Martin David Palmer -- Preprint -- Homology of configuration-mapping and -section spaces
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Homology of configuration-mapping and -section spaces
with Ulrike Tillmann
(v1: 2020)

arXiv: abspdf
v1

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, and describe a model for the stable homology. 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 and configuration-mapping spaces.