Bredon Cohomology, K theory and K homology of Pullbacks of groups

TL;DR

This paper develops spectral sequences to compute Bredon cohomology, equivariant K-theory, and K-homology for groups formed as pullbacks, applying these tools to crystallographic groups and conjectures like Baum-Connes and Farrell-Jones.

Contribution

It introduces an Eilenberg-Moore spectral sequence for Bredon cohomology of pullback group actions and applies it to compute K-theory and K-homology of specific crystallographic groups.

Findings

Computed equivariant K-theory and K-homology for a 6-dimensional crystallographic group.

Established vanishing results for negative algebraic K-theory of the group's integral ring.

Provided evidence supporting Baum-Connes and Farrell-Jones conjectures for the studied groups.

Abstract

We develop an Eilenberg-Moore spectral sequence to compute Bredon cohomology of spaces with an action of a group given as a pullback. Using several other spectral sequences, and positive results on the Baum-Connes Conjecture, we are able to compute Equivariant K-theory and K-Homology of the reduced group C*-algebra of a 6-dimensional crystallographic group $\Gamma$ introduced by Vafa and Witten. We also use positive results on the Farrell-Jones Conjecture to give a vanishing result for the negative algebraic K-theory of the integral group ring of $\Gamma$.