Configuration-mapping spaces and homology stability
Research in the Mathematical Sciences vol. 8 (2021) Article number: 38 (45 pp.)
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Abstract For a given bundle ξ : E → M over a manifold, configuration-section spaces on ξ parametrise finite subsets z ⊆ M equipped with a section of ξ defined on M ∖ 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. |
Here is the companion paper, on point-pushing actions for manifolds with boundary. |
Here are the notes of a talk that I gave, based on the results of this paper (and the companion paper), on 21 October 2020 at the Purdue Topology Seminar. The video of this talk is here.
Here is a poster on the results of this paper, from the BMC–BAMC (Glasgow) in April 2021. |