Topological K-theory for discrete groups and Index theory

TL;DR

This paper develops a comprehensive geometric and cohomological framework for the Baum-Connes assembly map for discrete groups, introducing explicit morphisms and formulas to compute pairings with cyclic cohomology.

Contribution

It constructs an explicit Chern-Baum-Connes assembly map and establishes a delocalised Riemann-Roch theorem, providing new tools for index theory and cyclic cohomology computations for discrete groups.

Findings

Constructed an explicit morphism from the Baum-Connes LHS to periodic cyclic homology.

Established a delocalised Riemann-Roch theorem for proper group actions.

Provided an index formula for pairings involving geometric cycles and group cocycles.

Abstract

We give 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 on manifolds, and cyclic periodic cohomology of the group algebra. Indeed, for any such group $\Gamma$ (without any further assumptions on it) we construct an explicit morphism from the Left-Hand side of the Baum-Connes assembly map to the periodic cyclic homology of the group algebra. This morphism, called here the Chern-Baum-Connes assembly map, allows to give a proper and explicit formulation for a Chern-Connes pairing with the periodic cyclic cohomology of the group algebra. 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 $\Gamma$-proper actions, the construction of a Chern morphism between the Left-Hand side of Baum-Connes and a delocalised cohomology group associated to $\Gamma$ which is an isomorphism once tensoring with $\mathbb{C}$, and the construction of an explicit cohomological assembly map between the delocalised cohomology group associated to $\Gamma$ and the homology group $H_*(\Gamma,F\Gamma)$. We then give an index theoretical formula for the above mentioned pairing (for any $\Gamma$) 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. As part of our results we prove that left-Hand side group used in this paper is isomorphic to the usual analytic model for the left-hand side of the assembly map.