Entire cyclic homology of stable continuous trace algebras

A central result in this paper is the computation of the entirecyclic homology of canonical smooth subalgebras of stable continuoustrace C*-algebras having smooth manifolds M as their spectrum.More precisely, the entire cyclic homology is shown to be canonicallyisomorphic to the continuous periodic cyclic homology for thesealgebras. By an earlier result of the authors, one concludesthat the entire cyclic homology of the algebra is canonicallyisomorphic to the twisted de Rham cohomology of M.     

Hopf-Cyclic Homology and Relative Cyclic Homology of Hopf-Galois Extensions

Let H be a Hopf algebra and let M_s (H) be the category of allleft H-modules and right H-comodules satisfying appropriatecompatibility relations. An object in M_s (H) will be calleda stable anti-Yetter–Drinfeld module (over H) or a SAYDmodule, for short. To each M M_s (H) we associate, in a functorialway, a cyclic object Z_* (H, M). We show that our constructioncan be used to compute the cyclic homology of the underlyingalgebra structure of H and the relative cyclic homology of H-Galoisextensions. Let K be a Hopf subalgebra of H. For an arbitrary M M_s (K)we define a right H-comodule structure on so that becomes an object in M_s (H). Under some assumptions on K and M we computethe cyclic homology of . As a direct application of this result, we describe the relativecyclic homology of strongly graded algebras. In particular,we calculate the cyclic homology of group algebras and quantumtori. Finally, when H is the enveloping algebra of a Lie algebra g,we construct a spectral sequence that converges to the cyclichomology of H with coefficients in a given SAYD module M. Wealso show that the cyclic homology of almost symmetric algebrasis isomorphic to the cyclic homology of H with coefficientsin a certain SAYD module. 2000 Mathematics Subject Classification16E40 (primary), 16W30 (secondary).     

A higher Lefschetz formula for flat bundles

In this paper, we prove a fixed point formula for flat bundles. To this end, we use cyclic cocycles which are constructed out of closed invariant currents. We show that such cyclic cocycles are equivariant with respect to isometric longitudinal actions of compact Lie groups. This enables us to prove fixed point formulae in the cyclic homology of the smooth convolution algebra of the foliation.

     

Cerf Theory for Graphs

We develop a deformation theory for k-parameter families ofpointed marked graphs with fixed fundamental group F_n. Applicationsinclude a simple geometric proof of stability of the rationalhomology of Aut(F_n), computations of the rational homology insmall dimensions, proofs that various natural complexes of freefactorizations of F_n are highly connected, and an improvementon the stability range for the integral homology of Aut(F_n).     

Excision in Cyclic Type Homology of Frechet Algebras

It is proved that every topologically pure extension of Fréchetalgebras 0 I A A/I 0 such that I is strongly H-unital hasthe excision property in continuous (co)homology of the followingtypes: bar, naive-Hochschild, Hochschild, cyclic, and periodiccyclic. In particular, the property holds for every extensionof Fréchet algebras such that I has a left or right boundedapproximate identity.     

Topological Hochschild Homology

In the appendix to [20] Waldhausen discussed a trace map tr:K(R)HH(R),from the algebraic K-theory of a ring to its Hochschild homology,which can be used to obtain information about K(R) from HH(R).In [1] Bökstedt described a factorization of this tracemap. The intermediate functor THH(HR) is called the topologicalHochschild homology of the Eilenberg–MacLane spectrumHR associated with R, because it is constructed similarly toHochschild homology with the tensor product replaced by thesmash product of spectra.     

The first homology of the group of equivariant diffeomorphisms and its applications

Let V be a representation space of a finite group G. We determinethe group structure of the first homology of the equivariantdiffeomorphism group of V. Then we can apply it to the calculationof the first homology of the corresponding automorphism groupsof smooth orbifolds, compact Hausdorff foliations, codimensionone or two compact foliations and the locally free S¹-actionson 3-manifolds. Received November 5, 2007.     

The homotopy invariance of the string topology loop product and string bracket

Let Mⁿ be a closed, oriented, n-manifold, and LM its free loopspace. In [Chas and Sullivan, ‘String topology’,Ann. of Math., to appear] a commutative algebra structure inhomology, H_*(LM), and a Lie algebra structure in equivarianthomology , were defined.In this paper, we prove that these structures are homotopy invariantsin the following sense. Let f:M₁ M₂ be a homotopy equivalenceof closed, oriented n-manifolds. Then the induced equivalence,Lf:LM₁ LM₂ induces a ring isomorphism in homology, and an isomorphismof Lie algebras in equivariant homology. The analogous statementalso holds true for any generalized homology theory h_* thatsupports an orientation of the M_i. Received February 5, 2007.     

On Towers Approximating Homological Localizations

Our object of study is the natural tower which, for any givenmap f:AB and each space X, starts with the localization of Xwith respect to f and converges to X itself. These towers canbe used to produce approximations to localization with respectto any generalized homology theory E_*, yielding, for example,an analogue of Quillen's plus-construction for E_*. We discussin detail the case of ordinary homology with coefficients inZ/p or Z[1/p]. Our main tool is a comparison theorem for nullificationfunctors (that is, localizations with respect to maps of theform f:Apt), which allows us, among other things, to generalizeNeisendorfer's observation that p-completion of simply-connectedspaces coincides with nullification with respect to a Moorespace M(Z[1/p], 1).     

Cyclic group actions on polynomial rings

We consider a cyclic group of order pⁿ, acting on a module incharacteristic p, and show how to reduce the calculation ofthe symmetric algebra to that of the exterior algebra.     

