Homological stability of topological moduli spaces

TL;DR

This paper develops a homological stability framework for topological moduli spaces using a canonical resolution, extending previous algebraic methods to geometric contexts and establishing new stability results.

Contribution

It introduces a new canonical resolution for graded modules over $E_2$-algebras, enabling homological stability proofs for various topological moduli spaces.

Findings

Homological stability for moduli spaces of high-dimensional manifolds.

Stability of twisted homology groups with finite degree coefficient systems.

New results on representation stability for ordered variants of moduli spaces.

Abstract

Given a graded $E_1$-module over an $E_2$-algebra in spaces, we construct an augmented semi-simplicial space up to higher coherent homotopy over it, called its canonical resolution, whose graded connectivity yields homological stability for the graded pieces of the module with respect to constant and abelian coefficients. We furthermore introduce a notion of coefficient systems of finite degree in this context and show that, without further assumptions, the corresponding twisted homology groups stabilize as well. This generalizes a framework of Randal-Williams and Wahl for families of discrete groups. In many examples, the canonical resolution recovers geometric resolutions with known connectivity bounds. As a consequence, we derive new twisted homological stability results for e.g. moduli spaces of high-dimensional manifolds, unordered configuration spaces of manifolds with labels in a fibration, and moduli spaces of manifolds equipped with unordered embedded discs. This in turn implies representation stability for the ordered variants of the latter examples.