On kernels of homological representations of mapping class groups

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.