Stable reduction and topological invariants of complex polynomials

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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. 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:²→ where the n points on a fibre are its points at infinity.