A Chern Character in Cyclic Homology

Abstract. In this paper, we show that, under natural homological conditions, Hopf cyclichomology theory has excision. Mathematics Subject Classifications (2000): 16W30, 19D55, 57T05, 58B34. Key words: cyclic homology, Hopf algebras. 1. Introduction Cyclic cohomology theory for Hopf algebras was introduced by Connesand Moscovici in their ground breaking paper on transverse index theory[4] (cf. also [3, 5, 6] for a recent survey and new results). This theory can beregarded as the right noncommutative analogue of Lie algebra homologybecause of existence of a characteristic map for actions of Hopf algebrasand since it reduces to Lie algebra homology for Hopf algebras associ-ated to Lie algebras. This should be compared with the role played byordinary cyclic cohomology of noncommutative algebras and its reductionto de Rham homology for smooth commutative algebras [2]. The alge-braic underpinnings of Hopf cyclic theory however turned out to be morecomplicated. It is now understood as a special case of a general theorydeveloped in the series of papers [11, 12, 15, 16] (with the help of [1])for (co)algebras endowed with a (co)action of a Hopf algebra and where,among other things, the right notion of coefficients was introduced (cf. also[14] for a recent survey). This theory was later extended by the first author[13] to bialgebras and to a much larger class of coefficients. This extensionis rather surprising since existence of a bijective antipode was essential inconstructing the cyclic and cocyclic complexes developed for Hopf cyclichomology of both variants. By bialgebra or Hopf cyclic homology we nowmean the theory expounded in these papers mentioned above.The main results of the present paper are two excision theorems(Theorems 4.13 and 7.7) for Hopf cyclic homology of module coal-gebras and comodule algebras, respectively. A basic property of cyclic(co)homology for algebras, and coalgebras is excision. If by cyclic theory

This article surveys some recent developments in the study of modules over crossed products R * G. As indicated in the title, we focus on three particular techniques, namely (1) characters or rather their ring theoretic analog, Hattori-Stallings ranks (2) cyclic homology and Connes’ “Chern characters”, an extension of Hattori-Stallings ranks, and (3) Moody’s induction theorem for Grothendieck groups G 0(R*G) where G is a polycyclic-by-finite group.

Homology of free loop spaces, cyclic homology and nonrational Poincare-Betti series in commutative algebra

We study the "quantized calculus" corresponding to the algebraic ideas related to "twisted cyclic cohomology" introduced in [12]. With very similar definitions and techniques as those used in [9], we define and study "twisted entire cyclic cohomology" and the "twisted Chern character" associated with an appropriate operator theoretic data called "twisted spectral data", which consists of a spectral triple in the conventional sense of noncommutative geometry [1] and an additional positive operator having some specified properties. Furthermore, it is shown that given a spectral triple (in the conventional sense) which is equivariant under the (co-) action of a compact matrix pseudogroup, it is possible to obtain a canonical twisted spectral data and hence the corresponding (twisted) Chern character, which will be invariant (in the usual sense) under the (co-)action of the pseudogroup, in contrast to the fact that the Chern character coming from the conventional noncommutative geometry need not to be invariant. In the last section, we also try to detail out some remarks made in [3], in the context of a new definition of invariance satisfied by the conventional (untwisted) cyclic cocycles when lifted to an appropriate larger algebra.

In this chapter, we introduce the reader to the homology theory for algebras known as cyclic homology. As we shall see in the next section, cyclic homology may be viewed as the “linearization” of K-theory, in the same sense in which the matrix ring M n (R) is the “linearization” of the general linear group GL(n, R). For motivation, it is useful to think of the case where the ring R is ℝ or ℂ. Then GL(n, R)is a Lie group, and the space BGL(n, R)+ giving rise to the higher K-groups is by its construction an H-space whose homology agrees with the homology of this Lie group (never mind for the moment that we are forgetting the topology!). On the other hand, one of the basic principles of Lie theory is that the best way to study Lie groups (which are “non-linear” objects) is often by way of their Lie algebras (which can be studied using linear algebra). For example, it is a famous result that for any compact Lie group G with Lie algebra g, the topological cohomology H●(G;ℝ) (here we are just thinking of G as a space and ignoring the group structure) is canonically isomorphic to the Lie algebra cohomology H●(g;ℝ), which can at least in principle be computed using only finite-dimensional linear algebra. This suggests that some construction with the Lie algebra M n (R) of GL(n, R)ought to yield a reasonable approximation to the K-theory of R, at least in the limit as n → ∞. While the actual construction of cyclic homology and of the Chern character which relates it to K-theory is a bit roundabout, this philosophy turns out to be basically correct.KeywordsExact SequenceCommutative RingHomology TheoryChern CharacterCyclic HomologyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A large part of research in noncommutative geometry is devoted to various far-reaching generalizations of the Atiyah–Singer index theory of elliptic operators on compact manifolds. Following Kasparov [K1], the notion of an elliptic operator on a manifold is replaced by that of a Fredholm module (or K-cycle) over an algebra of operators on Hilbert space. A number of index theorems for Fredholm modules associated to geometric data have been obtained by Kasparov [K1], Connes (see [C3] and the literature cited therein) and others. In order to handle concrete index problems it is not only necessary to dispose of an index theorem but also to provide an index formula which allows the explicit calculation of indices. The classical index formula of Atiyah and Singer is obtained from the index theorem by applying the Chern character in topological K-homology. This motivates the search for character formulas of Fredholm modules (K-cycles) that define a Chern character on the K-homology groups of an algebra of operators. It was this search for a Chern character in K-homology which led A. Connes to the invention of cyclic cohomology [C2]. He obtained various explicit character formulas [C1,2,3], which depend on the degree of analytic regularity (summability) of the given Fredholm module. Whereas the classical index problems of Atiyah–Singer are all finitely summable in the sense of [C2] there are many examples of Fredholm modules over (noncommutative) algebras which are infinite dimensional (not finitely summable). Characters of Fredholm modules have been calculated in many finitely summable cases, but as far as the author knows the character of a θ-summable, infinitedimensional (unbounded) Fredholm module [C1] has not yet been determined in a single case.

