Winding numbers of regular closed curves on aspherical surfaces
Masayuki Yamasaki
TL;DR
This paper introduces winding numbers for regular closed curves on surfaces with Euclidean or hyperbolic geometry, establishing a criterion for regular homotopy based on free homotopy and winding number equality.
Contribution
It defines winding numbers on aspherical surfaces and proves their role in classifying regular homotopy classes of closed curves.
Findings
Winding numbers are well-defined for curves on aspherical surfaces.
Two curves are regularly homotopic iff they are freely homotopic and share the same winding number.
Abstract
We define winding numbers of regular closed curves on surfaces with a nice euclidean or hyperbolic geometry. We prove that two regular closed curves are regularly homotopic if and only if they are freely homotopic and have the same winding number.
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Taxonomy
TopicsGeometric and Algebraic Topology · Mathematical Dynamics and Fractals · Homotopy and Cohomology in Algebraic Topology