Stable Reduction and Topological Invariants of Complex Polynomials

A topological invariant of a polynomial map p : X → B from a complex surface containing a curve C ⊂ X to a one-dimensional base is given by a rational second homology class in the compactification of the moduli space of genus g curves with n labeled points Mg,n. Here the generic fibre of p has genus g and intersects C in n points. In this paper we give an efficient method to calculate this homology class. We apply this to any polynomial in two complex variables p : C → C where the n points on a fibre are its points at infinity.