Homological stability for unordered configuration spaces

TL;DR

This paper proves homological stability for unordered configuration spaces in open manifolds with labels, improves stability ranges in rational homology for manifolds of dimension at least three, and explores stability phenomena in closed manifolds.

Contribution

It provides a self-contained proof of homological stability with optimal ranges, introduces new proofs for related complexes, and analyzes stability in closed manifolds over various fields.

Findings

Homological stability holds for configuration spaces in open manifolds with integral ranges up to 2n.

Rational homology stability range improves to n for manifolds of dimension at least three.

Stability of homology dimensions in closed manifolds holds for odd dimensions and certain fields.

Abstract

This paper consists of two related parts. In the first part we give a self-contained proof of homological stability for the spaces C_n(M;X) of configurations of n unordered points in a connected open manifold M with labels in a path-connected space X, with the best possible integral stability range of 2* \leq n. Along the way we give a new proof of the high connectivity of the complex of injective words. If the manifold has dimension at least three, we show that in rational homology the stability range may be improved to * \leq n. In the second part we study to what extent the homology of the spaces C_n(M) can be considered stable when M is a closed manifold. In this case there are no stabilisation maps, but one may still ask if the dimensions of the homology groups over some field stabilise with n. We prove that this is true when M is odd-dimensional, or when the field is F_2 or Q. It is known to be false in the remaining cases.