Spin Structures on Lefschetz Fibrations

Necessary and sufficient conditions for Lefschetz fibrationsover D² and S² to carry spin structures are presented. The conditionsare given in terms of vanishing cycles and homology sections.     

The Product Separability of the Generalized Free Product of Cyclic Groups

Let G be a group endowed with its profinite topology, then Gis called product separable if the profinite topology of G isHausdorff and, whenever H₁, H₂, ..., H_n are finitely generatedsubgroups of G, then the product subset H₁ H₂ ... H_n is closedin G. In this paper, we prove that if G=FxZ is the direct productof a free group and an infinite cyclic group, then G is productseparable. As a consequence, we obtain the result that if Gis a generalized free product of two cyclic groups amalgamatinga common subgroup, then G is also product separable. These resultsgeneralize the theorems of M. Hall Jr. (who proved the conclusionin the case of n=1, [3]), and L. Ribes and P. Zalesskii (whoproved the conclusion in the case of that G is a finite extensionof a free group, [6]).     

Cyclic, Separable and Semisimple Matrices in the Special Linear Groups Over a Finite Field

A matrix A with minimum polynomial m_A and characteristic polynomialc_A is said to be cyclic if m_A = c_A, semisimple if m_A has norepeated factors, and separable if it is both cyclic and semisimple.For any set T of matrices, we write C_T for the proportion ofcyclic matrices in T, SS_T for the proportion of semisimple matrices,and S_T for the proportion of separable matrices. We will writeC_GL(,q) for lim_dC_GL(d,q), and so on.     

Dihedral homology and Hermitian K-theory

We establish a rational isomorphism between certain relative versions of Hermitian K-theory and the dihedral homology of simplicial Hermitian rings. This is the dihedral analogue of Goodwillie's result for cyclic homology and algebraic K-theory. In particular, we describe involutions on (negative) cyclic homology and the K-theory of simplicial rings. We show that Goodwillie's map from K-theory to negative cyclic homology can be chosen to preserve involutions. By work of Burghelea and Fiedorowicz the invariants of the involution on K-theory can be identified with symmetric Hermitian K-theory. Finally, we show how the author's chain complex defining dihedral homology can be extended to the left to capture the invariants of the involution on negative cyclic homology.Supported by New Mexico State University, grant No. RC90-051.     

Unsolvable Problems About Small Cancellation and Word Hyperbolic Groups

We apply a construction of Rips to show that a number of algorithmicproblems concerning certain small cancellation groups and, inparticular, word hyperbolic groups, are recursively unsolvable.Given any integer k > 2, there is no algorithm to determinewhether or not any small cancellation group can be generatedby either two elements or more than k elements. There is a smallcancellation group E such that there is no algorithm to determinewhether or not any finitely generated subgroup of E is all ofE, or is finitely presented, or has a finitely generated secondintegral homology group.     

Symmetric Groups as Products of Abelian Subgroups

A proof is given that the full symmetric group over any infiniteset is the product of finitely many Abelian subgroups. In fact,289 subgroups suffice. Sharp bounds are also obtained on theminimal number k, such that the finite symmetric group S_n isthe product of k Abelian subgroups. Using this, S_n is provedto be the product of 72n^1/2(log n)^3/2 cyclic subgroups. 2000Mathematics Subject Classification 20B30, 20D40.     

Hochschild (Co)Homology Dimension

In 1989 Happel asked the question whether, for a finite-dimensionalalgebra A over an algebraically closed field k, gl.dim A < if and only if hch.dim A < . Here, the Hochschild cohomologydimension of A is given by hch.dim A := inf{n N₀ | dim HHⁱ(A) = 0 for i > n}. Recently Buchweitz, Green, Madsen andSolberg gave a negative answer to Happel's question. They founda family of pathological algebras A_q for which gl.dim A_q = but hch.dim A_q = 2. These algebras are pathological in manyaspects. However, their Hochschild homology behaviors are notpathological any more; indeed one has hh.dim A_q = = gl.dimA_q. Here, the Hochschild homology dimension of A is given byhh.dim A := inf{n N₀ | dim HH_i(A) = 0 for i > n}. This suggestsposing a seemingly more reasonable conjecture by replacing theHochschild cohomology dimension in Happel's question with theHochschild homology dimension: gl.dim A < if and only ifhh.dim A < if and only if hh.dim A = 0. The conjecture holdsfor commutative algebras and monomial algebras. In the casewhere A is a truncated quiver algebra, these conditions areequivalent to the condition that the quiver of A has no orientedcycles. Moreover, an algorithm for computing the Hochschildhomology of any monomial algebra is provided. Thus the cyclichomology of any monomial algebra can be read off when the underlyingfield is characteristic 0.     

Derived Subgroups of Products of an Abelian and a Cyclic Subgroup

Let G be a finite group and suppose that G = AB, where A andB are abelian subgroups. By a theorem of Ito, the derived subgroupG' is known to be abelian. If either of the subgroups A or Bis cyclic, then more can be said. The paper shows, for example,that G'/(G'A) is isomorphic to a subgroup of B in this case.     

Hopf type formulas for cyclic homology

We fit the cyclic homology of associative algebras into the context of cotriple homology of Barr and Beck. Consequently, we describe the cyclic homology of associative algebras in terms of the generalised Hopf type formulas. This Note is part of a joint project with Donadze about derived functors in cyclic (co)homology. To cite this article: N. Inassaridze, M. Ladra, C. R. Acad. Sci. Paris, Ser. I 346 (2008).