Equivariant Refinements

TL;DR

This paper proves that equivariant open covers of finite dimensional spaces can be refined to bounded dimension, aiding in the proof of the Farrell-Jones conjecture for certain linear groups.

Contribution

It introduces a method to produce equivariant refinements of covers with bounded dimension, generalizing previous constructions and impacting algebraic K-theory.

Findings

Equivariant refinements exist for open covers under finite group actions.

The results are instrumental in proving the Farrell-Jones conjecture for GL over finite fields.

Generalizes previous cover constructions to equivariant settings.

Abstract

We show that if an open cover of a finite dimensional space is equivariant with respect to some finite group action on the space then there is an equivariant refinement of bounded dimension. This will generalize some constructions of certain covers. Those generalizations play a key role in the proof of the Farrell-Jones conjecture for the general linear group over a finite field.