  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.
