Casson's invariant and gauge theory

Recently, Andrew Casson [7] (see [2] or [16]) defined an integer valued invariant for closed 3-manifolds with the homology of S 3 .Casson gave a topological definition of his invariant.An analytic definition of Casson's invariant is the subject of this article.Roughly speaking, Casson's invariant can be defined using gauge theory as an infinite dimensional generalization of the classical Euler characteristic.The article begins below with an introduction to the relevant geometry of the space of connections on a homology 3-sphere.In §2, the definition of an integer valued invariant of an oriented homology 3-sphere is given.The construction of Casson's integer valued invariant is reviewed in §3 where the main theorem is stated: These two invariants are equal.§ §4-8 are occupied with the proof of the main theorem.There is also a technical appendix. Gauge theory in 3 dimensions