ON GENERALIZED WINDING NUMBERS

Let M(m) be an oriented manifold, let N(m-1) be an oriented closed manifold, and let p be a point in M(m). For a smooth map f : N(m-1 ->)M(m), p is not an element of Im f, an invariant awin(p)(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 awin(p) estimates from below the number of passages of a wave front on M through a given point p is an element of M between two moments of time. The invariant awin(p) makes it possible to formulate an analog of the complex analysis Cauchy integral formula for meromorphic functions on complex surfaces of genus exceeding one.