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.

Definition

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.

Properties

The Casson invariant as a count of representations

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 \pi_1 (M). For a Heegaard splitting of M, the Casson invariant equals times the algebraic intersection of with.

Generalizations

Rational homology 3-spheres

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. 2. 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:.

Compact oriented 3-manifolds

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:

SU(N)

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, where \mathcal{A} is the space of SU(2) connections on M and \mathcal{G} is the group of gauge transformations. He regarded the Chern–Simons invariant as a S^1-valued Morse function on and used invariance under perturbations to define an invariant which he equated with the SU(2) Casson invariant. H. Boden and C. Herald (1998) used a similar approach to define an SU(3) Casson invariant for integral homology 3-spheres.