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)

arXiv: abspdf
v1

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 Chill(Σ) ⊆ M(Σ). Moreover, it may be untwisted on the Torelli group T(Σ) by passing to a Z-central extension, and, in the special case where we take coefficients in the Schrödinger representation of H, it may be untwisted on the full mapping class group M(Σ) by passing to a double covering. We illustrate our construction with several calculations for 2-point configurations, in particular for genus-1 separating twists.