Type of Culminating Activity
Master of Science in Mathematics
Uwe Kaiser, Ph.D.
The use of rotation numbers in the classification of regular closed curves in the plane up to regular homotopy sparked the investigation of winding numbers to classify regular closed curves on other surfaces. Chillingworth  defined winding numbers for regular closed curves on particular surfaces and used them to classify orientation preserving regular closed curves that are based at a fixed point and direction. We define geometrically a group structure of the set of equivalence classes of regular closed curves based at a fixed point and direction. We prove this group structure coincides with the one introduced by Smale  via a weak homotopy equivalence. The set of equivalence classes of orientation preserving regular closed curves is a subgroup. This thesis investigates the relationship between this subgroup and the winding number of each element. Specifically, it is proven that this subgroup is isomorphic to the direct product of the integers with the group of orientation preserving closed curves up to homotopy where the isomorphism sends an equivalence class to its winding number and corresponding homotopy class. Using this result, we describe the subgroup for several surfaces by depicting representatives of generators.
Zebedeo, Katherine Kylee, "Regular Homotopy of Closed Curves on Surfaces" (2012). Boise State University Theses and Dissertations. 326.