Configuration-mapping spaces and homology stability

TL;DR

This paper introduces configuration-section spaces on bundles over manifolds, generalizing Hurwitz spaces, and proves their homological stability under certain conditions, connecting topology with algebraic and geometric structures.

Contribution

It establishes homological stability for configuration-section spaces, extending the understanding of their topological properties and linking to Hurwitz space theory.

Findings

Configuration-section spaces are homologically stable with integral coefficients.

Stability holds for manifolds with boundary and small charges.

Connections are made to Hurwitz spaces and their properties.

Abstract

For a given bundle $\xi \colon E \to M$ over a manifold, configuration-section spaces on $\xi$ parametrise finite subsets $z \subseteq M$ equipped with a section of $\xi$ defined on $M \smallsetminus z$, with prescribed "charge" in a neighbourhood of the points $z$. These spaces may be interpreted physically as spaces of fields that are permitted to be singular at finitely many points, with constrained behaviour near the singularities. As a special case, they include the Hurwitz spaces, which parametrise branched covering spaces of the $2$-disc with specified deck transformation group. We prove that configuration-section spaces are homologically stable (with integral coefficients) whenever the underlying manifold $M$ is connected and has non-empty boundary and the charge is "small" in a certain sense. This has a partial intersection with the work on Hurwitz spaces of Ellenberg, Venkatesh and Westerland.