Casson Invariant and Gauge Theory

Abstract

In this chapter, we give an account of SU(2)-gauge theory in dimension three. We discuss C. Taubes’ gauge-theoretical definition of the Casson invariant as (roughly) the Euler number of the gradient field of the Chern-Simons function. The Chern-Simons function plays a central role in modern understanding of homology 3-spheres, so we discuss it in some detail. An infinite dimensional analogue of Morse theory applied to the Chern-Simons function produces the instanton Floer homology which will be discussed in the next chapter. This gauge-theoretical approach to the Casson invariant leads to several extensions in a direction different from that of Walker and Lescop. One of the extensions we discuss is the SU(3) Casson invariant of H. Boden and C. Herald. Another one is the Casson-type invariant for knots in integral homology spheres introduced by X.-S. Lin and C. Herald, and finally, the equivariant Casson invariant of integral homology spheres with a finite cyclic group action by O. Collin and the author.