Homological stability for configuration spaces of manifolds
type: Item
instance of: scholarly article
title: Homological stability for configuration spaces of manifolds
zbMATH Open document ID: 1244.55012
author: Thomas Church
published in: Inventiones Mathematicae
publication date: 2012-05-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1103.2441
review text: Let \(C_n(M)\) denote the configuration space of \(n\) distinct ordered points in \(M\). The author proves that the homology groups \(H_i(C_n(M);{\mathbb Q})\) are representation stable in the sense of \textit{T. Church} and \textit{B. Farb} [Representation theory and homological stability, \url{arXiv:1008.1368}]. It follows that the unordered configuration space \(B_n(M)\) satisfies the classical homological stability: \(H_i(B_n(M);\mathbb Q)\simeq H_i(B_{n+1}(M);\mathbb Q)\) for \(n>i\). This improves on the results of McDuff, Segal and others for open manifolds. Applied to closed manifolds, this provides examples where rational homological stability holds even though integral homological stability fails. The proofs use the notion of monotonicity for sequence of \(S_n\)-representations which provides a new mechanism for analysing stability using spectral sequences.
Mathematics Subject Classification ID: 55R80
zbMATH DE Number: 6037984
zbMATH Keywords: Configuration spaces
zbMATH Keywords: homological stability
reviewed by: Michael Farber
MaRDI profile type: Publication
OpenAlex ID: W1988649914
arXiv ID: 1103.2441
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DOI: 10.1007/S00222-011-0353-4
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description: scientific article; zbMATH DE number 6037984