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.References
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Bibliographic Information
- V. V. Chernov (Tchernov)
- Affiliation: Department of Mathematics, 6188 Kemeny Hall, Dartmouth College, Hanover, New Hampshire 03755
- Email: Vladimir.Chernov@dartmouth.edu
- Y. B. Rudyak
- Affiliation: Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, Florida 32611-8105
- Email: rudyak@math.ufl.edu
- Received by editor(s): November 14, 2006
- Published electronically: July 21, 2009
- © Copyright 2009 American Mathematical Society
- Journal: St. Petersburg Math. J. 20 (2009), 837-849
- MSC (2000): Primary 55M25; Secondary 53Z05, 57R35
- DOI: https://doi.org/10.1090/S1061-0022-09-01075-9
- MathSciNet review: 2492365