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      Heisenberg homology on surface configurations

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          Abstract

          We study the action of the mapping class group of \(\Sigma = \Sigma_{g,1}\) on the homology of configuration spaces with coefficients twisted by the discrete Heisenberg group \(\mathcal{H} = \mathcal{H}(\Sigma)\), or more generally by any representation \(V\) of \(\mathcal{H}\). In general, this is a twisted representation of the mapping class group \(\mathfrak{M}(\Sigma)\) and restricts to an untwisted representation on the Chillingworth subgroup \(\mathrm{Chill}(\Sigma) \subseteq \mathfrak{M}(\Sigma)\). Moreover, it may be untwisted on the Torelli group \(\mathfrak{T}(\Sigma)\) by passing to a \(\mathbb{Z}\)-central extension, and, in the special case where we take coefficients in the Schr\"odinger representation of \(\mathcal{H}\), it may be untwisted on the full mapping class group \(\mathfrak{M}(\Sigma)\) by passing to a double covering. We illustrate our construction with several calculations for \(2\)-point configurations, in particular for genus-\(1\) separating twists.

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          Author and article information

          Journal
          01 September 2021
          Article
          2109.00515
          91585156-6a16-4baf-ad9f-426f088eb15d

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          57K20, 55R80, 55N25, 20C12
          61 pages, 13 figures
          math.GT math.AT

          Geometry & Topology
          Geometry & Topology

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