Secondary Homological Stability for Unordered Configuration Spaces

TL;DR

This paper establishes secondary homological stability for unordered configuration spaces of connected manifolds, including closed manifolds, by constructing a new chain-level stabilization map to handle periodic homology.

Contribution

It introduces a novel chain-level stabilization map that proves secondary homological stability for unordered configuration spaces, especially addressing the challenging case of closed manifolds.

Findings

Proves secondary homological stability for unordered configuration spaces.

Constructs a new chain-level stabilization map for configuration spaces.

Addresses the case of closed manifolds with periodic homology.

Abstract

Secondary homological stability is a recently discovered stability pattern for the homology of a sequence of spaces exhibiting homological stability in a range where homological stability does not hold. We prove secondary homological stability for the homology of the unordered configuration spaces of a connected manifold. The main difficulty is the case that the manifold is closed because there are no obvious maps inducing stability and the homology eventually is periodic instead of stable. We resolve this issue by constructing a new chain-level stabilization map for configuration spaces.