Decoupling generalised configuration spaces on surfaces

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.