aa r X i v : . [ m a t h . A T ] M a r EIGHT FLAVOURS OF CYCLIC HOMOLOGY
K. CIELIEBAK AND E. VOLKOV
Abstract.
We introduce eight “flavours” of cyclic homology of a mixed com-plex and study their properties. In particular, we determine their behaviourwith respect to Chen’s iterated integrals. Introduction
Cyclic homology of an algebra was introduced in the mid-1980s by B. Tsygan [20]and A. Connes [9]. It can be seen as an algebraic counterpart to the S -equivarianthomology of a space with a circle action. Since then, cyclic homology has been gen-eralized to differential graded algebras (dgas), A ∞ -algebras, and beyond. Moreover,several other versions of cyclic homology have emerged: negative cyclic homology(Jones [16]), periodic cyclic homology (Goodwillie [12]), and completed negativecyclic homology (Jones and Petrack [17]).Cyclic homology is related to the homology of loop spaces in two different ways.First, for each connected space X , a suitable version of cyclic homology of singularchains on the based loop space of X (made a dga by the Pontrjagin product) isisomorphic to the S -equivariant homology of its free loop space LX . The secondrelation, which was the starting point of this paper, goes back to the work byK.T. Chen [5, 6]. For a simply connected manifold X , Chen showed that thesingular cohomology of its based loop space can be computed in terms of iteratedintegrals of differential forms. Getzler, Jones and Petrack [11] extended this resultto the homology and the S -equivariant homology of the free loop space LX .The goal of the present paper is a systematic study of the different versions of cyclichomology and their relation to loop space homology via Chen’s iterated integrals.The natural setting is that of a mixed complex , introduced by Kassel [18] (andpopularized by Getzler, Jones and Petrack [11] under the name dg-Λ-module).This is a graded vector space C = L k ∈ Z C k together with two anticommutingdifferentials δ, D of degrees | δ | = 1 and | D | = −
1. The two main examples are theHochschild complex ( C ( A ) , d H , B ) of a dga A with its Connes operator B , and thesingular cochain complex ( C ∗ ( Y ) , d, P ) of an S -space with its contraction P bythe circle action. See Section 2 for details.For a mixed complex ( C, δ, D ), the map δ u = δ + uD defines a differential of degree1 on the space C [[ u, u − ]] of formal power series in a degree 2 variable u and itsinverse. This complex has five subcomplexes C [[ u, u − ], C [ u, u − ]], C [ u, u − ], C [[ u ]]and C [ u ], corresponding to power series that are polynomial in u − etc, and twoquotient complexes C [[ u − ]] = C [[ u, u − ]] /uC [[ u ]] and C [ u − ] = C [ u, u − ] /uC [ u ].These give rise to eight versions (or “flavours”) of cyclic homology that we denoteby HC ∗ [[ u,u − ]] etc.The eight versions of cyclic homology are in general all different, and they are all in-variant under homotopy equivalences of mixed complexes. However, only the threeversions HC ∗ [ u − ] , HC ∗ [[ u ]] and HC ∗ [[ u,u − ] are invariant under quasi-isomorphisms of mixed complexes (Proposition 2.3). These correspond to the positive, negativeand periodic cyclic homologies in [16], and we will refer to them as the classicalversions .For the Hochschild complex of a dga (or more generally of a cyclic cochain complex),each version of cyclic homology is either trivial or agrees with one of the 3 classicalversions (Corollary 2.14). Moreover, the version HC ∗ [ u − ] agrees with Connes’ ver-sion HC ∗ λ , defined as the d H -homology of the Hochschild complex modulo cyclicpermutations.For the singular cochain complex of a smooth S -space Y (such as the free loopspace of a manifold), the version HC ∗ [[ u ]] of cyclic homology agrees with the sin-gular cohomology H ∗ S ( Y ) of its Borel space ( Y × ES ) /S (Jones [16]). The(non-classical) version HC ∗ [[ u,u − ]] satisfies fixed point localization (Jones and Pe-track [17]), whereas the version HC ∗ [[ u,u − ] for a free loop space Y = LX dependsonly on the fundamental group of X (Goodwillie [12]).Consider now a manifold X with its de Rham dga Ω ∗ ( X ). According to Getzler,Jones and Petrack [11] (see also Proposition 3.1 below), Chen’s iterated integralsdefine a morphism of mixed complexes I : (cid:0) C (Ω ∗ ( X )) , d H , B (cid:1) → (cid:0) C ∗ ( LX ) , d, P (cid:1) . For X simply connected this map is a quasi-isomorphism, so it induces isomor-phisms on the three classical versions of cyclic homology. On the other five versionsit does not induce an isomorphism in general. The main result of this paper (Corol-lary 3.6) asserts that for a simply connected manifold X the cyclic (or Connes)variant I λ of Chen’s iterated integral induces an isomorphism I λ ∗ : HC ∗ λ (Ω ∗ ( X )) ∼ = −→ H ∗ S ( LX, x )between the reduced Connes version of cyclic homology of the de Rham complexand the S -equivariant cohomology of LX relative to a fixed constant loop x .All the preceding results have counterparts for cyclic cohomology . For example(Corollary 4.11), for a simply connected manifold X the map J λ adjoint to I λ induces an isomorphism J λ ∗ : H S ∗ ( LX, x ) ∼ = −→ HC λ ∗ (Ω ∗ ( X )).The motivation for this article comes from string topology . This term refers toalgebraic structures on loop space homologies introduced by Chas and Sullivanin [3] and subsequent work. One of the puzzles in string topology concerns theappropriate versions of loop space homology on which these structures are defined.In the non-equivariant case, it has recently turned out that the Chas–Sullivan loopproduct and the Goresky–Hingston coproduct both naturally live on H ∗ ( LX ) if themanifold X has vanishing Euler characteristic, and on H ∗ ( LX, x ) otherwise [8].In the equivariant case, Chas and Sullivan described in [4] an involutive Lie bialge-bra structure on H S ∗ ( LX, X ), the S -equivariant homology of the loop space LX relative to the subset X ⊂ LX of constant loops. In [7] it was conjectured thata chain-level version of this structure exists on HC λ ∗ (Ω ∗ ( X )), the Connes versionof cyclic cohomology of the de Rham complex, so the question arose what thiscorresponds to on the loop space side. This question became more pressing whencomputations of examples showed that HC λ ∗ (Ω ∗ ( X )) can be nontrivial in negativedegrees, and thus cannot correspond to any version of loop space homology. Thesolution is provided by Corollary 4.11: the negative degree part in HC λ ∗ (Ω ∗ ( X ))only comes from the homology of a point, and after dividing this out (i.e., passingto reduced homology) it becomes isomorphic to H S ∗ ( LX, x ). In particular, this IGHT FLAVOURS OF CYCLIC HOMOLOGY 3 exhibits loop space homology relative to a point as the natural space supportingthe involutive Lie bialgebra structure from [4].
Acknowledgement.
We thank P. Hajek for many stimulating discussions.2.
Cyclic homology of mixed complexes
In this section we introduce the 8 versions of cyclic homology associated to a mixedcomplex and discuss their properties. Moreover, we will establish the followingdiagram in which all maps are functors and the upper right square and trianglecommute, whereas the lower pentagon commutes for simply connected manifolds.(1) dga (cid:15) (cid:15) / / A ∞ -algebra (cid:15) (cid:15) manifold de Rham ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ / / loop space ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ cyclic cochain complex / / Connes double complex (cid:15) (cid:15) S -space singular cochains / / mixed complex cyclic homologies (cid:15) (cid:15) R [ u ]-module2.1. Mixed complexes.
Definition . A mixed complex ( C, δ, D ) is a Z -graded R -vector space C ∗ = M k ∈ Z C k with two linear maps δ : C ∗ → C ∗ +1 and D : C ∗ → C ∗− satisfying δ = 0 , D = 0 , δD + Dδ = 0 . A morphism between mixed complexes is a linear map f : C ∗ → e C ∗ satisfying e δf = f δ, e Df = f D. It is called a quasi-isomorphism if it induces an isomorphism on homology H ∗ ( C, δ ) → H ∗ ( e C, e δ ). A homotopy between two morphisms f, g : C ∗ → e C ∗ is a linear map H : C ∗ → e C ∗− such that δH + Hδ = f − g and DH + HD = 0. A mor-phism f : C ∗ → e C ∗ is called a homotopy equivalence if there exists a morphism g : e C ∗ → C ∗ such that f g and gf are homotopic to the identity. Every homotopyequivalence is a quasi-isomorphism but not vice versa.Let u be a formal variable of degree | u | = 2. To a mixed complex ( C, δ, D ) weassociate the cochain complex C [[ u, u − ]] = M k ∈ Z C k [[ u, u − ]] , δ u := δ + uD. where C k [[ u, u − ]] denotes the space of formal power series P i ∈ Z c i u i with c i ∈ C k − i . We emphasize that C [[ u, u − ]] is not the usual tensor product of C with Most of this section works for modules over a commutative ring with unit instead of R -vectorspaces. K. CIELIEBAK AND E. VOLKOV R [[ u, u − ]]. Note that δ u has degree +1. This complex has seven sub/quotientcomplexes of interest, all equipped with the differential induced by δ u :(2) C [[ u, u − ] , C [ u, u − ]] , C [ u, u − ] , C [[ u ]] , C [ u ] ,C [[ u − ]] := C [[ u, u − ]] /uC [[ u ]] = C [ u, u − ]] /uC [ u ] ,C [ u − ] := C [[ u, u − ] /uC [[ u ]] = C [ u, u − ] /uC [ u ] . Here the complexes in the first line are the obvious subcomplexes of C [[ u, u − ]],where C [[ u, u − ] denotes power series in u and polynomials in u − etc., and theremaining two complexes are quotients. Note that C [ u, u − ] = u − C [ u ] , C [[ u, u − ] = u − C [[ u ]] , where the right hand sides denote the localization of C [ u ] (resp. C [[ u ]]) at themultiplicative set { , u, u , . . . } . We denote the homology of C [[ u, u − ]] with respectto δ u by HC ∗ [[ u,u − ]] := H ∗ (cid:16) C [[ u, u − ]] , δ u (cid:17) , and similarly for the other versions with the obvious notation. By construction, allthe chain complexes and thus also their homologies are modules over the polyno-mial ring R [ u ]. Moreover, the versions C [[ u, u − ]], C [[ u, u − ], C [ u, u − ]], C [ u, u − ]and their cohomologies are modules over the larger ring R [ u, u − ] of Laurent poly-nomials. We will use the following names for some versions of cyclic homology: • HC ∗ [[ u ]] Borel version ; • HC ∗ [[ u,u − ] Goodwillie version ; • HC ∗ [ u − ] nilpotent version ; • HC ∗ [[ u,u − ]] Jones–Petrack version .The first three of these versions will also be called the classical versions . Remark . (a) The explanations for the preceding names are the following (seelater in this section for details): HC ∗ [[ u ]] applied to cochains on an S -space yieldsthe cohomology of its Borel construction; HC ∗ [[ u,u − ] satisfies Goodwillie’s theorem:applied to cochains on a loop space LX it depends only on π ( X ); HC ∗ [ u − ] isthe version for which the action of u is nilpotent; HC ∗ [[ u,u − ]] applied to a smooth S -space satisfies the fixed point localization theorem of Jones and Petrack.(b) The notion of a mixed complex was introduced by Kassel [18]. Its name reflectsthe fact that δ has degree +1 while D has degree −
1. Mixed complexes also appearin [11] under the name dg- Λ -module .(c) Let us emphasize that our eight versions of HC ∗ correspond to the cyclic ho-mology of the mixed complex ( C, δ, D ). We write them with an upper ∗ becausethe differential δ u has degree +1, in the same way that the homology of a cochaincomplex is denoted by H ∗ . Our convention avoids unnecessary minus signs in themain examples, but care has to be taken when comparing it to other appearancesof cyclic homology in the literature. Example . Consider the mixed complex with C k := R in each degree k ∈ Z and trivial differentials d = D = 0. Then in each degree k and for each version { u, u − } we have HC k { u,u − } = C k { u,u − } = R { u, u − } , so all eight versions of cyclichomology are pairwise non-isomorphic as R [ u ]-modules. IGHT FLAVOURS OF CYCLIC HOMOLOGY 5
Quasi-isomorphism invariance.
A morphism f between mixed complexes( C, δ, D ) and ( e C, e δ, e D ) induces homomorphisms f ∗ between all versions of homol-ogy defined above as modules over R [ u ] resp. R [ u, u − ]. Cleary f ∗ is an isomor-phism if f is a homotopy equivalence. We say that a version of homology is quasi-isomorphism invariant if the induced map f ∗ is an isomorphism whenever f is aquasi-isomorphism of mixed complexes. Proposition 2.3.
The classical versions HC ∗ [[ u ]] , HC ∗ [[ u,u − ] and HC ∗ [ u − ] ofcyclic homology are quasi-isomorphism invariants of mixed complexes, whereas theother versions are not.Proof. The quasi-isomorphism invariance of the 3 classical versions is proved in [16,Lemma 2.1] in the special case of cyclic chain complexes, and in [21, Proposition 2.4]in the more general context of S -complexes (cf. Remark 2.4 below). Examples 2.27,2.28 and 2.29 below in conjunction with Lemma 2.30 show that the other 5 homologygroups are not quasi-isomorphism invariant. (cid:3) Remark . Much of the preceding discussion can be generalized to S -complexesas defined in [21]. This generalization is relevant if one wants to include symplectichomology in this framework, but will not be further discussed in this paper. Exact sequences.
The eight versions of cyclic homology are connected by variousexact sequences fitting into commuting diagrams.
