An Overview of the Casson Invariant

Abstract

In this thesis, I present an exposition on the Casson invariant, an invariant of integral homology 3-spheres. The thesis is intended for students who have studied basic topology: I cover the necessary background material on 3-manifolds and knots. This includes a definition of the Rokhlin invariant, which the Casson invariant generalizes. I describe the Casson invariant both axiomatically and through its explicit construction in terms of counting points in representation spaces of the fundamental group of a space. I also survey recent work on the Casson invariant, including generalizations such as the Casson-Walker invariant and applications to the cosmetic surgery conjecture.