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

Abstract

We develop a spectral sequence of Eilenberg-Moore type 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 C * -algebra of a 6-dimensional crystallographic group Γ introduced by Vafa and Witten. We also use positive results on the Farrell-Jones conjecture to give a vanishing result for the algebraic K-theory of the group ring of the group Γ in negative degrees.

A Kunneth formula for Bredon cohomology of pullbacks and twisted K-theory of some 6-dimensional orbifolds

“…This paper continues the work started in [BJV13] where some calculations concerning pullbacks of groups were made. We clarify some questions that appear in that work about the computation of twisted K-theory for pullbacks.…”

Bredon homology of Artin groups of dihedral type

“…4.3.4. The homomorphism g 1 2 when n is even: This homomomorphism is defined in equation (5) and, according to Propositions 4.12, 4.13 and 4.15, its first component is given by…”

“…The left-hand side of the conjecture can be approached by means of a G-equivariant version of the Atiyah-Hirzebruch spectral sequence, which converges to the Farrell-Jones K-homology, and whose E 2 -page is the Bredon homology of the classifying space EG with coefficients in the K-theory of the group rings of the virtually cyclic subgroups of G. This strategy has been used in the past with great success in the context of the Baum-Connes Conjecture; however, as the level of complexity increases when passing from the family of finite groups to the bigger family of virtually cyclic groups, there are not a lot of examples of explicit computations of Bredon homology in the Farrell-Jones framework ( [5], [32]). We remark here that the knowledge of the Bredon homology groups has always been of independent interest, because of the prominent role that they play in equivariant homotopy and the dimension theory of groups.…”