On kernels of homological representations of mapping class groups

TL;DR

This paper investigates the kernels of homological representations of mapping class groups, linking them to twisted intersection pairings, and identifies conditions under which these representations are faithful, especially for configurations with multiple points.

Contribution

It introduces a new approach to analyze kernels of homological representations using twisted intersection pairings and establishes faithfulness results for certain subgroups.

Findings

Kernel elements identified in twisted intersection form for specific configurations.

Faithfulness of representations on subgroups is established, including pure braid groups.

Connections made between homological representations and Lawrence representations.

Abstract

We study the kernels of representations of mapping class groups of surfaces on twisted homologies of configuration spaces. We relate them with the kernel of a natural twisted intersection pairing: if the latter kernel is trivial then the representation is faithful. As a main example, we study the representations $\rho_{n}$ of $\mathrm{Mod}(\Sigma_{g,1})$ based on a Heisenberg local system on the $n$ points configuration space of $\Sigma_{g,1}$, introduced by Blanchet--Palmer--Shaukat, and some of their specializations. In the one point configuration case, or when the Heisenberg group is quotiented by an element of its center, we find kernel elements in the twisted intersection form. On the other hand, for $n>2$ configuration points and the full Heisenberg local system, we identify subrepresentations of subgroups of $\mathrm{Mod}(\Sigma_{g,1})$ with Lawrence representations. In particular, we find one of these subgroups, which is isomorphic to a pure braid group on $g$ strands, on which the representations $\rho_n$ are faithful.