Twisted homological stability for 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.

v3: 25 pages

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Cited by in corpus (13)

• Representations of categories of G-maps
• Higher order representation stability and ordered configuration spaces of manifolds
• Homological stability of topological moduli spaces
• Improved homological stability for configuration spaces after inverting 2
• Topological representations of motion groups and mapping class groups -- a unified functorial construction
• Homological stability for moduli spaces of disconnected submanifolds, I
• A comparison of twisted coefficient systems
• Configuration-mapping spaces and homology stability
• Stability for moduli spaces of manifolds with conical singularities
• Vertical configuration spaces and their homology
• Homology stability for asymptotic monopole moduli spaces
• Motivic homological stability of configuration spaces
• Configuration spaces of clusters as $E_d$-algebras