Heisenberg homology on surface configurations

Christian Blanchet, Martin Palmer, Awais Shaukat

Published 2021-09-01Version 1

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.