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: The Casson invariant is unique (with respect to the above properties) up to an overall multiplicative constant.
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).
The representation space of a compact oriented 3-manifold M is defined as
denotes the space of irreducible SU(2) representations of
, the Casson invariant equals
times the algebraic intersection of
Kevin Walker found an extension of the Casson invariant to rational homology 3-spheres.
A Casson-Walker invariant is a surjective map λCW from oriented rational homology 3-spheres to Q satisfying the following properties: 1. λ(S3) = 0.
For every 1-component Dehn surgery presentation (K, μ) of an oriented rational homology sphere M′ in an oriented rational homology sphere M: where: Note that for integer homology spheres, the Walker's normalization is twice that of Casson's:
Christine Lescop defined an extension λCWL of the Casson-Walker invariant to oriented compact 3-manifolds.
It is uniquely characterized by the following properties: The Casson–Walker–Lescop invariant has the following properties: In 1990, C. Taubes showed that the SU(2) Casson invariant of a 3-homology sphere M has a gauge theoretic interpretation as the Euler characteristic of
is the group of gauge transformations.
(Taubes (1990)) H. Boden and C. Herald (1998) used a similar approach to define an SU(3) Casson invariant for integral homology 3-spheres.