Homological stability for configuration spaces of manifolds

TL;DR

This paper proves that the homology groups of configuration spaces of manifolds exhibit representation stability, leading to classical homological stability results for unordered configurations, with new techniques involving monotonicity of representations.

Contribution

It introduces the concept of monotonicity for S_n--representations and applies it to establish homological stability for configuration spaces of all connected orientable manifolds.

Findings

Homology groups H_i(C_n(M); Q) are representation stable for all connected orientable manifolds.

Unordered configuration spaces B_n(M) satisfy classical homological stability for n > i.

New method using spectral sequences and monotonicity of representations.

Abstract

Let C_n(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H_i(C_n(M); Q) are representation stable in the sense of [Church-Farb]. Applying this to the trivial representation, we obtain as a corollary that the unordered configuration space B_n(M) satisfies classical homological stability: for each i, H_i(B_n(M); Q) is isomorphic to H_i(B_{n+1}(M); Q) for n > i. This improves on results of McDuff, Segal, and others for open manifolds. Applied to closed manifolds, this provides natural examples where rational homological stability holds even though integral homological stability fails. To prove the main theorem, we introduce the notion of monotonicity for a sequence of S_n--representations, which is of independent interest. Monotonicity provides a new mechanism for proving representation stability using spectral sequences. The key technical point in the main theorem is that certain sequences of induced representations are monotone.