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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

On generalized winding numbers


Authors: V. V. Chernov (Tchernov) and Y. B. Rudyak
Original publication: Algebra i Analiz, tom 20 (2008), nomer 5.
Journal: St. Petersburg Math. J. 20 (2009), 837-849
MSC (2000): Primary 55M25; Secondary 53Z05, 57R35
Published electronically: July 21, 2009
MathSciNet review: 2492365
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Abstract | References | Similar Articles | Additional Information

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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Additional 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

DOI: http://dx.doi.org/10.1090/S1061-0022-09-01075-9
PII: S 1061-0022(09)01075-9
Keywords: Affine winding number, linking number, invariant
Received by editor(s): November 14, 2006
Published electronically: July 21, 2009
Article copyright: © Copyright 2009 American Mathematical Society