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 $f\colon N^{m-1}\to M^m$, $p\notin\operatorname{Im}f$, an invariant $\operatorname{awin}_p(f)$ 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 $\operatorname{awin}_p$ estimates from below the number of passages of a wave front on $M$ through a given point $p\in M$ between two moments of time. The invariant $\operatorname{awin}_p$ makes it possible to formulate an analog of the complex analysis Cauchy integral formula for meromorphic functions on complex surfaces of genus exceeding one.