Decoupling generalised configuration spaces on surfaces

TL;DR

This paper proves a decoupling theorem for generalized configuration spaces on surfaces, showing their homology can be described by simpler components and establishing homological stability with respect to genus increases.

Contribution

It introduces a decoupling theorem for homotopy quotients of generalized configuration spaces on surfaces, enabling analysis of their homology and stability properties.

Findings

Homology of these spaces is described by the product of moduli space and generalized configuration space.

Established homological stability for these spaces as the genus increases.

Identified the stable homology and related it to Diff-equivariant homological stability.

Abstract

The configuration space of k points on a manifold carries an action of its diffeomorphism group. The homotopy quotient of this action is equivalent to the classifying space of diffeomorphisms of a punctured manifold, and therefore admits results about homological stability. Inspired by the works of Segal, McDuff, Bodigheimer, and Salvatore, we look at generalised configuration spaces where particles have labels and even partially summable labels, in which points are allowed to collide whenever their labels are summable. These generalised configuration spaces also admit actions of the diffeomorphism group and we look at their homotopy quotients. Our main result is a decoupling theorem for these homotopy quotients on surfaces: in a range, their homology is completely described by the product of the moduli space of surfaces and a generalised configuration space of points in $\mathbb{R}^\infty$. Using this result, we show these spaces admit homological stability with respect to increasing the genus, and we identify the stable homology. This can be interpreted as a Diff-equivariant homological stability for factorization homology. In addition, we use this result to study the group completion of the monoid of moduli spaces of configurations on surfaces.