For any countable discrete group $\Gamma$ (without any further assumptions on it) I will first explain how to give a proper and explicit formulation for a Chern-Baum-Connes pairing between the topological K-theory group $K^*_{top}(\Gamma)$ (the left hand side of the Baum-Connes assembly map) and the periodic cyclic cohomology group $HP^*(\mathbb{C}\Gamma)$.

Several theorems are needed to get to the above mentioned pairing. 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 cyclic periodic homology of the group algebra.

At the end of the talk I will give (and try to sketch the proof) an explicit index theoretical formula for the above mentioned pairing in terms of Burghelea's computation of cyclic periodic cohomology of $\Gamma$.

This talk is based in a joint work with Bai-Ling Wang (ANU) and Hang Wang (ECNU).