Twisted homological stability for configuration spaces

Let $M$ be an open, connected manifold. A classical theorem of McDuff and Segal states that the sequence $\lbrace C_n (M) \rbrace$ of configuration spaces of $n$ unordered, distinct points in $M$ is homologically stable with coefficients in $\mathbb{Z}$ — in each degree, the integral homology is eventually independent of $n$. The purpose of this paper 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 $\lbrace C_n (M) \rbrace$ 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.

Keywords

configuration space, homological stability, polynomial twisted coefficients

2010 Mathematics Subject Classification

55R80, 57N65

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Received 17 December 2014

Received revised 17 December 2017

Published 16 May 2018