Topological K-theory for discrete groups and index theory

Abstract

For any countable discrete group Γ (without any further assumptions on it) we first construct an explicit morphism from the Left-Hand side of the Baum-Connes assembly map, Ktop⁎(Γ), to the periodic cyclic homology of the group algebra, HP⁎(CΓ). This morphism, called the Chern-Baum-Connes assembly map, allows in particular to give a proper and explicit formulation for a Chern-Connes pairingKtop⁎(Γ)×HP⁎(CΓ)⟶C.Several theorems are needed to formulate the Chern-Baum-Connes assembly map. In particular we establish a delocalised Riemann-Roch theorem, the wrong way functoriality for periodic delocalised cohomology for Γ-proper actions, the construction of a Chern morphism between the Left-Hand side of Baum-Connes and a delocalised cohomology group associated to Γ which is an isomorphism once tensoring with C, and the construction of an explicit cohomological assembly map between the delocalised cohomology group associated to Γ and the homology group H⁎(Γ,FΓ) (where FΓ is the complex vector space freely generated by the set of elliptic elements in Γ). This last group H⁎(Γ,FΓ) identifies, by the work of Burghelea, as a direct factor of the cyclic periodic homology of the group algebra.Moreover, in this paper Ktop⁎(Γ) stands for the topological K-theory for discrete groups as originally proposed by Baum and Connes in their first paper, where wrong way functoriality in equivariant K-theory was used for its construction. As part of our results we prove that this model is indeed isomorphic to the analytic model for the left-hand side of the assembly map.Our main final theorem gives an explicit geometric index theoretical formula for the above mentioned pairing (without any further assumptions on Γ) in terms of pairings of invariant forms, associated to geometric cycles and given in terms of delocalized Chern and Todd classes, and currents naturally associated to group cocycles using Burghelea's computation. This gives a complete solution, for discrete countable groups, to the problem of defining and computing a geometric pairing between the left hand side of the Baum-Connes assembly map, given in terms of geometric cycles associated to proper actions of the discrete group on manifolds, and cyclic periodic cohomology of the group algebra.