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

We examine the action of the fundamental group Γ of an mpunctured Riemann surface 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 B̃ on m strings on a suitable space of m × m matrices. This action is determined by a family of cocycles Sc : B̃ → GLm(Z[Γ]) parametrized by distinguished configurations c of embedded paths from the regular value to the critical values.