Proposition 2.5 (Hood and Jones [15]) . For every mixed complex ( C, δ, D ) thereexists a commuting diagram with exact rows and columns · · · · · · HC ∗ +1[ u − ] id −−−−→ = HC ∗ +1[ u − ] y D y D · · · HC ∗− u ] · u −−−−→ HC ∗ [ u ] u =0 −−−−→ H ∗ ( C, δ ) D ∗ −−−−→ HC ∗− u ] · · · = y id y i ∗ y i ∗ = y id · · · HC ∗− u ] · u −−−−→ HC ∗ [ u,u − ] p ∗ −−−−→ HC ∗ [ u − ] D ∗ −−−−→ HC ∗− u ] · · · y p ∗ · u y · u HC ∗ +2[ u − ] id −−−−→ = HC ∗ +2[ u − ] · · · · · · and similarly for the [ u, u − ]] , [[ u, u − ]] and [[ u, u − ] versions. Here D means D applied to the constant term in u and the other maps are the obvious ones. We will refer to the second vertical and the first horizontal sequences as the
Gysin(or Connes) exact sequences , and to the first vertical and the second horizontalsequences (which are equivalent in view of the periodicity HC ∗− u,u − ] ∼ = HC ∗ [ u,u − ] )as the tautological exact sequences . K. CIELIEBAK AND E. VOLKOV
Proof.
This diagram appears in [15, Figure 1]. It follows from the commutingsquare of short exact sequences0 0 0 y y y −−−−→ uC [ u ] −−−−→ C [ u ] −−−−→ C [ u ] /uC [ u ] −−−−→ = y id y i y i −−−−→ uC [ u ] −−−−→ C [ u, u − ] p −−−−→ C [ u, u − ] /uC [ u ] −−−−→ y y p y p −−−−→ C [ u, u − ] /C [ u ] id −−−−→ = C [ u, u − ] /C [ u ] −−−−→ y y C [ u ] /uC [ u ] = C , C [ u, u − ] /uC [ u ] = C [ u − ], and C [ u, u − ] /C [ u ] = u − C [ u − ]. (cid:3) Other homologies.
Given a mixed complex (
C, δ, D ), the chain complex (
C, δ )has two natural subcomplexes im D ⊂ ker D ⊂ C which together with their quotientcomplexes fit into the commuting diagram with exact rows and columns0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) ker D/ im D (cid:15) (cid:15) / / im D (cid:15) (cid:15) / / ( C, δ ) / / C/ im D (cid:15) (cid:15) / / / / ker D (cid:15) (cid:15) / / ( C, δ ) (cid:15) (cid:15) / / C/ ker D (cid:15) (cid:15) / / D/ im D (cid:15) (cid:15) IGHT FLAVOURS OF CYCLIC HOMOLOGY 7
On homology this yields the following commuting diagram with exact rows andcolumns H ∗− (ker D/ im D ) δ ∗ (cid:15) (cid:15) H ∗ (ker D/ im D ) (cid:15) (cid:15) H ∗ (ker D/ im D ) δ ∗ (cid:15) (cid:15) · · · H ∗ (im D ) (cid:15) (cid:15) / / H ∗ ( C, δ ) / / H ∗ ( C/ im D ) (cid:15) (cid:15) δ ∗ / / H ∗ +1 (im D ) · · · (cid:15) (cid:15) · · · H ∗ (ker D ) (cid:15) (cid:15) / / H ∗ ( C, δ ) / / H ∗ ( C/ ker D ) δ ∗ (cid:15) (cid:15) δ ∗ / / H ∗ +1 (ker D ) · · · (cid:15) (cid:15) H ∗ (ker D/ im D ) H ∗ +1 (ker D/ im D ) H ∗ +1 (ker D/ im D )A morphism f of mixed complexes induces homomorphisms between all these ho-mologies, which are isomorphisms if f is a homotopy equivalence. Note that themap D in Proposition 2.5 naturally factors through chain maps (where D hasdegree − D : ( C [[ u − ]] , δ u ) π −→ ( C/ im D, δ ) D −→ (im D, δ ) ι −→ ( C [ u ] , δ u ) . Cocyclic and cyclic objects.
One source of mixed complexes are cycliccochain complexes which we introduce in this subsection. For more backgroundsee [19, Section 6.1].
Cocyclic objects. A cocyclic object in some category is a sequence of objects C n , n ∈ N , with morphisms • δ i : C n − → C n , i = 0 , . . . , n (faces), • σ j : C n +1 → C n , j = 0 , . . . , n (degeneracies), • τ n : C n → C n (cyclic operators)satisfying the following relations: δ j δ i = δ i δ j − for i < j,σ j σ i = σ i σ j +1 for i ≤ j,σ j δ i = δ i σ j − for i < j, id for i = j or i = j + 1 ,δ i − σ j for i > j + 1 ,τ n δ i = δ i − τ n − for 1 ≤ i ≤ n, τ n δ = δ n ,τ n σ i = σ i − τ n +1 for 1 ≤ i ≤ n, τ n σ = σ n τ n +1 ,τ n +1 n = id . Forgetting τ n we have a cosimplicial object , and forgetting the σ j a pre-cocyclicobject . Example . A topological space X gives rise to a cocyclic space by setting C n ( X ) := X n +1 = X × · · · × X ( n + 1 times) and δ i ( x , . . . , x n − ) := ( x , . . . , x i , x i , . . . , x n − ) for 0 ≤ i ≤ n − ,δ n ( x , . . . , x n − ) := ( x , . . . , x n − , x ) ,σ j ( x , . . . , x n +1 ) := ( x , . . . , b x i , . . . , x n +1 ) ,τ n ( x , . . . , x n ) := ( x , . . . , x n , x ) . K. CIELIEBAK AND E. VOLKOV
For a subspace Y ⊂ X we can define a cosimplicial space by C n ( X, Y ) := X n × Y and the same operations δ i , σ j (appropriately viewing elements in Y as elements in X via the inclusion). Cyclic objects.
Dualizing the notion of a cocyclic object, we obtain that of acyclic object. A cyclic object in some category is a sequence of objects C n , n ∈ N ,with morphisms • d i : C n → C n − , i = 0 , . . . , n (faces), • s j : C n → C n +1 , j = 0 , . . . , n (degeneracies), • t n : C n → C n (cyclic operators)satisfying the following relations: d i d j = d j − d i for i < j,s i s j = s j +1 s i for i ≤ j,d i s j = s j − d i for i < j, id for i = j or i = j + 1 ,s j d i − for i > j + 1 ,d i t n = t n − d i − for 1 ≤ i ≤ n, d t n = d n ,s i t n = t n +1 s i − for 1 ≤ i ≤ n, s t n = t n +1 s n ,t n +1 n = id . Forgetting t n we have a simplicial object , and forgetting the s j a pre-cyclic object .Note that if F : A → B is a contravariant functor, then a cocyclic object in thecategory A gives rise to a cyclic object F ( A ) in the category B and vice versa.The cyclic structure gives rise to the extra degeneracy s n +1 := t n +1 s n : C n → C n +1 satisfying the following relations on C n : s i s n +1 = s n +2 s i − for 1 ≤ i ≤ n + 1 , s s n +1 = s n +2 s n +1 ,d i s n +1 = s n d i − for 1 ≤ i ≤ n, d n +1 s n +1 = t n , d s n +1 = id ,s n +1 t n = t n +1 s n − . Example . For a manifold X denote by C n (Ω ∗ ( X )) := Ω ∗ ( X n +1 ) the differen-tial graded algebra of differential forms on the ( n + 1)-fold product of X . Thisbecomes a cyclic object in the category of differential graded vector spaces (i.e.,chain complexes) with the operations d i := δ ∗ i , s j := σ ∗ j , t n := τ ∗ n induced fromthose of Example 2.6. For a submanifold Y ⊂ X we can define a simplicial chaincomplex by C n (Ω ∗ ( X ) , Ω ∗ ( Y )) := Ω ∗ ( X n × Y ) and the same operations d i , s j (ap-propriately viewing forms on X as forms on Y via restriction). Note that Ω ∗ ( Y ) isan Ω ∗ ( X )-bimodule. Example . A differential graded algebra ( dga ) A over R with unit 1 gives rise toa cyclic object in the category of differential graded R -modules by setting C n ( A ) := A ⊗ ( n +1) and d i ( a | · · · | a n ) := ( a | · · · | a i a i +1 | · · · | a n ) for 0 ≤ i ≤ n − ,d n ( a | · · · | a n ) := ( − deg a n (deg a + ··· +deg a n − ) ( a n a | a | · · · | a n − ) ,s j ( a | · · · | a n ) := ( a | · · · | a j | | a j +1 | · · · | a n ) for 0 ≤ i ≤ n,t n ( a | · · · | a n ) := ( − deg a n (deg a + ··· +deg a n − ) ( a n | a | · · · | a n − ) . IGHT FLAVOURS OF CYCLIC HOMOLOGY 9
Here for t n we use the convention in [19], which differs the one in [16]. The extradegeneracy becomes s n +1 ( a | · · · | a n ) = (1 | a | · · · | a n ) . For a differential graded A -bimodule M we can define a simplicial differential graded R -module by C n ( A, M ) := A ⊗ n ⊗ M and the same operations d i , s j (using thebimodule structure).For the de Rham complex, the constructions in Examples 2.7 and 2.8 give riseto two cyclic cochain complexes Ω ∗ ( X n +1 ) and Ω ∗ ( X ) ⊗ n +1 which we will refer toas the analytic and algebraic versions, respectively. They are compatible in thefollowing sense. Lemma 2.9.
The exterior cross products φ n : Ω ∗ ( X ) ⊗ n +1 → Ω ∗ ( X n +1 ) , ( a | · · · | a n ) a × · · · × a n define a morphism of cyclic cochain complexes.Proof. Each φ n is clearly a chain map with respect to the exterior derivative. Recallthat a × · · · × a n = π ∗ a ∧ · · · ∧ π ∗ n a n for the canonical projections π i : X n → X , i = 0 , . . . , n . Note that the π i and π ∗ i are compositions of the degeneracies in Examples 2.6 and 2.7, respectively. Usingthis, one deduces that φ is a map of cyclic cochain complexes. For example, therelations π i − τ n = π i for 1 ≤ i ≤ n, π n τ n = π imply compatibility of φ with t n = τ ∗ n : φ t n ( a | · · · | a n ) = ( − deg a n (deg a + ··· +deg a n − ) a n × a × · · · × a n − = ( − deg a n (deg a + ··· +deg a n − ) π ∗ a n ∧ π ∗ a ∧ · · · ∧ π ∗ n a n − = ( − deg a n (deg a + ··· +deg a n − ) τ ∗ n ( π ∗ n a n ∧ π ∗ a ∧ · · · ∧ π ∗ n − a n − )= τ ∗ n ( π ∗ a ∧ · · · ∧ π ∗ n a n )= t n φ ( a | · · · | a n ) . (cid:3) From cyclic cochain complexes to mixed complexes.
Consider a cycliccochain complex ( C n , d i , s j , t n , d ), where ( C n , d ) are Z -graded cochain complexes for n ∈ N with differential of degree +1. Thus the operations d i , s j and t n commutewith d and satisfy the relations in Section 2.2. We define a Z -graded R -vector space( C, | · | ) by C := M n ≥ C n , | c | := deg( c ) − n for c ∈ C n . We define the following operations on homogeneous elements c ∈ C n :(4) b ′ ( c ) := ( − | c | +1 n − X i =0 ( − i d i ( c ) ∈ C n − ,b ( c ) := ( − | c | +1 n X i =0 ( − i d i ( c ) ∈ C n − ,t := ( − n t n : C n → C n ,N := 1 + t + · · · + t n : C n → C n ,s ( c ) := ( − | c | s n +1 ( c ) ∈ C n +1 ,B := (1 − t ) sN : C n → C n +1 . The operations have the following degrees with respect to the grading on C : | d | = | b | = | b ′ | = 1 , | t | = | N | = 0 , | s | = | B | = − . Straightforward computations yield the following relations:(5) ker(1 − t ) = im N, ker N = im (1 − t ) , (1 − t ) b ′ = b (1 − t ) ,b ′ N = N b, ( b ′ ) = b = db + bd = db ′ + b ′ d = 0 ,ds + sd = 0 , b ′ s + sb ′ = 1 , bs + sb ′ = (1 − t ) . Remark . The signs in (4) are chosen for compatibility under Chen’s iteratedintegral in Section 3. An alternative sign convention is given in [19] (in the case ofa cyclic module):(6) e b ′ := n − X i =0 ( − i d i : C n → C n − , e b := n X i =0 ( − i d i : C n → C n − , e t := ( − n t n : C n → C n , e N := 1 + t + · · · + t n : C n → C n , e s := s n +1 : C n → C n +1 , e B := (1 − e t ) e s e N : C n → C n +1 . Note that e t = t and e N = N are unchanged. In this case we need to modify thedifferential d to make it anticommute with e b , e b ′ and e s , so we set e d ( c ) := ( − | c | +1 d ( c ) . It is straightforward to check that with these signs the relations (5) continue tohold. Moreover, the two sign conventions are intertwined by the chain isomorphismΦ : ( C, e d ) −→ ( C, d ) defined on homogeneous elements by Φ( c ) := ( − | c | ( | c | +1) / c .Consider the space C [[ θ, θ − ]] of Laurent series in a formal variable θ of degree | θ | = 1, as usual understood in the graded sense so that elements of degree k aresums P i ∈ Z c i θ i with | c i | = k − i . We view this as a double complex where powers of θ increase in the horizontal direction and the columns are copies of C , see Figure 1.We define a differential δ θ := δ ver + δ hor : C [[ θ, θ − ]] → C [[ θ, θ − ]] IGHT FLAVOURS OF CYCLIC HOMOLOGY 11 θ − θ − θ θ Nd + b − t − ( d + b ′ ) Figure 1.
