The Casson invariant and applications

Abstract

This paper familiarizes the reader with basic concepts and results in the topology of 3 and 4-manifolds before introducing the Rohlin invariant, a mod 2 invariant of oriented integral homology 3-spheres Y. The paper proceeds to introduce the Casson invariant as a signed count of irreducible SU(2) representations of the fundamental group of Y and contains proofs of its uniqueness, simple computational examples, a sketch of a proof of its existence, and generalization to the Casson-Walker invariant for rational homology 3-spheres. The paper concludes by highlighting the role played by the Casson invariant in a couple of applications: combinatorial triangulations of manifolds and the cosmetic surgery conjecture for 2-bridge knots.