On generalized winding numbers

Abstract

Let Mm be an oriented manifold, let Nm−1 be an oriented closed manifold, and let p be a point in Mm. For a smooth map f: Nm−1→Mm, p ∈ Im f, an invariant awinp(f) is introduced, which can be regarded as a generalization of the classical winding number of a planar curve around a point. It is shown that awinp estimates from below the number of passages of a wave front on M through a given point p ∈ M between two moments of time. The invariant awinp makes it possible to formulate an analog of the complex analysis Cauchy integral formula for meromorphic functions on complex surfaces of genus exceeding one. © 2009 American Mathematical Society.