The double complex C [[ θ, θ − ]]of degree 1 with(7) δ ver ( cθ n ) := ( ( d + b )( c ) θ n n even , − ( d + b ′ )( c ) θ n n odd ,δ hor ( cθ n ) := ( N ( c ) θ n +1 n even , (1 − t )( c ) θ n +1 n odd . The relations (5) imply that this is indeed a double complex. Its total differential δ θ induces differentials on all the seven sub- and quotient complexes C [[ θ, θ − ] etc,defined as in (2) with u replaced by θ . The double complex C [[ θ, θ − ]] has thefollowing two key properties:(i) its odd columns are contractible with contracting homotopy − s , i.e., ( d + b ′ ) s + s ( d + b ′ ) = id;(ii) its rows are exact (cf. [19] Theorem 2.1.5). Lemma 2.11.
Let ( C n , d i , s j , t n , d ) be a cyclic cochain complex. Then ( C, d H := d + b, B ) is a mixed complex. Moreover, the chain map k : C [[ u, u − ]] −→ C [[ θ, θ − ]] , X j c j u j X j ( c j θ j + sN c j θ j +1 ) induces an isomorphism on homology as an R [ u, u − ] -module, where u acts on C [[ θ, θ − ]] as multiplication by θ . Similarly for all the other versions of homology.Moreover, these isomorphisms fit into a commuting diagram (8) · · · HC ∗− u ] ∼ = k ∗ (cid:15) (cid:15) · u / / HC ∗ [ u,u − ] ∼ = k ∗ (cid:15) (cid:15) p ∗ / / HC ∗ [ u − ] ∼ = k ∗ (cid:15) (cid:15) B / / HC ∗− u ] · · · ∼ = k ∗ (cid:15) (cid:15) · · · HC ∗− θ ] · θ / / HC ∗ [ θ,θ − ] r ∗ p ∗ / / HC ∗ [ θ − ] B / / HC ∗− θ ] · · · and similarly for all other tautological and Connes exact sequences.Proof. The proof is an easy adaptation of the arguments in [19, Section 2.1]: that d + b and B are anticommuting differentials and k is a chain map follows by directcomputation based on (5), and that k induces an isomorphism on homology followsfrom contractibility of the odd columns of the double complex C [[ θ, θ − ]] and [19,Lemma 2.1.6]. To derive the diagram (8), consider the commuting diagram of shortexact sequences0 / / uC [ u ] k (cid:15) (cid:15) / / C [ u, u − ] k (cid:15) (cid:15) p / / C [ u − ] q (cid:15) (cid:15) / / / / θ C [ θ ] / / C [ θ, θ − ] p / / θC [ θ − ] r (cid:15) (cid:15) / / C [ θ − ] , where q is the induced map between the quotients and r is the map dividing outthe θ -column. Since that column is contractible, the induced map r ∗ on homologyis an isomorphism and we obtain on homology the diagram · · · HC ∗− u ] ∼ = k ∗ (cid:15) (cid:15) · u / / HC ∗ [ u,u − ] ∼ = k ∗ (cid:15) (cid:15) p ∗ / / HC ∗ [ u − ] ∼ = q ∗ (cid:15) (cid:15) B / / HC ∗− u ] · · · ∼ = k ∗ (cid:15) (cid:15) · · · HC ∗− θ ] · θ / / HC ∗ [ θ,θ − ] p ∗ / / H ∗ ( θC [ θ − ]) ∼ = r ∗ (cid:15) (cid:15) / / HC ∗− θ ] · · · HC ∗ [ θ − ] B ♣♣♣♣♣♣♣♣♣♣♣ Using r ∗ q ∗ = k ∗ we obtain (8). (cid:3) The homology H ( C, d H := d + b ) is the Hochschild homology and the homologies HC [[ u,u − ]] etc are the various versions of cyclic homology of the cyclic cochaincomplex. Connes’ version of cyclic homology.
Due to the relations (5) we have shortexact sequences of cochain complexes0 −→ (cid:0) im (1 − t ) , d + b (cid:1) −→ ( C, d + b ) −→ (cid:0) C/ im (1 − t ) , d + b (cid:1) −→ , −→ (im N, d + b ′ ) −→ ( C, d + b ′ ) −→ ( C/ im N, d + b ′ ) −→ . IGHT FLAVOURS OF CYCLIC HOMOLOGY 13
The first one induces a long exact sequence · · · H ∗ (cid:0) im (1 − t ) , d + b (cid:1) −→ H ∗ ( C, d + b ) −→ H ∗ (cid:0) C/ im (1 − t ) , d + b (cid:1) −→ H ∗ +1 (cid:0) im (1 − t ) , d + b (cid:1) · · · and the second one, by acyclicity of ( C, d + b ′ ), an isomorphism H ∗ ( C/ im N, d + b ′ ) ∼ = H ∗ +1 (im N, d + b ′ ) . Moreover, the chain isomorphisms N : (cid:0) C/ im (1 − t ) , d + b (cid:1) ∼ = −→ (im N, d + b ′ ) and1 − t : ( C/ im N, d + b ′ ) ∼ = −→ (cid:0) im (1 − t ) , d + b (cid:1) induce isomorphisms N : H ∗ (cid:0) C/ im (1 − t ) , d + b (cid:1) ∼ = −→ H ∗ (im N, d + b ′ ) , − t : H ∗ ( C/ im N, d + b ′ ) ∼ = −→ H ∗ (cid:0) im (1 − t ) , d + b (cid:1) . Definition . Connes’ version of cyclic homology of the cyclic cochain complex( C n , d i , s j , t n , d ) is HC ∗ λ := H ∗ +1 (cid:0) C/ im (1 − t ) , d + b (cid:1) ∼ = H ∗ +1 (im N, d + b ′ ) ∼ = H ∗ ( C/ im N, d + b ′ ) ∼ = H ∗ (cid:0) im (1 − t ) , d + b (cid:1) . The following lemma identifies Connes’ version of cyclic homology with two of theother versions.
Lemma 2.12.
Let ( C n , d i , s j , t n , d ) be a cyclic cochain complex. Then HC ∗ [ u,u − ]] = HC ∗ [ θ,θ − ]] = 0 and we have a commuting diagram (9) HC ∗ [[ u − ]] ∼ = k ∗ (cid:15) (cid:15) B ∼ = / / HC ∗− u ] ∼ = k ∗ (cid:15) (cid:15) HC ∗ [[ θ − ]] ∼ = e ∗ (cid:15) (cid:15) B ∼ = / / HC ∗− θ ] H ∗ (cid:0) C/ im (1 − t ) , d + b (cid:1) N ∗ ∼ = / / H ∗ (im N, d + b ′ ) ∼ = ((1 − t ) s ) ∗ O O where the upper square arises from (8) and e : C [[ θ − ]] → C/ im (1 − t ) is theprojection onto the -th column.Proof. Consider the double complex with exact rows ( C [ θ, θ − ]] , δ θ ). Standardzigzag arguments as in [19] (see also the proof of Lemma 2.13 below) show thatits total homology vanishes and the map e : C [[ θ − ]] → C/ im (1 − t ) inducesan isomorphism on homology. In view of (8) this yields the diagram (9), wherecommutativity of the lower square follows from the definition of B . The maps B are isomorphism because the third terms in the corresponding tautological exactsequences vanish. Since N ∗ is an isomorphism, this implies that the map ((1 − t ) s ) ∗ is an isomorphism as well. It is important to note, however, that these argumentsfail for other versions of cyclic homology. (cid:3) The following refinement of Lemma 2.12 will be crucial in the following.
Lemma 2.13.
Let ( C n , d i , s j , t n , d ) be a cyclic cochain complex. Then the canonicalinclusion φ : C [ θ − ] → C [[ θ − ]] induces an isomorphism on homology φ ∗ : HC ∗ [ θ − ] ∼ = −→ HC ∗ [[ θ − ]] . The same holds true for the inclusions C [ u − ] → C [[ u − ]] , C [ u, u − ] → C [ u, u − ]] ,and C [[ u, u − ] → C [[ u, u − ]] . Proof.
Consider the sequence of chain maps(10) ( C [ θ − ] , δ θ ) φ −−−−→ ( C [[ θ − ]] , δ θ ) e −−−−→ ( C/ im (1 − t ) , d + b ) . Since e induces an isomorphism on homology by Lemma 2.12, it suffices to showthat the composition eφ induces an isomorphism ( eφ ) ∗ on homology.We call an element c ∈ θ − C [[ θ − ]] a resolution of an element x ∈ C if δ θ c = x (in particular, all nonzero powers of θ − cancel). Since the rows of C [[ θ − ]] areexact, an element x ∈ C admits a resolution if and only if ( d + b ) x = 0 and x ∈ im (1 − t ) = ker N . Claim.
Every x ∈ C with ( d + b ) x = 0 and x ∈ im (1 − t ) admits a resolution c which is polynomial in θ − , i.e. c ∈ θ − C [ θ − ].Let us assume the claim for the moment and finish the proof. We begin withsurjectivity of ( eφ ) ∗ . Let y ∈ C represent a closed element in ( C/ im (1 − t ) , d + b ).Then ( d + b ) y is ( d + b )-closed and lies in im (1 − t ), so by the claim it has apolynomial resolution c ∈ θ − C [ θ − ]. Then δ θ ( y − c ) = N yθ + ( d + b ) y − δ θ c = N yθ ≡ ∈ C [ θ − ] = C [ θ, θ − /θC [ θ ] , so the element y − c is closed in C [ θ − ]. Since eφ ( y − c ) = y , this proves surjectivityof ( eφ ) ∗ .Next we show that ( eφ ) ∗ is injective. Since e ∗ is an isomorphism, it suffices to showthat φ ∗ is injective. Let a closed element c = P nk =0 c − k θ − k ∈ C [ θ − ] represent anexact element in C [[ θ − ]], i.e., δ θ a = c for some a = P ∞ k =0 a − k θ − k ∈ C [[ θ − ]]. Wemay assume that n is odd by allowing the last coefficient c − n to be zero. Then( d + b ) a − n − is ( d + b )-closed and lies in im (1 − t ), so by the claim it has a polynomialresolution z = P Nk =1 z − k θ − . Now˜ a := n +1 X k =0 a − k θ − k − zθ − n − ∈ C [ θ − ]satisfies δ θ ˜ a = c + ( d + b ) a − n − θ − n − − δ θ z θ − n − = c, so c is exact in C [ θ − ] and injectivity is proved.It remains to prove the claim. By a slight abuse of notation, we will denote by δ ver and δ hor the maps C → C defined as in (7) but without the factors of θ . Recall theformula c = Σ ∞ k =1 c − k θ − k for a resolution of x ∈ C . We will write c − = δ − ( x ) and c − ( k +1) = − δ − δ ver c − k for k ≥ , where by δ − we mean some preimage under δ hor . The main point of the proof isa certain special choice of the preimage. To explain it, recall that C = L ∞ n =0 C n .The maps t, N, d preserve C n , whereas b and b ′ map C n +1 to C n . In particular, δ hor preserves C n . Let us define the weight w ( c ) of a nonzero element c ∈ C as thesmallest integer w such that c ∈ L wn =0 C n , and set w (0) := −
1. By the precedingdiscussion we can choose the preimages under δ hor such that(11) w ( c − k − ) ≤ w ( c − k ) for all k ≥ . This condition can be improved further. For this, note that the map N : C n → C n acts on its image as multiplication by 1 + n . Therefore N − on the image of N canbe taken to be multiplication by (1 + n ) − . In particular, it maps d -closed elementsto d -closed ones. From this we can deduce the following strict (!) inequality:(12) w ( c − k − ) < w ( c − k ) for k odd and c − k = 0 . IGHT FLAVOURS OF CYCLIC HOMOLOGY 15
Indeed, for k odd we obtain from (11) w ( c − k − ) = w ( δ − δ ver δ − δ ver c − k ) ≤ w ( δ ver δ − δ ver c − k ) = w (cid:0) ( d + b ) N − ( d + b ′ ) c − k (cid:1) . To compute the last term we write out( d + b ) N − ( d + b ′ ) c − k = dN − dc − k + dN − b ′ c − k + bN − dc − k + bN − b ′ c − k , where the first summand dN − dc k = 0 vanishes since N − maps closed elementsto closed ones. All other summands have weight strictly less than that of c − k dueto the presence of either b or b ′ (or both), and inequality (12) follows.Inequalities (11) and (12) imply that the weight w ( c − k ) strictly decreases eachtime k increases by 2, hence c − k = 0 for all sufficiently large k and the resolutionis polynomial. This proves the claim and thus the isomorphism φ ∗ : HC ∗ [ θ − ] ∼ = −→ HC ∗ [[ θ − ]] . The statement about the inclusion C [ u − ] → C [[ u − ]] follows from thisand Lemma 2.11, and the statement about the other two inclusions follows fromthe tautological exact sequence in Proposition 2.5 and the 5-lemma. (cid:3) The preceding two lemmas are summarized in
Corollary 2.14.
For a cyclic cochain complex ( C n , d i , s j , t n , d ) the different ver-sions of cyclic homology are given by HC ∗ [ u − ] ∼ = HC ∗ [[ u − ]] ∼ = HC ∗− λ ∼ = HC ∗− u ] , HC ∗ [[ u ]] ,HC ∗ [[ u,u − ] ∼ = HC ∗ [[ u,u − ]] , HC ∗ [ u,u − ] = HC ∗ [ u,u − ]] = 0 . Behaviour under maps of cyclic cochain complexes.
According to Proposi-tion 2.15, only the 3 classical versions of cyclic homology are invariant under quasi-isomorphisms of mixed complexes. By contrast, with respect to quasi-isomorphismsof cyclic cochain complexes we have
Proposition 2.15.
