In this talk, I will report on joint work with B.L. WANG (ANU Canberra) et H. WANG (ECNU Shanghai). Given a discrete group \(G\), I will tell you about the construction of a group morphism (explicit and geometric) from the “Topological K-theory of \(G\)”, the left-hand side topological model from the Baum-Connes assembly map to the periodic cyclic homology of the group algebra. This morphism is called the Chern assembly map because as I will tell you it is constructed by assembling Chern characters. The Chern assembly map allows to give an explicit and precise formulation for the pairing between the left-hand side of BC and the periodic cyclic cohomology of the group but also since it is given in an explicit way, it allows to give a formula for such a pairing. There is no further assumption on the group (maybe just countable for simplicity), the use of groupoids and in particular of deformation groupoids is
essential in our approach.