In this paper, we introduce a paracyclic version of S -modules. These new objects are called para- S -modules . Paracyclic modules and parachain complexes give rise to para- S -modules much in the same way as cyclic modules and mixed complexes give rise to S -modules. More generally, para- S -modules provide us with a natural framework to get analogues for paracyclic modules and parachain complexes of various constructions and equivalence results for cyclic modules or mixed complexes. The datum of a para- S -module does not provide us with a chain complex, and so notions of homology and quasi-isomorphisms do not make sense. However, chain maps and chain homotopies continue to make sense, and so by equivalences we actually mean chain homotopy equivalences. We establish some generalizations for para- S -modules and parachain complexes of the basic perturbation lemma of differential homological algebra. These generalizations provide us with general recipes for converting deformation retracts of Hochschild chain complexes into deformation retracts of para- S -modules. By using ideas of Kassel this then allows us to get comparison results between the various para- S -modules associated with para-precyclic modules, and between them and Connes’ cyclic chain complex. These comparison results lead us to alternative descriptions of Connes’ periodicity operator. This has some applications in periodic cyclic homology. We also describe the counterparts of these results in cyclic cohomology. In particular, we obtain an explicit way to convert a periodic (b, B) -cocycle into a cohomologous periodic cyclic cocycle.

On the cohomology of free and twisted loop spaces

The pull back of a flat bundle E → X along the evaluation map π : LX → X from the free loop space LX to X comes equipped with a canonical automorphism given by the holonomies of E. This construction naturally generalizes to flat Z-graded connections on X. Our main result is that the restriction of this holonomy automorphism to the based loop space Ω∗X of X provides an A∞ quasi-equivalence between the dg category of flat Z-graded connections on X and the dg category of representations of C•(Ω∗X), the dg algebra of singular chains on Ω∗X. Dedicated to James Dillon Stasheff, with gratitude.

Using similarities between topological K-theory and periodic cyclic ho- mology we show that, after tensoring with C, for certain Frechet algebras the Chern character provides an isomorphism between these functors. This is applied to prove that the Hecke algebra and the Schwartz algebra of a reductive p-adic group have isomorphic periodic cyclic homology. Later an appendix was added, to deal with infinite direct products of algebras. Mathematics Subject Classification (2000) 19D55, 19L10, 22E50, 46L80

In this PhD thesis, we construct an explicit algebraic model over Z of the cochains of the free loop space of a 1-connected space X. We start from an enriched Adams-Hilton model of X, which can be obtained relatively easily when X is the realisation of a simplicial set. Note it is not supposed that the Steenrod algebra acts trivially on X. The second part is dedicated to the construction of a model of the cochains of mapping spaces XY. where X is r-connected and Y is a CW-complex that has dimension less or equal to r. The space X must possess commutative models for the cochains of each Ωk X for k ≤ r. We first construct an algebraic model for the cochains of XSn ∀n ≤ r, then we then glue all of them to obtain a model of the cochains of XY. We give examples for each of these situations. The techniques used here rely heavily on the concept of a twisted bimodule. A description of this can be found in [DH99b].

We show that the bivariant Chern character in entire cyclic cohomology constructed in a previous paper in terms of superconnections and heat kernel regularization, retracts on periodic cocycles under some finite summability conditions. The trick is a bivariant generalization of the Connes-Moscovici method for finitely summable K-cycles. This yields concrete formulas for the Chern character of p-summable quasihomomorphisms and invertible extensions, analogous to those of Nistor. The latter formulation is completely algebraic and based on the universal extensions of Cuntz and Zekri naturally appearing in the description of bivariant K-theory.

For a single Dirac operator on a closed manifold the cocycle introduced by Jaffe-Lesniewski-Osterwalder [19] (abbreviated here to JLO), is a representative of Connes' Chern character map from the K-theory of the algebra of smooth functions on the manifold to its entire cyclic cohomology. Given a smooth fibration of closed manifolds and a family of generalized Dirac operators along the fibers, we define in this paper an associated bivariant JLO cocycle. We then prove that, for any l ≥ 0, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the Cl+1 topology and functions on the base manifold with the Cl topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.

The cyclic homology of an exact category

To any path connected topological space X X , such that rk ⁡ H i ( X ) > ∞ \operatorname {rk} H_i(X) >\infty for all i ≥ 0 i\geq 0 , are associated the following two sequences of integers: b i = rk ⁡ H i ( Ω X ) b_i= \operatorname {rk} H_i(\Omega X) and r i = rk ⁡ π i + 1 ( X ) r_i= \operatorname {rk} \pi _{i+1}(X) . If X X is simply connected, the Milnor-Moore theorem together with the Poincaré-Birkoff-Witt theorem provides an explicit relation between these two sequences. If we assume moreover that H i ( X ; Q ) = 0 H_i(X;\mathbb Q)=0 , for all i ≫ 0 i\gg 0 , it is a classical result that the sequence of Betti numbers ( b i ) (b_i) grows polynomially or exponentially, depending on whether the sequence ( r i ) (r_i) is eventually zero or not. The purpose of this note is to prove, in both cases, that the r t h r^{\mathrm {th}} Betti number b r b_r is controlled by the immediately preceding ones. The proof of this result is based on a careful analysis of the Sullivan model of the free loop space X S 1 X^{S^1} .