Hochschild homology and all versions of cyclic homology areinvariant under quasi-isomorphisms of cyclic cochain complexes, i.e. morphismsinducing an isomorphism on d -cohomology.Proof. For Hochschild homology and the classical versions HC [[ u ]] , HC [[ u,u − ]] and HC [ u − ] this is an immediate consequence of Proposition 2.3. By Corollary (2.14),this covers all the 8 verstions of cyclic homology. (cid:3) One important application of this proposition concerns the analytic and algebraiccyclic cochain complexes in Examples 2.7 and 2.8 built from the de Rham algebraΩ ∗ ( X ) of a manifold X . Recall from Lemma 2.9 that the exterior cross products φ n : Ω ∗ ( X ) ⊗ ( n +1) → Ω ∗ ( X n +1 ) define a morphism of cyclic cochain complexes.Since φ n induces an isomorphism on d -homology for all n by the K¨unneth formula,Proposition 2.15 implies Corollary 2.16 (Algebraic vs. analytic cyclic homology) . For a manifold X , themap of mixed complexes φ : (cid:16)M n ≥ Ω ∗ ( X ) ⊗ ( n +1) , d + b, B (cid:17) → (cid:16)M n ≥ Ω ∗ ( X n +1 ) , d + b, B (cid:17) induces isomorphisms on Hochschild homology and on all versions of cyclic homol-ogy. (cid:3) From differential graded algebras to mixed complexes.
Recall fromExample 2.8 that each dga (
A, d ) canonically gives rise to a cyclic cochain com-plex with C n ( A ) = A ⊗ ( n +1) . By Lemma 2.11 it gives rise to a mixed complex( C ( A ) = L n ≥ C n ( A ) , d + b, B ) and a double complex ( C [[ θ, θ − ]] , δ θ ). The homol-ogy HH ( A ) := H ( C ( A ) , δ ) is the Hochschild homology of (
A, d ) and the homologies HC [[ u,u − ]] ( A ) etc are the various versions of cyclic homology of ( A, d ). In this sub-section we will study these cyclic homologies in more detail.Note that a morphism of dgas canonically induces a morphism of cyclic cochaincomplexes. A quasi-isomorphism of dgas induces a quasi-isomorphism of cycliccochain complexes (by the K¨unneth formula and exactness of the tensor functoron vector spaces), so by Proposition 2.15 it induces isomorphisms on Hochschildhomology and all versions of cyclic homology. Moreover, by Corollary 2.14 allversions of cyclic homology of (
A, d ) are either zero or isomorphic to one of the 3classical versions.
Remark . The invariance of all versions of cyclic homology under quasi-isomorphismsof dgas implies, in particular, that the cyclic homology of a commutative dga with H ( A ) = R can be computed using its Sullivan minimal model. Normalized and reduced cyclic homology of a dga.
Consider a dga A with itsassociated cyclic cochain complex ( C n ( A ) = A ⊗ ( n +1) , d i , s j , t n , d ) as in Example 2.8.As in [19] we denote by D n ( A ) ⊂ C n ( A ) the linear subspace spanned by words( a | a | · · · | a n ) with a i = 1 for some i ≥ degenerate ). Sowe get a short exact sequence0 → D n ( A ) → C n ( A ) → C n ( A ) → C n ( A ) = A ⊗ ¯ A ⊗ n , ¯ A = A/ ( R · . Note that the subspaces D n ( A ) ⊂ C n ( A ) define a simplicial subcomplex but nota cyclic one, i.e., it is preserved under the operations d i , s j , d but not under t n .Nonetheless, one easily checks that the direct sums D ( A ) = L n ≥ D n ( A ) and C ( A ) = L n ≥ C n ( A ) give rise to a short exact sequence of mixed complexes0 −→ (cid:0) D ( A ) , d + b, B (cid:1) −→ (cid:0) C ( A ) , d + b, B (cid:1) p −→ (cid:0) C ( A ) , d + b, B (cid:1) −→ B = sN . The quotient C ( A ) is called the normalized mixedcomplex . It is a standard fact (see e.g. [19, Proposition 1.15], which holds moregenerally for cyclic chain complexes) that ( D ( A ) , d + b ) has vanishing homology,so p : C ( A ) → C ( A ) is a quasi-isomorphism of mixed complexes. We denote theresulting versions of normalized cyclic homology by HC ∗ [ u,u − ] ( A ) etc.The normalized mixed complex C ( A ) of a dga A has the mixed subcomplex C ( R )given by C ( R ) = R and C n ( R ) = 0 for n ≥ reduced mixed complex C red ( A ) by the short exact sequence0 −→ C ( R ) −→ C ( A ) −→ C red ( A ) −→ . We denote the resulting versions of reduced cyclic homology by HC ∗ [ u,u − ] ( A ) etc.There is also a reduced Connes version HC λ ( A ) defined as the ( d + b )-homology ofthe quotient C λ ( A ) by all words containing 1 in some entry.A dga A is called augmented if A = R · ⊕ A IGHT FLAVOURS OF CYCLIC HOMOLOGY 17 for a dg ideal A ⊂ A . For example, the de Rham complex Ω ∗ ( X ) of a connectedmanifold X is augmented with A = Ω ∗ ( X, x ) the forms restricting to zero on abasepoint x . If A is augmented the above exact sequence splits as a direct sum ofmixed complexes(13) C ( A ) = C ( R ) ⊕ C red ( A ) , where C red0 ( A ) = A and C red n ( A ) = A ⊗ A ⊗ n for n ≥
1. It follows that each version { u, u − } of normalized cyclic homology splits as(14) HC { u,u − } ( A ) = HC { u,u − } ( R ) ⊕ HC { u,u − } ( A ) . Note that for A augmented the reduced Connes version is given by HC λ ( A ) = HC λ ( A ), viewing A as a non-unital dga. Proposition 2.18 (normalized cyclic homology) . For a dga ( A, d ) , the projection p : C ( A ) → C ( A ) induces isomorphisms from Hochschild homology HH ( A ) and theclassical versions HC [[ u ]] ( A ) , HC [[ u,u − ] ( A ) and HC [ u − ] ( A ) of cyclic homology tothe corresponding homologies of the normalized mixed complex. The same holds forthe versions HC ∗ [[ u − ]] ( A ) and HC ∗ [[ u,u − ]] ( A ) if A is augmented. The remaining versions of cyclic homology are in general not isomorphic to their normalized cylichomology.Proof. The first assertion follows directly from Proposition 2.3 because p is a quasi-isomorphism, and the last assertion follows from Example 2.27. For the secondassertion, suppose that A = R · ⊕ A is augmented and denote by p : A → A theprojection. Viewing A as a non-unital dga, it has an associated Hochschild complex( C ( A ) , d + b ) and double complexes ( C ( A ) { θ, θ − } , δ θ ), where { θ, θ − } stands forany of the eight versions [ θ ], [ θ, θ − ]] etc. Note that the reduced cyclic complex hasa canonical splitting as a vector space C red ( A ) = C ( A ) ⊕ sC ( A ) , where sC ( A ) is generated by words with 1 in the 0-th slot and elements from A inall others. Now the main observation is that the map r (( x + sy ) u n ) := xθ n + yθ n − defines a chain isomorphism r : (cid:16)(cid:0) C ( A ) ⊕ sC ( A ) (cid:1) { u, u − } , d + b + Bu (cid:17) ∼ = −→ (cid:16) C ( A ) { θ, θ − } , δ θ (cid:17) . Indeed, the map is clearly bijective, and the chain map property follows from r ( d + b + Bu ) (cid:0) ( x + sy ) u n (cid:1) = r (cid:16) ( d + b )( x + sy ) u n + B ( x + sy ) u n +1 (cid:17) = r (cid:16)(cid:0) dx + + dsy + bx − sb ′ y + (1 − t ) y (cid:1) u n + sN xu n +1 (cid:17) = (cid:0) ( d + b ) x + (1 − t ) y (cid:1) θ n − ( d + b ′ ) yθ n − + N xθ n +1 and δ θ r (cid:0) ( x + sy ) u n (cid:1) = δ θ ( xθ n + yθ n − )= ( d + b ) xθ n − ( d + b ′ ) yθ n − + N xθ n +1 + (1 − t ) yθ n = (cid:0) ( d + b ) x + (1 − t ) y (cid:1) θ n − ( d + b ′ ) yθ n − + N xθ n +1 . Combining this with the splitting (13), we obtain a chain isomorphismid ⊕ r : (cid:16) C ( A ) { u, u − } , d + b + Bu (cid:17) ∼ = −→ C ( R ) { u, u − } ⊕ (cid:16) C ( A ) { θ, θ − } , δ θ (cid:17) . The induced isomorphisms on homology for the [ u − ] and [[ u − ]] versions fit intothe commuting diagram HC ∗ [ u − ]( A ) φ ∗ ∼ = (cid:15) (cid:15) p ∗ ∼ = / / HC ∗ [ u − ] ( A ) φ ∗ (cid:15) (cid:15) (id ⊕ r ) ∗ ∼ = / / R [ u − ] ⊕ HC ∗ [ θ − ] ( A ) id ⊕ φ ∗ ∼ = (cid:15) (cid:15) HC ∗ [[ u − ]] ( A ) p ∗ / / HC ∗ [[ u − ]] ( A ) (id ⊕ r ) ∗ ∼ = / / R [ u − ] ⊕ HC ∗ [[ θ − ]] ( A ) . Here the vertical maps φ ∗ come from Lemma 2.13 and the upper horizontal map p ∗ is an isomorphism by the first assertion above. It follows that the lower horizontalmap p ∗ is an isomorphism as well. An analogous argument for the [[ u, u − ] and[[ u, u − ]] versions concludes the proof. (cid:3) Corollary 2.19 (reduced cyclic homology) . For an augmented dga ( A, d ) we havea canonical splitting HC ∗ [[ u ]] ( A ) = HC ∗ [[ u ]] ( R ) ⊕ HC ∗ [[ u ]] ( A ) , and similarly for the [[ u, u − ] , [ u − ] , [[ u − ]] and [[ u, u − ]] versions. Moreover, HC ∗ [ u,u − ]] ( A ) = 0 and the connecting homomorphism in the corresponding tauto-logical exact sequence splits into isomorphisms HC ∗ +1[[ u − ]] ( A ) ∼ = HC ∗ λ ( A ) = HC ∗ λ ( A ) ∼ = HC ∗ [ u ] ( A ) . Proof.
The first assertion follows immediately from the splitting (14) and Proposi-tion 2.18. The second assertion follows from the following variant of the diagramin Lemma 2.12 in reduced cyclic homology: HC ∗ [[ u − ]] ( A ) ∼ = r ∗ (cid:15) (cid:15) B / / HC ∗− u ] ( A ) ∼ = r ∗ (cid:15) (cid:15) HC ∗ [[ θ − ]] ( A ) ∼ = e ∗ (cid:15) (cid:15) B ∼ = / / HC ∗− θ ] ( A ) H ∗ (cid:0) C ( A ) / im (1 − t ) , d + b (cid:1) N ∗ ∼ = / / H ∗ (im N, d + b ′ ) . ∼ = s ∗ O O Here B = sN , the maps r are the chain isomorphisms from the proof of Propo-sition 2.18, and the other maps are isomorphisms by Lemma 2.12 applied to thecyclic complex of A . Since H ∗ +1 (cid:0) C ( A ) / im (1 − t ) , d + b (cid:1) = HC ∗ λ ( A ) = HC ∗ λ ( A ),this proves the chain of isomorphisms. Vanishing of HC ∗ [ u,u − ]] ( A ) now follows fromthe tautological exact sequence. (cid:3) From S -spaces to mixed complexes. In this subsection we recall themixed complexes arising from S -spaces and their properties. Topological S -spaces. Let Y be a topological space with an S -action φ : S × Y → Y . We make the singular cochain complex ( C ∗ ( Y ) , d ) a mixed complexby introducing the second differential P : C ∗ ( Y ) → C ∗− ( Y ) as follows. Let × : C m ( X ) ⊗ C n ( Y ) → C m + n ( X × Y ) be the Eilenberg-MacLane shuffle product,associating to f : ∆ m → X and g : ∆ n → Y the map f × g : ∆ m × ∆ n → X × Y ,( p, q ) ( f ( p ) , g ( q )) with a canonical subdivision of ∆ m × ∆ n into simplices (usingshuffles of the variables, see e.g. [14]). The shuffle product with the fundamental IGHT FLAVOURS OF CYCLIC HOMOLOGY 19 cycle F S : ∆ → S defines a map C ∗ ( Y ) → C ∗ +1 ( S × Y ), c F S × c , whosecomposition with φ ∗ : C ∗ ( S × Y ) → C ∗ ( Y ) gives a map(15) Q : C ∗ ( Y ) → C ∗ +1 ( Y ) , c φ ∗ ( F S × c ) . The operator P : C ∗ ( Y ) → C ∗− ( Y ) is dual to Q , i.e. h P α, c i := h α, φ ∗ ( F S × c ) i , α ∈ C ∗ ( Y ) , c ∈ C ∗− ( Y ) . Lemma 2.20. ( C ∗ ( Y ) , d, P ) is a mixed complex whose homology H ∗ ( Y, d ) is thesingular cohomology of Y . Proof.
To show P = 0, we compute for c ∈ C ∗ ( Y ) using associativity of the shuffleproduct Q c = φ ∗ ( F S × φ ∗ ( F S × c )) = φ ∗ ( ψ ∗ ( F S × F S ) × c ) , where ψ : S × S → S is the product ( σ, τ ) σ + τ . Now by definition ofthe shuffle product F S × F S is the difference of the singular simplices f, g : ∆ → S × S given by f ( s, t ) = ( s, t ) and g ( s, t ) = ( t, s ). This shows that ψ ∗ ( F S × F S ) =0 ∈ C ( S ), thus Q = 0 and therefore P = 0. For anticommutation of d and P note that ∂ ( F S × c ) = ∂F S × c − F S × ∂c = − F S × c because ∂F S = 0. Applying φ ∗ and dualizing we find ∂Q + Q∂ = 0 and dP + P d = 0.The statement about cohomology is clear by definition. (cid:3)
We denote the cyclic homologies of the mixed complex ( C ∗ ( Y ) , d, P ) by H ∗ [[ u,u − ]] ( Y )etc. Relation to the Borel construction.
