Martin David Palmer -- Preprint -- Heisenberg homology on surface configurations
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Heisenberg homology on surface configurations
with Christian Blanchet and Awais Shaukat
(v1: 2021, v2: 2021)

arXiv: abspdf
v2v1

Abstract

We study the action of the mapping class group of Σ = Σg,1 on the homology of configuration spaces with coefficients twisted by the discrete Heisenberg group H = H(Σ), or more generally by any representation V of H.
In general, this is a twisted representation of the mapping class group M(Σ) and restricts to an untwisted representation on the Chillingworth subgroup. We also show how this may be modified to produce an untwisted representation of the Torelli group. Moreover, in the special case where we take coefficients in the Schrödinger representation of H, we show how this action induces an untwisted representation of the stably universal central extension of the full mapping class group M(Σ), as well as a native representation of a large subgroup of the mapping class group that we will call the Morita subgroup.
We illustrate our constructions with several calculations in the case of 2-point configurations, in particular for genus-1 separating twists.

Here are the notes of a talk that I gave, based on the results of this paper, on 12 November 2021 at the IMAR Topology Seminar.