On the equivariant $K$- and $KO$-homology of some special linear groups

Abstract

We compute the equivariant $KO$-homology of the classifying space for proper actions of ${\text{SL}}_3(\mathbb{Z})$ and ${\text{GL}}_3(\mathbb{Z})$. We also compute the Bredon homology and equivariant $K$-homology of the classifying spaces for proper actions of ${\text{PSL}}_2(\mathbb{Z}[\frac{1}{p}])$ and ${\text{SL}}_2(\mathbb{Z}[\frac{1}{p}])$ for each prime $p$. Finally, we prove the Unstable Gromov-Lawson-Rosenberg Conjecture for a large class of groups whose maximal finite subgroups are odd order and have periodic cohomology.

21 pages. Corrected typos and added some more examples in Section 6. To appear in Algebraic and Geometric Topology

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Cited by in corpus (1)

• Cohomology of Fuchsian Groups and Non-Euclidean Crystallographic Groups