Martin David Palmer -- Paper -- Twisted homological stability for configuration spaces
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Twisted homological stability for configuration spaces

Homology, Homotopy and Applications vol. 20 no. 2 (2018) pp. 145-178
arXiv: abspdf


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.

Extended notes (22 Dec 2017)

Here (pdf) is a collection of extended remarks, explanations and examples that complement the main paper, but which have not been included (or which have been abridged) in the paper in order to keep it reasonably compact.