Segal's spectral sequence in twisted equivariant K-theory for proper and discrete actions

TL;DR

This paper applies Segal's spectral sequence to twisted G-equivariant K-theory for proper, discrete group actions, revealing its relation to Bredon cohomology and analyzing differentials, including cases with finite order twists.

Contribution

It demonstrates the isomorphism between the second page of Segal's spectral sequence and Bredon cohomology with twisted coefficients, and explores the third differential phenomena.

Findings

Second page is isomorphic to Bredon cohomology with twisted coefficients

Analysis of the third differential in the spectral sequence

Recovery of known results for finite order twists in discrete torsion

Abstract

We use a spectral sequence developed by Graeme Segal in order to understand the twisted G-equivariant K-theory for proper and discrete actions. We show that the second page of this spectral sequence is isomorphic to a version of Bredon cohomology with local coefficients in twisted representations. We furthermore explain some phenomena concerning the third differential of the spectral sequence, and we recover known results when the twisting comes from finite order elements in discrete torsion.