According to the Borel construction, the S -equivariant cohomology of a topological S -space Y is defined as H ∗ S ( Y ) := H ∗ ( Y × S ES ) , where ES → BS is the universal S -bundle and Y × S ES = ( Y × ES ) /S is the quotient by the diagonal circle action. In the following proposition the firstisomorphism is shown in the discussion following Lemma 5.1 in [16], and the secondisomorphism holds because the complex C ∗ ( Y ) lives in nonnegative degrees. Proposition 2.21 (Jones [16]) . For each topological S -space Y we have canonicalisomorphisms H ∗ S ( Y ) ∼ = −−−−→ H ∗ [ u ] ( Y ) ∼ = −−−−→ H ∗ [[ u ]] ( Y ) . (cid:3) Projection to a point.
For an S -space Y with a fixed point y consider theinclusion and projection pt ι −→ Y π −→ pt , ι (pt) = y . These maps are S -equivariant and satisfy πι = id, so they induce morphisms ofmixed complexes (cid:16) C ∗ (pt) = R , d = 0 , P = 0 (cid:17) π ∗ −→ ( C ∗ ( Y ) , d, P ) ι ∗ −→ (cid:16) C ∗ (pt) = R , d = 0 , P = 0 (cid:17) satisfying ι ∗ π ∗ = id. These maps are functorial and the induced maps on cyclichomology are compatible with the Gysin and tautological sequences in the obvious In this paper, all singular (co)homology is with R -coefficients. way. For example, one tautological sequence yields the following commuting dia-gram, where all the vertical maps π ∗ are injective and we have surjective verticalmaps ι ∗ in the other direction:(16) H ∗ [[ u,u − ] ( Y ) p ∗ −−−−→ H ∗ [ u − ] ( Y ) P −−−−→ H ∗− u ]] ( Y ) · u −−−−→ H ∗ +1[[ u,u − ] ( Y ) x π ∗ x π ∗ x π ∗ x π ∗ R [ u, u − ] p ∗ −−−−→ R [ u − ] −−−−→ R [ u ] · u −−−−→ R [ u, u − ] . For a loop space Y = LX , the first group in this diagram can be explicitly com-puted: Theorem 2.22 (Goodwillie [12]) . For a path connected space X , the group H ∗ [[ u,u − ] ( LX ) depends only on π ( X ) . In particular, for X simply connected the projection LX → pt to a point induces an isomorphism H ∗ [[ u,u − ] ( LX ) ∼ = H ∗ [[ u,u − ] (pt) = R [ u, u − ] . Smooth S -spaces. Let now Y be a differentiable space in the sense of Chen [6]equipped with a smooth circle action. For example, Y could be a finite dimensionalmanifold or an infinite dimensional Banach or Fr´echet manifold. Our main exampleis the space Y = LX of smooth maps S → X into a finite dimensional manifold X with the natural circle action φ ( s, γ ) := γ ( · + s ) by reparametrization. FollowingChen [6], to such a smooth S -space Y we associate a mixed complex (Ω ∗ ( Y ) , d, P )as follows. Let (Ω ∗ ( Y ) , d ) be the space of differential forms on Y with exteriorderivative. The inner product with the vector field v on Y generating the circleaction gives a map ι : Ω ∗ ( Y ) → Ω ∗− ( Y ). By Cartan’s formula, dι + ιd = L v is theLie derivative in direction of v . Let A : Ω ∗ ( Y ) → Ω ∗ ( Y ) be the operator averagingover the circle action. In view of the relations ιA = Aι and dA = Ad , Cartan’sformula implies that P := Aι : Ω ∗ ( Y ) → Ω ∗− ( Y )satisfies dP + P d = 0. The relation ι = 0 implies P = 0, so (Ω ∗ ( Y ) , d, P ) isindeed a mixed complex. It is shown in [6] that for “nice” smooth S -spaces (suchas loop spaces of manifolds) the homology of (Ω ∗ ( Y ) , d ) is isomorphic to the singularcohomology H ∗ ( Y ; R ).To a smooth S -space Y we can associate another canonical mixed complex (Ω ∗ S ( Y ) , d, ι ),where Ω ∗ S ( Y ) denotes the space of S -invariant forms. Proposition 2.23 (Jones–Petrack [17]) . The inclusion Ω ∗ S ( Y ) ֒ → Ω ∗ ( Y ) inducesa homotopy equivalence of mixed complexes (Ω ∗ S ( Y ) , d, ι ) → (Ω ∗ ( Y ) , d, P ) . In particular, it induces isomorphisms on all versions of cyclic cohomology.
Fixed point free S -actions. Let Y be a smooth S -space without fixed points.Then there exists a connection form, i.e., an invariant 1-form α ∈ Ω S ( Y ) satisfying α ( v ) = 1 for the vector field v generating the action. The wedge product with α then defines a chain homotopy H : Ω ∗ S ( Y ) → Ω ∗ +1 S ( Y ) satisfying ιH + Hι = id . Consider the tautological sequence H ∗ +1[ u,u − ]] ( Y ) p ∗ −−−−→ H ∗ +1[[ u − ]] ( Y ) P −−−−→ H ∗ [ u ] ( Y ) · u −−−−→ H ∗ +2[ u,u − ]] ( Y ) . The map P is the composition of the first two maps in H ∗ +1[[ u − ]] ( Y ) −−−−→ H ∗ (ker ι ) = H ∗ ( Y /S ) −−−−→ H ∗ [ u ] ( Y ) ∼ = −−−−→ H ∗ [[ u ]] ( Y ) , IGHT FLAVOURS OF CYCLIC HOMOLOGY 21 where the last map is an isomorphism because the complex Ω ∗ S ( Y ) lives in non-negative degrees. The homotopy formula for H shows that the double complex(Ω ∗ S ( Y )[ u, u − ]] , d + ιu ) for the computation of H ∗ [ u,u − ]] ( Y ) has exact rows. Nowa standard zigzag argument as in the proof of Lemma 2.12 yields Lemma 2.24.
Let Y be a smooth S -space without fixed points. Then H ∗ [ u,u − ]] ( Y ) =0 and we have canonical isomorphisms H ∗ +1[[ u − ]] ( Y ) ∼ = −−−−→ H ∗ ( Y /S ) ∼ = −−−−→ H ∗ [ u ] ( Y ) ∼ = −−−−→ H ∗ [[ u ]] ( Y ) . (cid:3) Remark . Consider a smooth S -space Y (possibly with fixed points). Thenthe canonical projection Y × ES → Y induces a morphism of mixed complexes(Ω ∗ S ( Y ) , d, ι ) → (Ω ∗ S ( Y × ES ) , d, ι )which induces an isomorphism on d -homology, and thus on the [[ u ]], [ u − ] and[[ u, u − ] versions of cyclic homology. Since Y × ES has no fixed points, applyingLemma 2.24 to it provides an alternative proof of Proposition 2.21 in the smoothcase. Fixed point localization.
The version H ∗ [[ u,u − ]] ( Y ) satisfies the following fixedpoint localizaton theorem. Theorem 2.26 (Jones–Petrack [17]) . Let Y be a smooth S -space whose fixed pointset F ⊂ Y is a smooth submanifold which has an S -invariant tubular neighbour-hood. Then the inclusion c : F ֒ → Y induces an isomorphism c ∗ : H ∗ [[ u,u − ]] ( Y ) ∼ = −→ H ∗ ( F ) ⊗ R [ u, u − ] . In particular, for a manifold X the inclusion c : X ֒ → LX of the constant loopsinduces an isomorphism c ∗ : H ∗ [[ u,u − ]] ( LX ) ∼ = −→ H ∗ ( X ) ⊗ R [ u, u − ] . Examples.
In this subsection we work out the different flavours of cyclichomology for some examples. In the case of a dga A we will use the equivalent signconvention (6), dropping the ∼ ’s, which spells out as b ′ ( a | . . . | a n ) = n − X i =0 ( − i ( a | . . . | a i a i +1 | . . . | a n ) ,b ( a | . . . | a n ) = b ′ ( a | . . . | a n ) + ( − n +deg a n (deg a + ··· +deg a n − ) ( a n a | a | . . . | a n − ) ,t ( a | . . . | a n ) = ( − n +deg a n (deg a + ··· +deg a n − ) ( a n | a | . . . | a n − ) ,s ( a | . . . | a n ) = (1 | a | . . . | a n ) , Example . Consider the trivial dga ( A = R , d =0) sitting in degree 0, which is the de Rham complex of a point. Its Hochschildcomplex C ( R ) has the basis 1 n := (1 | . . . |
1) of words with n | n | = 1 − n ,for n ≥
1. From the definitions (and being careful about signs) we read off the u u u u u u u u u u bbbb BB B B Figure 2.
Cyclic complex of the trivial dga ( A = R , d = 0)operations b (1 n ) = ( n − n ≥ , ,t (1 n ) = ( − n − n ,N (1 n ) = ( n n n odd , ,sN (1 n ) = ( n n +1 n odd , ,B (1 n ) = ( n n +1 n odd , . This gives us the mixed complex ( C = C ( R ) , b, B ). As shown in Figure 2, cyclesin the complex ( C [[ u, u − ]] , δ u = b + uB ) consist of zigzags in the lower half planestarting on the horizontal axis and extending indefinitely to the right. IGHT FLAVOURS OF CYCLIC HOMOLOGY 23
Thus it makes no difference whether we consider polynomials or power series in u − . Using this, we read off the Hochschild and cyclic homology groups (where [ i ]denotes degree shift by i ) H ( C, δ ) = R ,HC [ u ] = R [ u − ][ −
1] generated by 1 ∼ u ∼ · · · ,HC [ u,u − ] = HC [ u,u − ]] = 0 ,HC [ u − ] = HC [[ u − ]] = R [ u − ] generated by u − k u − k +1 + · · · + 1 k +1 , k ≥ ,HC [[ u ]] = R [ u ] generated by 1 + u + u · · · ,HC [[ u,u − ]] = HC [[ u,u − ] = R [ u, u − ] generated by 1 + u + u · · · . So we see two types of tautological sequences:0 / / HC ∗− u ]] · u / / HC ∗ [[ u,u − ] p ∗ / / HC ∗ [ u − ] / / / / R [ u ][ − · u / / R [ u, u − ] p ∗ / / R [ u − ] / / HC ∗ [ u,u − ] / / HC ∗ [ u − ] D ∼ = / / HC ∗− u ] / / HC ∗ +1[ u,u − ] / / R [ u − ] · ∼ = / / R [ u − ] · / / . The normalized complex is given by C ( R ) = R sitting in degree 0 and C n ( R ) = 0for n ≥
1, so the normalized cylic homologies are HC [ u ] = HC [[ u ]] = R [ u ] , HC [ u − ] = HC [[ u − ]] = R [ u − ] ,HC [ u,u − ] = HC [[ u,u − ] = HC [ u,u − ]] = HC [[ u,u − ]] = R [ u, u − ] . Example . Consider the trivial mixed complex( C = R , δ = D = 0), which is the singular cochain complex of a point viewed as an S -space. The homologies can be directly read off to be H ( C, δ ) = R ,HC [ u ] = HC [[ u ]] = R [ u ] ,HC [ u − ] = HC [[ u − ]] = R [ u − ] ,HC [ u,u − ] = HC [ u,u − ]] = HC [[ u,u − ] = HC [[ u,u − ]] = R [ u, u − ] . Here all tautological sequences look like (17) above.
Example ES ) . Consider the mixed complex (
C, δ, D ),where C = Λ[ α, β ] is the exterior algebra in two generators of degrees | α | = 1 and | β | = 2 with the operations defined by the Leibniz rule and δα = β, δβ = 0 , Dα = 1 , Dβ = 0 . This is the Cartan–Weil model for the classifying space ES = S ∞ with its standard S -action, see [1]. As shown in Figure 3, cycles in the complex ( C [[ u, u − ]] , δ u = δ + uD ) consist of zigzags in the upper half plane starting on the horizontal axisand extending indefinitely to the left. δδ Du − u − αu − βu − αβ αβαββ u Figure 3.
Cyclic complex of singular cochains on ES Thus it makes no difference whether we consider polynomials or power series in u .Using this, we read off the Hochschild and cyclic homology groups H ( C, δ ) = R ,HC [[ u ]] = HC [ u ] = R [ u ] generated by 1 (note that β ∼ u ) ,HC [[ u,u − ]] = HC [ u,u − ]] = 0 ,HC [[ u − ]] = R [ u ][1] generated by α (1 + u − β + u − β + · · · ,HC [[ u,u − ] = HC [ u,u − ] = R [ u, u − ] generated by 1 ,HC [ u − ] = R [ u − ] generated by 1 , u − , u − , . . . . So we see two types of tautological sequences: −−−−→ HC ∗− u ] · u −−−−→ HC ∗ [ u,u − ] p ∗ −−−−→ HC ∗ [ u − ] 0 −−−−→ = y = y = y −−−−→ R [ u ][ − · u −−−−→ R [ u, u − ] p ∗ −−−−→ R [ u − ] −−−−→ IGHT FLAVOURS OF CYCLIC HOMOLOGY 25 and HC ∗ [ u,u − ]] −−−−→ HC ∗ [[ u − ]] D −−−−→ ∼ = HC ∗− u ] −−−−→ HC ∗ +1[ u,u − ]]= y = y = y = y −−−−→ R [ u − ] · α α −−−−→ ∼ = R [ u − ] · −−−−→ . Lemma 2.30.
The mixed complexes in Examples (2.27) , (2.28) and (2.29) arequasi-isomorphic.Proof. The morphisms of mixed complexes( C ( R ) , δ = b, D = B ) → ( R , δ = D = 0) , n ( n = 1 , n ≥ R , δ = D = 0) → (Λ[ θ, ω ] , δ, D ) , H ( C, δ )equal R . (cid:3) Comparison of the homology groups in these three examples shows that the 5remaining homologies in Proposition 2.3 are not quasi-isomorphism invariants ofmixed complexes. 3.
