Lefschetz fibrations, intersection numbers, and representations of the framed braid group

TL;DR

This paper explores how the action of the fundamental group on homology in Lefschetz fibrations can be reconstructed from intersection numbers, revealing new connections with braid group representations and cocycles.

Contribution

It introduces a family of cohomologous 1-cocycles parametrized by configurations, linking intersection data with braid group actions and comparing with Magnus cocycles.

Findings

The action of the fundamental group can be recovered from intersection numbers of vanishing cycles.

A family of cocycles parametrized by configurations describes the braid group action on matrices.

Intersection numbers along straight lines encode all relevant information in the disc case.

Abstract

We examine the action of the fundamental group $\Gamma$ of a Riemann surface with $m$ punctures on the middle dimensional homology of a regular fiber in a Lefschetz fibration, and describe to what extent this action can be recovered from the intersection numbers of vanishing cycles. Basis changes for the vanishing cycles result in a nonlinear action of the framed braid group $\widetilde{\mathcal B}$ on $m$ strings on a suitable space of $m\times m$ matrices. This action is determined by a family of cohomologous 1-cocycles ${\mathcal S}_c:\widetilde{\mathcal B}\to GL_m({\mathbb{Z}}[\Gamma])$ parametrized by distinguished configurations $c$ of embedded paths from the regular value to the critical values. In the case of the disc, we compare this family of cocycles with the Magnus cocycles given by Fox calculus and consider some abelian reductions giving rise to linear representations of braid groups. We also prove that, still in the case of the disc, the intersection numbers along straight lines, which conjecturally make sense in infinite dimensional situations, carry all the relevant information.