Whitney-type Formula for Non-null-homotopic Curves on Aspherical Surfaces

TL;DR

This paper extends a winding number formula to non-null-homotopic curves on complete euclidean and hyperbolic surfaces, generalizing previous results on tori, and linking winding numbers with regular homotopy classes.

Contribution

It provides a Whitney-type formula for the winding number of non-null-homotopic curves on aspherical surfaces with Euclidean or hyperbolic geometry, broadening prior work on tori.

Findings

Derived a generalized Whitney-type formula for winding numbers

Connected winding number with regular homotopy classes on aspherical surfaces

Extended previous formulas from tori to more general surfaces

Abstract

In an earlier paper, I defined a new winding number of regular closed curves on complete euclidean/hyperbolic surfaces and showed that this winding number, together with the free homotopy class, determines the regular homotopy class. In this paper, I give a Whitney-type formula for the winding number of non-null-homotopic generic regular closed curves on surfaces with a complete euclidean or hyperbolic structure, generalizing the formula for curves on a torus by Tanio and Kobayashi.