Homological stability for configuration spaces of manifolds (Q421026)

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.