Equivariant K -homology of the classifying space for proper actions

The aim of this doctoral thesis is to explicitly compute the topological side of the Baum-Connes conjecture, that is, the equivariant K-homology of the classifying space for proper actions, for some discrete groups. This is achieved by means of the Bredon homology with coefficients in the representation ring of the corresponding classifying spaces.

In particular, we obtain the K-homology and Bredon homology for SL₃([special characters omitted]), GL₃([special characters omitted]) and lower-rank (up to three), right-angled and even Coxeter groups. On the process, we required a Künneth formula for Bredon homology; we present Künneth formulas for the Bredon homology of the product of a G-CW-complex and a H-CW-complex, the direct product of two groups and also versions for relative Bredon homology, and proper actions with coefficients in the representation ring.

We used the mathematical software GAP during our research. We have implemented routines to compute the Bredon homology, with coefficients in the representation ring, of a proper G-CW-complex, and of a Coxeter group from its Coxeter matrix.