ( π 2 )) ∼ = H 1  Cok er ( δ R ( π 2 ) )  and H 1 ( E ( q )) ∼ = H 1  Cok er ( δ E ( q ) )  as w ell as the iden tifications K 2 ( M 1 ) ∼ = H 2 ( E ( M 1 )) and K 2 ( C 1 ( H )) ∼ = H 2 ( E ( C 1 ( H ))) will b e suppressed. F urthermore w e will mak e use of the results in Corollary 4.3, Lemma 4.5 and Lemma 4.6 as w ell as of the observ ations preceeding the statemen t of the Theorem. W e 16 J. KAAD get exp ◦ τ 1 ◦ ch rel 2 = exp ◦ ( − T ◦ TR ◦ L ◦ ∂ ◦ h 2 ) = exp ◦ ˜ L ◦ ∂ ◦ h 2 = det ◦ D ∗ ◦ θ ◦ ∂ ◦ h 2 = det ◦ ∂ ◦ E ( R ) ∗ ◦ θ ◦ h 2 = det ◦ ∂ ◦ R ∗ ◦ θ But this is precisely t he comp osition d ◦ R ∗ ◦ θ : K rel 2 ( M 1 ) → C ∗ pro ving the theorem.  The main theorem can b e refined in the follow ing w ay . Let ρ F : A → M 1 b e the unital algebra homomorphism asso ciated with an o dd unital 2 - summable F redholm mo dule ( F , H ) ov er the C -algebra A . W e then define a C -subalgebra of L ( H ) b y E = { P ρ F ( a ) P + T | T ∈ L 1 ( H ) and a ∈ A } Letting B denote the quotien t o f E by L 1 ( H ) w e get an exact sequence X F : 0 − − − → L 1 ( H ) i − − − → E π − − − → B − − − → 0 W e then hav e the algebra homomorphism ι : B → C 1 ( H ) whic h is induced b y the inclusion ι : E → L ( H ) and the surjectiv e a lg ebra homomorphism R F : A → B a 7→ q ( P ρ F ( a ) P ) Here q : L ( H ) → C 1 ( H ) denotes the quotien t map. Corollary 4.8. The diagr am K 2 ( A ) K 2 ( B ) C / (2 π i ) Z C ∗ ✲ ( R F ) ∗ ❄ M F ❄ d ( X F ,ι ) ✲ exp is c ommutative. Pr o of. The result follows from Theorem 4.7 a nd the identit y R ◦ ρ F = ι ◦ R F : A → C 1 ( H ) using t he definition of the in v arian ts and the functorialit y of algebraic K -theory .  COMP ARISON OF SECONDAR Y INV ARI ANTS OF A LGEBRAIC K -THEOR Y 17 References [1] B. 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