Concept
In 3-dimensional topology, a part of the mathematical field of geometric topology, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson. Kevin Walker (1992) found an extension to rational homology 3-spheres, called the Casson–Walker invariant, and Christine Lescop (1995) extended the invariant to all closed oriented 3-manifolds. A Casson invariant is a surjective map λ from oriented integral homology 3-spheres to Z satisfying the following properties: λ(S3) = 0. Let Σ be an integral homology 3-sphere. Then for any knot K and for any integer n, the difference is independent of n. Here denotes Dehn surgery on Σ by K. For any boundary link K ∪ L in Σ the following expression is zero: The Casson invariant is unique (with respect to the above properties) up to an overall multiplicative constant. If K is the trefoil then The Casson invariant is 1 (or −1) for the Poincaré homology sphere. The Casson invariant changes sign if the orientation of M is reversed. The Rokhlin invariant of M is equal to the Casson invariant mod 2. The Casson invariant is additive with respect to connected summing of homology 3-spheres. The Casson invariant is a sort of Euler characteristic for Floer homology. For any integer n where is the coefficient of in the Alexander–Conway polynomial , and is congruent (mod 2) to the Arf invariant of K. The Casson invariant is the degree 1 part of the Le–Murakami–Ohtsuki invariant. The Casson invariant for the Seifert manifold is given by the formula: where Informally speaking, the Casson invariant counts half the number of conjugacy classes of representations of the fundamental group of a homology 3-sphere M into the group SU(2). This can be made precise as follows. The representation space of a compact oriented 3-manifold M is defined as where denotes the space of irreducible SU(2) representations of . For a Heegaard splitting of , the Casson invariant equals times the algebraic intersection of with .