On generalized winding numbers

Chernov (Tchernov), V.; Rudyak, Y.

doi:10.1090/S1061-0022-09-01075-9

On generalized winding numbers
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by V. V. Chernov (Tchernov) and Y. B. Rudyak

St. Petersburg Math. J. 20 (2009), 837-849

DOI: https://doi.org/10.1090/S1061-0022-09-01075-9

Published electronically: July 21, 2009

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Abstract:

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 $f : N^{m-1} {\to } M^m, p\notin \operatorname {Im} f$, an invariant $\mathrm {awin}_p(f)$ 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 $\mathrm {awin}_p$ 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 $\mathrm {awin}_p$ makes it possible to formulate an analog of the complex analysis Cauchy integral formula for meromorphic functions on complex surfaces of genus exceeding one.

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