Curves on Surfaces

Abstract

This project report concerns the study of simple closed curves on surfaces. Specifically, we consider the question of when a closed curves on a torus can be deformed to a simple closed curve. We also consider applications of this. In Chapter 2, we discuss preliminary results from Algebraic Topology. This allows us to formulate the main question as representing conjugacy classes in the fundamental group in terms of simple curves. In Chapter 3, we consider the simplest case, that of simple curves on the torus. We see that a non-trivial class can be represented by a simple curve if and only if it is primitive. We also consider applications. Chapter 4 is an exposition of Chillingworth’s theory of winding numbers, which gives a criterion for representing by a simple curve. In Chapter 5, we discuss various problems in low-dimensional topology that are related to the question of the existence of simple closed curves