Configuration-mapping spaces and homology stability

Abstract

For a given bundle \(\xi :E \rightarrow M\) over a manifold, configuration-section spaces 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 \(\mathbb {Z}\) 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.