3-Manifold Invariants

In Chapter 9 we discussed the geometry and topology of moduli spaces of gauge fields on a manifold. In recent years these moduli spaces have been extensively studied for manifolds of dimensions 2, 3, and 4 (collectively referred to as low-dimensional manifolds). This study was initiated for the 2-dimensional case in [17]. Even in this classical case, the gauge theory perspective provided fresh insights as well as new results and links with physical theories. We make only a passing reference to this case in the context of Chern–Simons theory. In this chapter, we mainly study various instanton invariants of 3-manifolds. The material of this chapter is based in part on [263]. The basic ideas come from Witten’s work on supersymmetric Morse theory. We discuss this work in Section 10.2. In Section 10.3 we consider gauge fields on a 3-dimensional manifold. The field equations are obtained from the Chern–Simons action functional and correspond to flat connections. Casson invariant is discussed in Section 10.4. In Section 10.5 we discuss the Z 8-graded instanton homology theory due to Floer and its relation to the Casson invariant. Floer’s theory was extended to arbitrary closed oriented 3-manifolds by Fukaya. When the first homology of such a manifold is torsion-free, but not necessarily zero, Fukaya also defines a class of invariants indexed by the integer s, 0 ≤ s