Twisted homological stability for configuration spaces

TL;DR

This paper extends classical homological stability results for configuration spaces of manifolds to include twisted coefficients, using spectral sequences and generalising prior symmetric group methods.

Contribution

It introduces a notion of finite-degree twisted coefficient systems for configuration spaces and proves their homological stability, broadening the scope of classical results.

Findings

Homological stability holds for twisted coefficients in configuration spaces.

Spectral sequence techniques are effective in proving twisted stability.

Generalises methods from symmetric groups to configuration spaces.

Abstract

Let M be an open, connected manifold. A classical theorem of McDuff and Segal states that the sequence of configuration spaces of n unordered, distinct points in M is homologically stable with coefficients in Z: in each degree, the integral homology is eventually independent of n. The purpose of this note is to prove that this phenomenon also holds for homology with twisted coefficients. We first define an appropriate notion of finite-degree twisted coefficient system for configuration spaces and then use a spectral sequence argument to deduce the result from the untwisted homological stability result of McDuff and Segal. The result and the methods are generalisations of those of Betley for the symmetric groups.