Casson invariant

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 found an extension to rational homology 3-spheres, called the Casson–Walker invariant, and Christine Lescop 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:

λ = 0.
Let Σ be an integral homology 3-sphere. Then for any knot K and for any integer n, the difference
For any boundary link K ∪ L in Σ the following expression is zero:

The Casson invariant is unique up to an overall multiplicative constant.

Properties

If K is the trefoil then
The Casson invariant is 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
The Casson invariant is the degree 1 part of the Le–Murakami–Ohtsuki invariant.
The Casson invariant for the integer Homology Sphere is given by the formula:

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 representations of. For a Heegaard splitting of, 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 λ_CW from oriented rational homology 3-spheres to Q satisfying the following properties:
1. λ = 0.
2. For every 1-component Dehn surgery presentation of an oriented rational homology sphere M′ in an oriented rational homology sphere M:
where:

m is an oriented meridian of a knot K and μ is the characteristic curve of the surgery.
ν is a generator the kernel of the natural map H₁ → H₁.
is the intersection form on the tubular neighbourhood of the knot, N.
Δ is the Alexander polynomial normalized so that the action of t corresponds to an action of the generator of in the infinite cyclic cover of M−''K'', and is symmetric and evaluates to 1 at 1.

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 λ_CWL of the Casson-Walker invariant to oriented compact 3-manifolds. It is uniquely characterized by the following properties:

If the first Betti number of M is zero,
If the first Betti number of M is one,
If the first Betti number of M is two,
If the first Betti number of M is three, then for a,''b,c'' a basis for, then
If the first Betti number of M is greater than three,.

The Casson–Walker–Lescop invariant has the following properties:

When the orientation of M changes the behavior of depends on the first Betti number of M: if is M with the opposite orientation, then
For connect-sums of manifolds

[SU(N)]

In 1990, C. Taubes showed that the SU Casson invariant of a 3-homology sphere M has a gauge theoretic interpretation as the Euler characteristic of, where is the space of SU connections on M and is the group of gauge transformations. He regarded the Chern–Simons invariant as a -valued Morse function on and used invariance under perturbations to define an invariant which he equated with the SU Casson invariant.

H. Boden and C. Herald used a similar approach to define an SU(3) Casson invariant for integral homology 3-spheres.