Abstract: We examine the action of the fundamental group $\Gamma$ of the $m$-punctured disc on the middle dimensional homology of a regular fiber in a Lefschetz fibration. We give an astract algebraic framework, based on the Picard-Lefschetz formula, to describe the extent to which 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. The intersection numbers along straight lines, which conjecturally make sense in infinite dimensional situations, contain all the relevant information. Our proof relies on the construction of a canonical non-abelian 1-cohomology class in $H^1(\widetilde {\mathcal B},GL_m(\mathbb{Z}[\Gamma]))$. This last construction remains valid for higher genus closed surfaces.