Chen’s iterated integral
In this section we study the behaviour of the various flavours of cyclic homologyunder Chen’s iterated integral, introduced by K.T. Chen in [5, 6].3.1.
Chen’s iterated integral as a map of mixed complexes.
Let X be aconnected oriented manifold and LX its free loop space. For each k ∈ N the Chenpairing (18) h· , ·i : Ω ∗ + k ( X k +1 ) × C ∗ ( LX ) −→ R between differential forms on the ( k + 1)-fold product X k +1 and singular chains on LX is defined as follows. Recall the standard simplex∆ k = { ( t , ..., t k ) ∈ R k | ≤ t ≤ t ≤ ... ≤ t k ≤ } , ∆ = { } with its face maps (parametrizing the boundary faces) δ j : ∆ k − −→ ∂ ∆ k definedby δ ( t , ..., t k − ) = (0 , t , ..., t k − ) , δ k ( t , ..., t k − ) = ( t , ..., t k − , ,δ j ( t , ..., t k − ) = ( t , ..., t j , t j , ..., t k − ) , j = 1 , ..., k − . We give ∆ k the induced orientation from R k . Then δ is orientation reversing, δ isorientation preserving, and in general δ j changes orientation by ( − j +1 . Considera simplex B and a continuous map f : B −→ LX.
For any k ≥ ev f : B × ∆ k −→ X k +1 , ev f ( p, t , ..., t k ) := (cid:0) f ( p )(0) , f ( p )( t ) , ..., f ( p )( t k ) (cid:1) . Given ω ∈ Ω ∗ ( X k +1 ) and such a map f , we define the pairing (18) as h ω, f i := Z B × ∆ k ev ∗ f ω. It gives rise to a degree preserving linear map,
Chen’s iterated integral (19) I : M n ≥ Ω ∗ + n ( X n +1 ) → C ∗ ( LX ) , ( Iω )( f ) := h ω, f i . Recall from Example 2.7 that the spaces Ω ∗ ( X n +1 ) for n ∈ N form a cycliccochain complex, so according to Lemma 2.11 they give rise to a mixed complex( L n ≥ Ω ∗ ( X n +1 ) , d H = d + b, B ). The other side of equation (19) carries thestructure of a mixed complex ( C ∗ ( LX ) , d, P ) provided by Lemma 2.20. Proposition 3.1.
Chen’s iterated integral I defines a morphism of mixed complexes I : (cid:16)M n ≥ Ω ∗ ( X n +1 ) , d H , B (cid:17) −→ ( C ∗ ( LX ) , d, P ) . Proof.
Consider ω ∈ Ω ∗ ( X k +1 ) and f : B → LX as above. For compatibility withthe differentials, we need to show(20) h d H ω, f i = h ω, ∂f i . Using Stokes’ theorem, we rewrite the left hand side as h d H ω, f i = Z B × ∆ k ev ∗ f dω + Z B × ∆ k − ev ∗ f ( bω )= Z ∂ ( B × ∆ k ) ev ∗ f ω + Z B × ∆ k − ev ∗ f ( bω )= Z ∂B × ∆ k ev ∗ f ω + ( − dim B Z B × ∂ ∆ k ev ∗ f ω + Z B × ∆ k − ev ∗ f ( bω )= h ω, ∂f i + (cid:18) ( − | ω | +1 Z B × ∂ ∆ k ev ∗ f ω + Z B × ∆ k − ev ∗ f ( bω ) (cid:19) . Here for the signs in the last step we have used that deg dω = dim( B × ∆ k ) andthus dim B = deg ω + 1 − k = | ω | + 1. So for (20) it remains to show(21) Z B × ∆ k − ev ∗ f ( bω ) = ( − | ω | Z B × ∂ ∆ k ev ∗ f ω. For this, observe that the face maps δ j : X k → X k +1 from Example (2.6) and theface maps δ j : ∆ k − → ∂ ∆ k satisfy for each j = 0 , . . . , k the relation ev f ◦ (id × δ j ) = δ j ◦ ev f : B × ∆ k − → X k +1 . Using this and the fact that δ j : ∆ k − → ∂ ∆ k changes orientation by ( − j +1 weobtain Z B × ∆ k − ev ∗ f δ ∗ j ω = Z B × ∆ k − (id × δ j ) ∗ ev ∗ f ω = ( − j +1 Z B × δ j (∆ k − ) ev ∗ f ω. Now recall that bω = P kj =0 ( − | ω | + j +1 δ ∗ j ω and ∂ ∆ k = S kj =0 δ j (∆ k − ). Multiplyingboth sides of the last displayed equation with ( − j + | ω | +1 and summing over j =0 , . . . , k thus yields equation (21).Next we show compatibility with the BV operators(22) ( I ◦ B )( ω )( f ) = ( P ◦ I )( ω )( f )for ω ∈ Ω ∗ ( X k ) and f : B → LX . We first discuss the left hand side of (22).Recall that Bω = (1 − t ) sN ω = ( − | ω | (1 − t ) σ ∗ N ω , where σ : X k +1 → X k isthe projection forgetting the zero-th factor and t = ( − k τ ∗ k with τ k : X k +1 → X k +1 given by τ k ( x , . . . , x k ) = ( x , . . . , x k , x ). It follows that the form tsN ω ∈ Ω ∗ ( X k +1 ) is independent of the variable x , so the contraction of its pullback IGHT FLAVOURS OF CYCLIC HOMOLOGY 27 ev ∗ f tsN ω with the coordinate vector field ∂ t on ∆ k is zero, and therefore theintegrand ev ∗ f tsN ω vanishes pointwise. This shows that I ( tsN ω ) = 0, and thus(23) ( I ◦ B )( ω )( f ) = h (1 − t ) sN ω, f i = ( − | ω | Z B × ∆ k d ev f ∗ N ω with d ev f := σ ◦ ev f : B × ∆ k −→ X k . By definition of ev f and σ we have(24) d ev f ( p, t , ..., t k ) = ( f ( p )( t ) , ..., f ( p )( t k )) . Let us now rewrite the right hand side of (22). Recall that the pushforward φ ∗ ( F S × f ) is given by the map ∆ × B → LX sending ( s, p ) to the loop f ( p )( · + s ).Switching the order of ∆ and B , let us define the map S f : B × S → LX by S f ( p, s )( t ) := f ( p )( t + s ). Since switching the order changes orientation by( − dim B , we obtain(25) ( P ◦ I )( ω )( f ) = h ω, φ ∗ ( F S × f ) i = ( − dim B Z B × (∆ × ∆ k ) ev ∗ S f ω. To proceed further we need to relate the two maps ev S f and d ev f and their domainsof definition. For this we split ∆ × ∆ k into the k + 1 regions R i := { ( s, t , . . . , t k ) | t i − ≤ − s ≤ t i } , i = 0 , . . . , k. Then S ki =0 R i = ∆ × ∆ k with R i ∩ R j of measure zero for i = j . In R i , thecondition t i − + s ≤ ≤ t i + s implies the ordering0 ≤ t i + s − ≤ · · · ≤ t k + s − ≤ s ≤ t + s ≤ · · · ≤ t i − + s ≤ , so R i is naturally diffeomorphic to ∆ k +1 via the map r i : R i −→ ∆ k +1 , r i ( s, t , ..., t k ) := ( t i + s − , . . . , t k + s − , s, t + s, . . . , t i − + s ) . Note that r i is orientation preserving if and only if ik is even. Unwrapping thedefinition of the various maps we find that they satisfy the relation ev S f = τ − ik ◦ d ev f ◦ (id × r i ) , with the permutation map τ k from above. Using t k = τ ∗ k and and invariance ofintegration with respect to id × r i , we deduce( − dim B Z B × R i ev ∗ S f ω = ( − dim B Z B × R i (id × r i ) ∗ d ev f ∗ t − ik ω = ( − dim B + ik Z B × ∆ k +1 d ev f ∗ ( t − ik ω ) . Since t = ( − k t k satisfies t k +1 = 1, we can write N = P ki =0 t − i = P ki =0 ( − ik t − ik .Summing up the previous displayed equation over i = 0 , . . . , k and using equa-tion (25) we therefore obtain( P ◦ I )( ω )( f ) = ( − dim B Z B × ∆ k +1 d ev f ∗ ( N ω ) . Now observe that dim B + k +1 = deg ω , and thus dim B = | ω |− ω ∈ Ω ∗ ( X k ).Comparing with equation (23), this finishes the proof of equation (22) and thus ofProposition 3.1. (cid:3) Chen’s iterated integral on Connes’ version.
On Connes’ version ofcyclic homology, Chen’s iterated integral is induced by the cyclic Chen pairing (26) h· , ·i cyc : Ω ∗ + k ( X k ) × C ∗ ( LX ) −→ R . It is defined on ω ∈ Ω ∗ + k ( X k ) and f : B → LX by h ω, f i cyc := ( − | ω | Z B × ∆ k cyc d ev f ∗ ω, where ∆ k cyc denotes the space of tuples ( t , . . . , t k ) ∈ ( S ) k in the cyclic order t ≤ t ≤ · · · ≤ t k ≤ t and d ev f is defined in (24). The Connes (or cyclic) versionof Chen’s iterated integral is the degree preserving map I λ : M n ≥ Ω ∗ + n ( X n ) −→ C ∗ ( LX ) , I λ ( ω )( f ) := h ω, f i cyc . Lemma 3.2. (a) I λ is given by the composition I λ : M n ≥ Ω ∗ + n ( X n ) B −→ M n ≥ Ω ∗ + n ( X n +1 ) I −→ C ∗ ( LX ) . In particular, I λ = I ◦ B : (cid:0) C ∗ (Ω( X )) , d + b (cid:1) −→ (cid:0) C ∗ ( LX ) , d (cid:1) is a chain map.(b) I λ vanishes on im (1 − t ) and thus descends to C ∗ λ (Ω( X )) = C ∗ +1 (Ω( X )) / im (1 − t ) .(c) The following diagram commutes, where π denotes the quotient projection ofthe constant term in u − : C ∗ +1 (Ω( X ))[ u − ] π −−−−→ C ∗ λ (Ω( X )) B −−−−→ C ∗ (Ω( X ))[[ u ]] I y y I λ y I C ∗ +1 ( LX )[ u − ] P π −−−−→ C ∗ ( LX ) ι −−−−→ C ∗ ( LX )[[ u ]] . (d) For smooth maps f : B → LX and σ : B → S define f σ : B → LX by f σ ( p )( t ) := f ( p )( t + σ ( p )) . Then for each ω ∈ Ω ∗ ( X k ) we have h I λ ω, f σ i = h I λ ω, f i . Proof.
For part (a) we decompose∆ n cyc = n [ i =1 ∆ ni , ∆ ni := { ( t , . . . , t n ) | t i ≤ ≤ t i +1 } where each ∆ ni is isomorphic to the standard simplex via the permutation map τ in − : ∆ n ∼ = → ∆ ni . Now for ω ∈ Ω ∗ + n ( X n ) and f : B → LX we compute h ω, f i cyc = ( − | ω | Z B × ∆ n cyc d ev f ∗ ω = ( − | ω | n X i =1 ( − ( n − i Z B × ∆ n d ev f ∗ ( τ in − ) ∗ ω = ( − | ω | Z B × ∆ n d ev f ∗ ( n X i =1 t i ω ) = ( − | ω | Z B × ∆ n d ev f ∗ ( N ω ) = h Bω, f i , where the last equality follows from (23). This proves I λ = I ◦ B , and this is achain map because I (by Proposition 3.1) and B are chain maps.Part (b) follows from B (1 − t ) = 0, which holds because B = (1 − t ) sN and N (1 − t ) = 0. In part (c) commutativity of the first square follows by applying π to I λ = I ◦ B = P ◦ I (which holds by Proposition 3.1), and commutativity of the IGHT FLAVOURS OF CYCLIC HOMOLOGY 29 second square follows from part (a). For part (d), note that ev f σ = ev f ◦ ρ with theorientation preserving diffeomorphism ρ : B × ∆ k cyc → B × ∆ k cyc , ( p, t , . . . , t k ) (cid:16) p, t + σ ( p ) , . . . , t k + σ ( p ) (cid:17) , so part (d) follows from invariance of integration under diffeomorphisms. (cid:3) Consider the following diagram: LX × ES π (cid:15) (cid:15) pr / / LXB e g sssssssssss g / / LX × S ES Using this, we define h I λ ω, g i := h I λ ω, pr ◦ e g i for ω ∈ Ω ∗ ( X k ) and e g : B → LX × ES any lift of g : B → LX × S ES , whichexists for each simplex B because π : LX × ES → LX × S ES is a circle bundle.Any other lift of g is of the form e g σ ( p ) = σ ( p ) · e g ( p ) for a smooth map σ : B → S (where · denotes the circle action on LX × ES ), and applying Lemma 3.2 to pr ◦ e g σ = ( pr ◦ e g ) σ we see that the definition does not depend on the lift e g anddefines a chain map I λ : (cid:0) C ∗ λ (Ω( X )) , d + b (cid:1) −→ (cid:0) C ∗ ( LX × S ES ) , d (cid:1) . Note that the cohomology of the right hand side is the equivariant cohomology H ∗ S ( LX ) = H ∗ ( LX × S ES , d ) of LX defined via the Borel construction. Passingto homology in Lemma 3.2(c) we thus obtain the following commuting diagram,where the maps π ∗ and ι ∗ are isomorphisms by Lemma 2.12 and Proposition 2.21,respectively:(27) HC ∗ +1[ u − ] (Ω( X )) π ∗ −−−−→ ∼ = HC ∗ λ (Ω( X )) B ∗ −−−−→ HC ∗ [[ u ]] (Ω( X )) y I ∗ y I λ ∗ y I ∗ H ∗ +1[ u − ] ( LX ) P ∗ −−−−→ H ∗ S ( LX ) ι ∗ −−−−→ ∼ = H ∗ [[ u ]] ( LX ) . Chen’s iterated integral for simply connected manifolds.
