The analytical assembly map and index theory

Abstract

In this talk I want to present a recent result of my own about a geometric interpretation of the abstract analytical assembly map as it appears in the BaumConnes conjecture. The setup is as follows: Let G be a countable discrete group, then the Baum-Connes conjecture predicts the so-called analytical assembly map K ∗ (EG) A // K∗(C rG) to be an isomorphism of graded abelian groups between the G-equivariant analytical K-homology of the classifying space for proper G-actions and the K-theory of the reduced group C∗-algebra. In the case where the group G is torsion-free there is an isomorphism K ∗ (EG) = K ∗ (EG) ∼= K∗(BG). Moreover one can define a Mishchenko-Fomenko index map K∗(BG) MF // K∗(C rG) and it is natural to study the relation between these two resulting maps. This is precisely my result: Theorem. For every countable, discrete, torsion-free group G the diagram K∗(BG) MF // K∗(C rG)