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 which plays the role of a noncommutative quotient in Conness theory The cyclic homology of this algebra was studied extensively in a series of papers It is well known that this homology admits a decomposition into a direct sum cf 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 term is the homology of the group with coecients in certain homology groups cf These latter homology groups are the cyclic homology groups of the algebra if the conjugacy class is the identity element In this paper we studie this spectral sequence and the generalized version to negative and periodique cyclic homology The main result cf is the description of the dierential in the E term More generally we dene characteristic classes for the action of an algebra on a mixed complex When the mixed complex is the naturel one associated to a G algebra A and the operation is that of an automorphism group G of A these classes determine the dierential d 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