In this sub-section we assume in addition that X is simply connected. This has two importantconsequences. First, by Goodwillie’s Theorem 2.22 the projection LX → pt to apoint induces an isomorphism H ∗ [[ u,u − ] ( LX ) ∼ = R [ u, u − ]. For the second conse-quence, recall that composition of the map I from Proposition 3.1 with the mor-phism φ from Corollary 2.16 yields a morphism of mixed complexes (still denoted I ) I : (cid:0)L n ≥ Ω ∗ ( X ) ⊗ n +1 , d H , B (cid:1) → ( C ∗ ( LX ) , d, P ) . The following theorem is provedin [11] and attributed to Chen, see also Jones [16].
Theorem 3.3 (Getzer, Jones and Petrack [11]) . Let X be a simply connectedmanifold. Then Chen’s iterated integral I : (cid:16) C ∗ (Ω( X )) , d H , B (cid:17) −→ ( C ∗ ( LX ) , d, P ) induces an isomorphism on Hochschild homology, hence defines a quasi-isomorphismof mixed complexes. Corollary 3.4.
In the situation of Theorem 3.3, Chen’s iterated integral I inducesisomorphisms on the classical [[ u ]] , [[ u, u − ] and [ u − ] versions of cyclic homology.On the other five versions it does not induce an isomorphism in general. Proof.
The first assertion follows directly from Theorem 3.3 combined with Propo-sition 2.3. For the second assertion, comparison of Examples 2.27 and 2.28 showsthat I does not induce isomorphisms on the [ u ], [ u, u − ] and [ u, u − ]] versions for X = { pt } . The remaining two versions fit into the commuting diagram of tauto-logical sequences · · · HC ∗ [[ u ]] (Ω( X )) I ∗ ∼ = (cid:15) (cid:15) / / HC ∗ [[ u,u − ]] (Ω( X )) I ∗ (cid:15) (cid:15) / / HC ∗ [[ u − ]] (Ω( X )) · · · I ∗ (cid:15) (cid:15) · · · H ∗ [[ u ]] ( LX ) / / H ∗ [[ u,u − ]] ( LX ) / / H ∗ [[ u − ]] ( LX ) · · · Now HC ∗ [[ u,u − ]] (Ω( X )) ∼ = HC ∗ [[ u,u − ] (Ω( X )) ∼ = H ∗ [[ u,u − ] ( LX ) ∼ = R [ u, u − ] wherethe first isomorphism follows from Corollary 2.14, the second one is given by I ∗ ,and the last one follows from Goodwillie’s theorem 2.22. On the other hand, byTheorem 2.26 we have H ∗ [[ u,u − ]] ( LX ) ∼ = H ∗ ( X ) ⊗ R [ u, u − ]. For X noncontractiblethese two R [ u ]-modules differ, hence the middle map I ∗ in the diagram is notan isomorphism. It follows from the five-lemma that the right map I ∗ is not anisomorphism either. (cid:3) The following theorem, which is the main result of this paper, describes the be-haviour of Connes’ version under Chen’s iterated integral.
Theorem 3.5.
Let X be a simply connected manifold. Then the kernel and imageof the iterated integral I λ ∗ : HC ∗ λ (Ω( X )) → H ∗ S ( LX ) are given by ker I λ ∗ = span { [1] , [1 ] , [1 ] , . . . } ∼ = R [ u − ] , im I λ ∗ = ker (cid:16) ι ∗ : H ∗ S ( LX ) → R [ u ] (cid:17) ∼ = H ∗ S ( LX ) . R [ u ] . Proof.
Consider the commuting diagram HC ∗ λ (Ω( X )) I λ ∗ / / H ∗ S ( LX ) ι ∗ ∼ = (cid:15) (cid:15) H ∗ +1[[ u,u − ] ( LX ) ∼ = (cid:15) (cid:15) p ∗ / / H ∗ +1[ u − ] ( LX ) π I − ∗ ∼ = O O P ∗ / / H ∗ [[ u ]] ( LX ) ι ∗ (cid:15) (cid:15) · u / / H ∗ +2[[ u,u − ] ( LX ) ∼ = (cid:15) (cid:15) R [ u, u − ] p ∗ / / R [ u − ] π ∗ O O / / R [ u ] · u / / R [ u, u − ] . Here the lower two rows follow from (16) for Y = LX where π ∗ is injective, ι ∗ is surjective, and the outer vertical maps are isomorphisms by Goodwillie’s Theo-rem 2.22. The upper square follows from diagram (27), where I ∗ is an isomorphismby Corollary 3.4 and we have abbreviated ι P as P . We read offker P ∗ = π ∗ R [ u − ] = span { u − k | k ∈ N } , im P ∗ = ker (cid:16) ι ∗ : H ∗− u ]] ( LX ) → R [ u ] (cid:17) , and thereforeker I λ ∗ = π ∗ I − ∗ ker P ∗ = π ∗ span { u − k u − k +1 + · · · + 1 k +1 | k ∈ N } = span { [1] , [1 ] , [1 ] , . . . } , im I λ ∗ = im P ∗ = ker (cid:16) ι ∗ : H ∗− S ( LX ) → R [ u ] (cid:17) . IGHT FLAVOURS OF CYCLIC HOMOLOGY 31
Here the description of I − ∗ ker P ∗ and its image under π ∗ follow from Exam-ple 2.27. (cid:3) Corollary 3.6.
Let X be a simply connected manifold. Denote by HC ∗ λ (Ω( X )) thereduced Connes version of cyclic homology of the de Rham complex of X , and by H ∗ S ( LX, x ) the S -equivariant cohomology of LX relative to a fixed constant loop x . Then Chen’s iterated integral induces an isomorphism I λ ∗ : HC ∗ λ (Ω( X )) ∼ = −→ H ∗ S ( LX, x ) . Proof.
Passing to reduced homologies, the commuting diagram in the proof of The-orem 3.5 simplifies to HC ∗ λ (Ω( X )) I λ ∗ / / H ∗ S ( LX.x ) ι ∗ ∼ = (cid:15) (cid:15) / / H ∗ +1[ u − ] ( LX, x ) π I − ∗ ∼ = O O P ∗ ∼ = / / H ∗ [[ u ]] ( LX, x ) / / (cid:3) Computations for spheres.
For a simply connected manifold X , the S -equivariant cohomology of LX can be computed via minimal models as follows, seee.g. [2]. Let ( M = Λ[ x , x , . . . ] , d ) be the minimal model of X . To it we asso-ciate a mixed complex ( LM, d, B ) by setting LM := Λ[ x , ¯ x , x , ¯ x · · · ] with newgenerators x i of degrees deg x i = deg x i −
1, defining B as the derivation satisfy-ing Bx i = x i and Bx i = 0, and extending d from M to LM by the requirement dB + Bd = 0, i.e., by defining dx i := − B ( dx i ). Then ( LM [ u ] , d u = d + uB ) is theminimal model for the Borel space LX × S ES , so it computes H ∗ S ( LX ).Recall that a manifold X is formal over R (in the sense of rational homotopy theory)if its de Rham dga Ω ∗ ( X ) is connected to its cohomology H ∗ ( X ) by a zigzag ofquasi-isomorphisms, cf. [13]. By Proposition 2.15, all versions of cyclic homologyof Ω ∗ ( X ) can then be computed using the dga (with trivial differential) H ∗ ( X ). Example . The sphere S n with n ≥ a ] with deg a = n and da = 0. So the minimal model of LS n × S ES isΛ[ a, ¯ a, u ] with differential d u ¯ a = 0 and d u a = u ¯ a . It follows that H ∗ S ( LS n ) = Λ[¯ a, u ] . h u ¯ a i = ¯ a R [¯ a ] ⊕ R [ u ] , where u acts trivially on the first summand. Relative to a point x ∈ S n thisbecomes H ∗ S ( LS n , x ) = ¯ a R [¯ a ] , deg ¯ a = n − . Let us now compute the reduced Connes version of cyclic homology of Ω ∗ ( S n ).Since S n is formal, we can compute this from its cohomology H ( S n ) = R ⊕ R v ,where deg v = n , or rather the reduced homology H ( S n ) = R v . For k ≥ v k the word with k letters v . Since t ( v k ) = ( − ( k − n ( k − v k = v k , each v k survives in the quotient by cyclic permutation, and by definition of HC ∗ λ (withtrivial Hochschild differential) we get HC ∗ λ (Ω( S n )) = HC ∗ λ ( H ( S n )) = v R [ v ] , | v | = n − . This is compatible with Theorem 3.5 if I λ ∗ sends v to ¯ a . Example . The sphere S n with n ≥ a, b ] with deg a = n , deg b = 2 n − da = 0, db = a . Sothe minimal model of LS n × S ES is Λ[ a, ¯ a, b, ¯ b, u ] with differential d u a = u ¯ a , d u b = a + u ¯ b , d ¯ a = 0 and d ¯ b = − a ¯ a . It follows (cf. [2]) that H ∗ S ( LS n ) = ¯ a (cid:16) Λ[¯ a, u ] . h u, a, a + u ¯ b i (cid:17) ⊕ R [ u ] = ¯ a R [¯ b ] ⊕ R [ u ] , where u acts trivially on the first summand. Relative to a point x ∈ S n thisbecomes H ∗ S ( LS n , x ) = ¯ a R [¯ b ] , deg ¯ a = n − , deg ¯ b = 2 n − . Again, we can compute the reduced Connes version of cyclic homology of Ω ∗ ( S n )from H ( S n ) = R v , where deg v = n . Since t ( v k ) = ( − ( k − n ( k − v k =( − k − v k , the word v k survives in the quotient by cyclic permutation iff k isodd, and by definition of HC ∗ λ (with trivial Hochschild differential) we get HC ∗ λ (Ω( S n )) = HC ∗ λ ( H ( S n )) = v R [ v ] , | v | = n − . This is compatible with Theorem 3.5 if I λ ∗ sends v i +1 to ¯ a ¯ b i .4. Duality
Generalities.
As before, all complexes are over the ground field R . Let ( C = L k ∈ Z C k , δ ) be a cochain complex. We dualize it in such a way that the result isa cochain complex as well, i.e. (cid:16) C ∨ := M k ∈ Z Hom( C − k , R ) , δ ∗ (cid:17) . Suppose now that ( C j , δ j ), j = 1 , h , i : C ⊗ C −→ R is a bilinear pairing of degree 0 such that the differentials are mutual adjoints withrespect to the pairing, h δ x, y i = h x, δ y i . The pairing naturally gives rise to two maps f : C −→ C ∨ , f : C −→ C ∨ by f ( x ) := h x, ·i and f ( y ) := h· , y i . Since the differentials are adjoints, the maps f and f are chain maps. Denoting by ι j : C j ֒ → ( C ∨ j ) ∨ j = 1 , f = f ∗ ◦ ι and f = f ∗ ◦ ι . We say that a graded vector space E = L k ∈ Z E k is graded finite dimensional ifeach E k is finite dimensional. Lemma 4.1.
In the above setting, assume that the homologies of C and C aregraded finite dimensional. Then f is a quasi-isomorphism if and only if f is.Proof. For each chain complex C over R we have a commuting diagram H ( C ) ι H (cid:15) (cid:15) Hι / / H (cid:0) ( C ∨ ) ∨ (cid:1) ∼ = (cid:15) (cid:15) (cid:0) H ( C ) ∨ (cid:1) ∨ ∼ = / / H ( C ∨ ) ∨ , IGHT FLAVOURS OF CYCLIC HOMOLOGY 33 where Hι is the map on homology induced by the canonical embedding ι : C ֒ → ( C ∨ ) ∨ , ι H is the canonical embedding for H ( C ), and the two isomorphisms comefrom the universal coefficient theorems. If H ( C ) is graded finite dimensional, thenthe map ι H is an isomorphism (because finite dimensional spaces are reflexive),hence so is Hι . This shows that in the situation of the lemma both canonicalembeddings ι and ι are quasi-isomorphisms. In view of equation (28) this im-plies the lemma, recalling that the dual of a quasi-isomorphism is again a quasi-isomorphism. (cid:3) Duality of mixed complexes.
We now generalize the preceding discussionto mixed complexes. The dual mixed complex to (
C, δ, D ) is defined as ( C ∨ , δ ∗ , D ∗ ).Suppose now that ( C j , δ j , D j ), j = 1 , h , i : C ⊗ C −→ R is a bilinear pairing of degree zero such that both differentials are mutual adjointswith respect to the pairing, h δ x, y i = h x, δ y i , and h D x, y i = h x, D y i . This implies that the maps f and f are morphisms of mixed complexes, andLemma 4.1 yields Corollary 4.2.
In the above setting assume that the homologies H ( C , δ ) and H ( C , δ ) are graded finite dimensional. Then f is a quasi-isomorphism of mixedcomplexes if and only if f is. (cid:3) Let now (
C, δ, D ) be a mixed complex. We want to investigate the relation betweenthe total complex of its dual and the dual of its total complex. For concreteness,let us consider the version C [ u − ]. We define a degree zero pairing(29) h , i : C ∨ [[ u ]] − k ⊗ C [ u − ] k → R , h φ, c i := X i ≥ φ i ( c − i )where φ = P i ≥ φ i u i with φ i ∈ ( C ∨ ) − k − i = Hom( C k +2 i , R ), and c = P i ≥ c − i u − i with c − i ∈ C k +2 i . Note that the sum P i ≥ φ i ( c − i ) is finite because only finitelymany c − i are nonzero. Direct computation yields Lemma 4.3.
