Casson invariant

Definition

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 Σ + 1 m ⋅ K denotes 1 m 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.

Properties

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 ϕ 1 ( K ) is the coefficient of z 2 in the Alexander-Conway polynomial ∇ K ( z ) , and is congruent (mod 2) to the Arf invariant of K.

The Casson invariant is the degree 1 part of the LMO invariant.

The Casson invariant for the Seifert manifold Σ ( p , q , r ) is given by the formula:

where

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 R ( M ) = R i r r ( M ) / S O ( 3 ) where R i r r ( M ) denotes the space of irreducible SU(2) representations of π 1 ( M ) . For a Heegaard splitting Σ = M 1 ∪ F M 2 of M , the Casson invariant equals ( − 1 ) g 2 times the algebraic intersection of R ( M 1 ) with R ( M 2 ) .

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(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:

λ C W ( M ′ ) = λ C W ( M ) + ⟨ m , μ ⟩ ⟨ m , ν ⟩ ⟨ μ , ν ⟩ Δ W ′ ′ ( M − K ) ( 1 ) + τ W ( m , μ ; ν )

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(∂N(K), Z) → H₁(M−K, Z).

⟨ ⋅ , ⋅ ⟩ is the intersection form on the tubular neighbourhood of the knot, N(K).

Δ is the Alexander polynomial normalized so that the action of t corresponds to an action of the generator of H 1 ( M − K ) / Torsion in the infinite cyclic cover of M−K, and is symmetric and evaluates to 1 at 1.

τ W ( m , μ ; ν ) = − s g n ⟨ y , m ⟩ s ( ⟨ x , m ⟩ , ⟨ y , m ⟩ ) + s g n ⟨ y , μ ⟩ s ( ⟨ x , μ ⟩ , ⟨ y , μ ⟩ ) + ( δ 2 − 1 ) ⟨ m , μ ⟩ 12 ⟨ m , ν ⟩ ⟨ μ , ν ⟩

where x, y are generators of H₁(∂N(K), Z) such that ⟨ x , y ⟩ = 1 , v = δy for an integer δ and s(p, q) is the Dedekind sum.

Note that for integer homology spheres, the Walker's normalization is twice that of Casson's: λ C W ( M ) = 2 λ ( M ) .

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,

where Δ is the Alexander polynomial normalized to be symmetric and take a positive value at 1.

If the first Betti number of M is two,

where γ is the oriented curve given by the intersection of two generators S 1 , S 2 of H 2 ( M ; Z ) and γ ′ is the parallel curve to γ induced by the trivialization of the tubular neighbourhood of γ determined by S 1 , S 2 .

If the first Betti number of M is three, then for a,b,c a basis for H 1 ( M ; Z ) , then

If the first Betti number of M is greater than three, λ C W L ( M ) = 0 .

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

If the orientation of M, then if the first Betti number of M is odd the Casson–Walker–Lescop invariant is unchanged, otherwise it changes sign.

For connect-sums of manifolds

SU(N)

In 1990, C. Taubes showed that the SU(2) Casson invariant of a 3-homology sphere M has gauge theoretic interpretation as the Euler characteristic of A / G , where A is the space of SU(2) connections on M and G is the group of gauge transformations. He led Chern–Simons invariant as a S 1 -valued Morse function on A / G and pointed out that the SU(3) Casson invariant is important to make the invariants independent on perturbations. (Taubes (1990))

Boden and Herald (1998) defined an SU(3) Casson invariant.