Homologie cyclique du produit croise algebrique et groupes de surfaces

Abstract

Let a group G act on an associative algebra A. One can form the algebraic crossed product A ⋊ G (cf. 1.3), which plays the role of a “noncommutative quotient” in Connes's theory [6]. The cyclic homology of this algebra was studied extensively in a series of papers [4, 8, 10, 16, 17]. It is well known that this homology admits a decomposition into a direct sum (cf. 1.4) where the summands are indexed by conjugacy classes of elements of the group. Every direct summand is a limit of a spectral sequence whose E 2 term is the homology of the group with coefficients in certain homology groups (cf. 1.5). These latter homology groups are the cyclic homology groups of the algebra if the conjugacy class is the identity element. In this paper we study this spectral sequence, and the generalized version to negative and periodique cyclic homology. The main result (cf. 2.6) is the description of the differential in the E 2 term. More generally we define characteristic classes for the action of an algebra on a mixed complex. When the mixed complex is the natural one associated to a G-algebra A, and the operation is that of an automorphism group G of A, these classes determine the differential d 2 of this spectral sequence. We apply this result to obtain a description of the periodic cyclic homology of a crossed product by the fundamental group of a Riemann surface (cf. 3.1).