The pairing (29) induces via ι ( φ )( c ) = h φ, c i a chain isomorphism ι : (cid:16) C ∨ [[ u ]] , δ ∗ + uD ∗ (cid:1) ∼ = −→ (cid:16) C [ u − ] ∨ , ( δ + uD ) ∗ (cid:17) respecting the R [ u ] -module structures with | u | = 2 on both sides. Similarly, weobtain the chain isomorphisms C ∨ [[ u − ]] ∼ = C [ u ] ∨ and C ∨ [[ u, u − ]] ∼ = C [ u, u − ] ∨ . (cid:3) Finally, note that for a mixed complex (
C, δ, D ) and its dual we have a commutingdiagram of chain maps (with respect to δ ∗ )im D ∗ (cid:15) (cid:15) ∼ = / / (im D ) ∨ D ∗ ∼ = / / ( C/ ker D ) ∨ (cid:15) (cid:15) ker D ∗ φ ∼ = / / ( C/ im D ) ∨ where the maps D ∗ and φ have degree −
1. On homology this yields(30) H (im D ∗ ) (cid:15) (cid:15) ∼ = / / H (im D ) ∨ ( D ∗ ) ∗ ∼ = / / H ( C/ ker D ) ∨ (cid:15) (cid:15) H (ker D ∗ ) φ ∗ ∼ = / / H ( C/ im D ) ∨ . Equivariant homology of S -spaces. Let Y be a topological S -space. Itwas shown in the proof of Lemma 2.20 that( C −∗ ( Y ) , ∂, Q )is a mixed complex, where ( C ∗ ( Y ) , ∂ ) is the singular chain complex and Q is themap (15) induced by the circle action. Note that we grade the singular chainsnegatively to give ∂ and Q degrees 1 and −
1, respectively. The homology of thismixed complex is the (negatively graded) singular homology H −∗ ( Y ), and its dualis the mixed complex ( C ∗ ( Y ) , d, P ) in Lemma 2.20. We denote the cyclic homologyof ( C −∗ ( Y ) , ∂, Q ) by H [[ u ]] −∗ ( Y ) etc.Lemma 5.1 in [16] and the fact that C −∗ ( Y ) lives in nonpositive degrees imply thefollowing dual version of Proposition 2.21. Proposition 4.4 (Jones [16]) . For each topological S -space Y we have canonicalisomorphisms H S −∗ ( Y ) ∼ = H [ u − ] −∗ ( Y ) ∼ = H [[ u − ]] −∗ ( Y ) . (cid:3) As in Section 2.5, for an S -space Y with a fixed point y the inclusion and projec-tion pt ι −→ Y π −→ pt, ι (pt) = y , induce the following commuting diagram, whereall the vertical maps ι ∗ are injective and we have surjective vertical maps π ∗ in theother direction:(31) H [[ u,u − ] −∗ ( Y ) p ∗ −−−−→ H [ u − ] −∗ ( Y ) Q −−−−→ H [[ u ]] −∗− ( Y ) · u −−−−→ H [[ u,u − ] −∗ +1 ( Y ) x ι ∗ x ι ∗ x ι ∗ x ι ∗ R [ u, u − ] p ∗ −−−−→ R [ u − ] −−−−→ R [ u ] · u −−−−→ R [ u, u − ] . Remark . Lemma 2.24 has the following dual version: If Y is a smooth S -spacewithout fixed points, then H [ u,u − ]] −∗ ( Y ) = 0 and we have canonical isomorphisms H [ u − ] −∗ ( Y ) ∼ = −−−−→ H [[ u − ]] −∗ +1 ( Y ) ∼ = −−−−→ H −∗ ( Y /S ) ∼ = −−−−→ H [ u ] −∗ ( Y ) . For Y with fixed points, applying this to Y × ES provides an alternative proof ofProposition 4.4 in the smooth case.4.4. Finiteness.
In this subsection we prove two finiteness results on homology.
Lemma 4.6.
Let ( A, d ) be a dga whose homology H ∗ ( A ) is graded finite dimen-sional. Assume in addition that dim H ( A ) = 1 and dim H ( A ) = 0 . Then theHochschild homology HH ∗ ( A ) is graded finite dimensional.Proof. Consider the word length filtration on the Hochschild complex. This filtra-tion is bounded from below and exhaustive, therefore the corresponding spectralsequence converges to HH ∗ ( A ). It is enough to show that graded finite dimension-ality holds for the second page. The first page computes to E p,q = H p ( A ⊗ ( q +1) , d )and the second page to E p,q = H q ( E , b ) p = H q ( E /D ( A ) , b ) p IGHT FLAVOURS OF CYCLIC HOMOLOGY 35 where q denotes the word length degree, p the degree in A , and D ( A ) is the acyclicsubcomplex generated by words with 1 in some positive slot considered in Sec-tion 2.4. We will show that the desired finite dimensionality holds even before wetake the homology with respect to b . Fix some degree k = p − q for the chaincomplex E /D and write out the degree k part of the complex,( E /D ) k = M p − q = k ( E /D ) p,q Since H ( A ) = 0 and we have factored out D ( A ), we have ( E /D ) p,q = 0 for p < q .So the sum runs over p that satisfy p ≥ q , in other words k = p − q ≥ q . Thisleaves us with only finitely many options for q . Therefore, we have only finitelymany nonzero summands in ( E /D ) k and thus dim( E /D ) k < ∞ . (cid:3) Note that Lemma 4.6 applies in particular to A = Ω ∗ ( X ) for a simply connectedmanifold X . Lemma 4.7. If X is a simply connected manifold, then H ∗ ( LX ) and H S ∗ ( LX ) are graded finite dimensional.Proof. Consider the Sullivan minimal model M = (Λ[ x , x , . . . ] , d ) for X , whereΛ[ x , x , . . . ] is the free graded commutative algebra on generators x i of degreesdeg x i ≥
2. Moreover, by [10, Proposition 12.2] there are only finitely manygenerators x i of any given degree. Then the minimal model for LX is LM =(Λ[ x , ¯ x , x , ¯ x , . . . ] , δ ), with deg ¯ x i = deg x i − δ . Sinceall generators of LM have strictly positive degrees, LM is graded finite dimensional,hence so is its homology H ∗ ( LX ). Graded finite dimensionality of H S ∗ ( LX ) fol-lows by the same argument from its minimal model (Λ[ x , ¯ x , x , ¯ x , . . . , u ] , δ u ),with deg u = 2 and a suitable differential δ u . (cid:3) Cyclic cohomology.
Consider a mixed complex (
C, δ, D ) and its dual mixedcomplex ( C ∨ , δ ∗ , D ∗ ). The cyclic cohomology of C is defined as HC [ u,u − ]] ∗ := H ( C ∨ ) −∗ [ u,u − ]] , and similarly for the other seven versions. Lemma 4.3 and the universal coefficienttheorem yield(32) HC [[ u ]] k = H ( C ∨ ) − k [[ u ]] ∼ = (cid:0) ( HC [ u − ] ) ∨ (cid:1) − k = Hom( HC k [ u − ] , R ) , and similarly for the other two versions in Lemma 4.3. Thus results about polyno-mial versions of cyclic cyclic homology dualize to results about the correspondingpower series versions of cyclic cohomology.Consider now a cyclic cochain complex ( C n , d i , s j , t n , d ) with its associated mixedcomplex ( C := L n ≥ C n , d + b, B ). Its Connes’ version of cyclic cohomology isdefined as HC λ ∗ := H −∗− (cid:0) ( C/ im (1 − t )) ∨ , d ∗ + b ∗ (cid:1) = H −∗− (cid:0) ker(1 − t ) ∗ , d ∗ + b ∗ (cid:1) ∼ = H −∗− (cid:0) C ∨ / im (1 − t ∗ ) , d ∗ + b ∗ (cid:1) , where the last isomorphism is induced by the inverse of the chain isomorphism N ∗ : C ∨ / im (1 − t ∗ ) ∼ = −→ im N ∗ = ker(1 − t ∗ ) ⊂ C ∨ . Recall from Corollary 2.14 theseries of canonical isomorphisms(33) HC ∗ [ u − ] ∼ = HC ∗ [[ u − ]] ∼ = HC ∗− λ ∼ = HC ∗− u ] . In view of equation (32), dualizing the first, third and fourth terms yields theisomorphisms HC [[ u − ]] ∗− ∼ = HC λ ∗− ∼ = HC [[ u ]] ∗ . Moreover, the proof of Lemma 2.12 (which uses only exactness of the rows in the θ double complex) carries over to C ∨ to yield the isomorphisms HC [[ u − ]] ∗− ∼ = HC λ ∗− ∼ = HC [ u ] ∗ . Combining these, we have proved
Lemma 4.8.
For a cyclic cochain complex ( C n , d i , s j , t n , d ) , the canonical mapson cyclic cohomology give the series of isomorphisms HC [[ u − ]] ∗− ∼ = HC λ ∗− ∼ = HC [ u ] ∗ ∼ = HC [[ u ]] ∗ . (cid:3) Note that this series of isomorphisms differs from (33) in the degrees and by theappearance of the [ u ] rather than the [ u − ] version.4.6. Chen’s iterated integral on cyclic cohomology.
Let X be a manifoldand Ω( X ) its de Rham dga. Recall the mixed complexes (cid:0) C ∗ (Ω( X )) , d + b, B (cid:1) fromCorollary 2.16, and (cid:16) C −∗ ( LX ) , ∂, Q (cid:1) from Section 4.3 for Y = LX . Let h· , ·i : C ∗ (Ω( X )) ⊗ C −∗ ( LX ) → R be the Chen pairing 18 from Section 3. By the proof of Proposition 3.1 this pair-ing respects the structures of mixed complexes, so it induces two maps of mixedcomplexes: Chen’s iterated integral I : C ∗ (Ω( X )) → C ∗ ( LX ), and its adjoint J : (cid:0) C −∗ ( LX ) , ∂, Q (cid:1) → (cid:0) C −∗ (Ω( X )) , d ∗ + b ∗ , B ∗ (cid:1) . Similarly, the cyclic Chen pairing h , i cyc defined in (26) induces two chain maps:Connes’ version of Chen’s iterated integral I λ , and its adjoint J λ : (cid:0) C −∗ +1 ( LX ) , ∂ (cid:1) → (cid:0) C −∗ (Ω( X )) , d ∗ + b ∗ (cid:1) . Lemma 3.2 and the discussion following it dualize to
Lemma 4.9. (a) J λ is given by the composition of chain maps J λ : C −∗ +1 ( LX ) J −→ C −∗ +1 (Ω( X )) B ∗ −→ C −∗ (Ω( X )) . (b) J λ lands in (cid:0) C ∗ (Ω( X )) / im (1 − t ) (cid:1) ∨ = ker(1 − t ∗ ) = C λ −∗ +1 (Ω( X )) and inducesa map J λ : C −∗ ( LX × S ES ) → ker(1 − t ∗ ) . (c) The following diagram commutes: C [ u − ] −∗ +1 (Ω( X )) B ∗ π −−−−→ C λ −∗ +1 (Ω( X )) ι −−−−→ C [[ u ]] −∗ (Ω( X )) J x x J λ x J C [ u − ] −∗ +1 ( LX ) π −−−−→ C −∗ +1 ( LX ) ι Q −−−−→ C [[ u ]] −∗ ( LX ) . (cid:3) The map J λ induces a map on homology which we denote by J λ ∗ : H S ∗ ( LX ) → HC λ ∗ (Ω( X )) . IGHT FLAVOURS OF CYCLIC HOMOLOGY 37
Passing to homology in Lemma 4.9(c) we obtain the following commuting diagram,where the maps B ∗ and P ∗ are isomorphisms by Lemma 4.8 and Proposition 4.4,respectively:(34) HC [ u − ] −∗ +1 (Ω( X )) B ∗∗ π ∗ −−−−→ HC λ −∗ +1 (Ω( X )) ι ∗ −−−−→ ∼ = HC [[ u ]] −∗ (Ω( X )) x J ∗ x J λ ∗ x J ∗ H [ u − ] −∗ +1 ( LX ) π ∗ −−−−→ ∼ = H S −∗ +1 ( LX ) ι ∗ Q ∗ −−−−→ H [[ u ]] −∗ ( LX ) . The simply connected case.
Assume now in addition that X is simply con-nected. Then by Theorem 3.3, Chen’s iterated integral I : C ∗ (Ω( X )) → C ∗ ( LX )is a quasi-isomorphism of mixed complexes. Since the homologies HH ∗ (Ω( X )) (byLemma 4.6) and H ∗ ( LX ) (by Lemma 4.7) are graded finite dimensional, Corol-lary 4.2 applied to the Chen pairing yields Corollary 4.10.
Let X be a simply connected manifold. Then the dual Cheniterated integral J : C −∗ ( LX ) → C −∗ (Ω( X )) defines a quasi-isomorphism of mixedcomplexes, and therefore induces isomorphisms on the [[ u ]] , [[ u, u − ] and [ u − ] -versions of cyclic cohomology. (cid:3) As in the proof of Theorem 3.5, we obtain a commuting diagram H S ∗ ( LM ) J λ ∗ / / HC λ ∗ (Ω( X )) ι ∗ J − ∗ ∼ = (cid:15) (cid:15) H [[ u,u − ] ∗ ( LX ) ∼ = (cid:15) (cid:15) p ∗ / / H [ u − ] ∗ ( LX ) π ∗ ∼ = O O ι ∗ Q ∗ / / H [[ u ]] ∗− ( LX ) π ∗ (cid:15) (cid:15) · u / / H [[ u,u − ] ∗ +1 ( LX ) ∼ = (cid:15) (cid:15) R [ u, u − ] p ∗ / / R [ u − ] ι ∗ O O / / R [ u ] · u / / R [ u, u − ] . Here the lower two rows follow from (31) for Y = LX where ι ∗ is injective, π ∗ is surjective, and the outer vertical maps are isomorphisms by the dual version ofGoodwillie’s Theorem 2.22. The upper square follows from diagram (34), where J ∗ is an isomorphism by Corollary 4.10. We read offker J λ ∗ = π ∗ ι ∗ R [ u − ] , im J λ ∗ = ker (cid:16) π ∗ ι ∗ J − ∗ : HC λ ∗ (Ω( X )) → R [ u ] (cid:17) , and passing to reduced homologies as in the proof of Corollary 3.6 we conclude Corollary 4.11.
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