nLab Casson invariant

Idea

Let $M$ be an integral homology 3-sphere. Then one can consider the moduli space of equivalence classes of $SU(2)$-valued linear representations of $\pi_1(M)$, i.e. $SU(2)$-local systems. The Casson invariant counts them algebraically to yields an integral invariant of $M$.

There is a generalization to rational homology 3-spheres, so called Casson-Walker invariant. A holomorphic counterpart for (Calabi-Yau 3-folds) of the Casson invariant are essentially the Donaldson-Thomas invariants, or physically counting BPS states.

Floer homology of 3-manifolds, which uses the Chern-Simons functionals as Morse functions yields some related invariants as well.

• Wikipedia Casson invariant

• Kevin Walker, An extension of Casson invariant to rational homology spheres, Bull. AMS, 22, 2 (1990) 261–267; A generalization of Casson invariant, Annals of Math. Studies 126, Princeton Univ. Press 1992.; and the book review by W. Goldman: ps, dvi

• C. H. Taubes, Casson’s invariant and gauge theory, J. Diff. Geom 31 (1990) 547–599, euclid, MR1037415

• eom: Chern–Simons functional