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 (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:
 * λ(S3) = 0.
 * Let Σ be an integral homology 3-sphere. Then for any knot K and for any integer n, the difference
 * $$\lambda\left(\Sigma+\frac{1}{n+1}\cdot K\right)-\lambda\left(\Sigma+\frac{1}{n}\cdot K\right)$$
 * is independent of n. Here $$\Sigma+\frac{1}{m}\cdot K$$ denotes $$\frac{1}{m}$$ Dehn surgery on Σ by K.


 * For any boundary link K ∪ L in Σ the following expression is zero:
 * $$\lambda\left(\Sigma+\frac{1}{m+1}\cdot K+\frac{1}{n+1}\cdot L\right) -\lambda\left(\Sigma+\frac{1}{m}\cdot K+\frac{1}{n+1}\cdot L\right)-\lambda\left(\Sigma+\frac{1}{m+1}\cdot K+\frac{1}{n}\cdot L\right) +\lambda\left(\Sigma+\frac{1}{m}\cdot K+\frac{1}{n}\cdot L\right)$$

The Casson invariant is unique (with respect to the above properties) up to an overall multiplicative constant.

Properties

 * If K is the trefoil then
 * $$\lambda\left(\Sigma+\frac{1}{n+1}\cdot K\right)-\lambda\left(\Sigma+\frac{1}{n}\cdot K\right)=\pm 1$$.


 * 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
 * $$\lambda \left ( M + \frac{1}{n+1}\cdot K\right ) - \lambda \left ( M + \frac{1}{n}\cdot K\right ) = \phi_1 (K), $$
 * where $$\phi_1 (K)$$ is the coefficient of $$z^2$$ in the Alexander–Conway polynomial $$\nabla_K(z)$$, 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 $$\Sigma(p,q,r)$$ is given by the formula:
 * $$ \lambda(\Sigma(p,q,r))=-\frac{1}{8}\left[1-\frac{1}{3pqr}\left(1-p^2q^2r^2+p^2q^2+q^2r^2+p^2r^2\right)

-d(p,qr)-d(q,pr)-d(r,pq)\right]$$
 * where
 * $$d(a,b)=-\frac{1}{a}\sum_{k=1}^{a-1}\cot\left(\frac{\pi k}{a}\right)\cot\left(\frac{\pi bk}{a}\right)$$

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 $$\mathcal{R}(M)=R^{\mathrm{irr}}(M)/SU(2)$$ where $$R^{\mathrm{irr}}(M)$$ denotes the space of irreducible SU(2) representations of $$\pi_1 (M)$$. For a Heegaard splitting $$\Sigma=M_1 \cup_F M_2$$ of $$M$$, the Casson invariant equals $$\frac{(-1)^g}{2}$$ times the algebraic intersection of $$\mathcal{R}(M_1)$$ with $$\mathcal{R}(M_2)$$.

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 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:
 * $$\lambda_{CW}(M^\prime)=\lambda_{CW}(M)+\frac{\langle m,\mu\rangle}{\langle m,\nu\rangle\langle \mu,\nu\rangle}\Delta_{W}^{\prime\prime}(M-K)(1)+\tau_{W}(m,\mu;\nu)$$

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).
 * $$\langle\cdot,\cdot\rangle$$ 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 $$H_1(M-K)/\text{Torsion}$$ in the infinite cyclic cover of M−K, and is symmetric and evaluates to 1 at 1.
 * $$\tau_{W}(m,\mu;\nu)= -\mathrm{sgn}\langle y,m\rangle s(\langle x,m\rangle,\langle y,m\rangle)+\mathrm{sgn}\langle y,\mu\rangle s(\langle x,\mu\rangle,\langle y,\mu\rangle)+\frac{(\delta^2-1)\langle m,\mu\rangle}{12\langle m,\nu\rangle\langle \mu,\nu\rangle}$$
 * where x, y are generators of H1(∂N(K), Z) such that $$\langle x,y\rangle=1$$, 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: $$ \lambda_{CW}(M) = 2 \lambda(M) $$.

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:
 * If the first Betti number of M is zero,
 * $$\lambda_{CWL}(M)=\tfrac{1}{2}\left\vert H_1(M)\right\vert\lambda_{CW}(M)$$.


 * If the first Betti number of M is one,
 * $$\lambda_{CWL}(M)=\frac{\Delta^{\prime\prime}_M(1)}{2}-\frac{\mathrm{torsion}(H_1(M,\mathbb{Z}))}{12}$$
 * 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,
 * $$\lambda_{CWL}(M)=\left\vert\mathrm{torsion}(H_1(M))\right\vert\mathrm{Link}_M (\gamma,\gamma^\prime)$$
 * where γ is the oriented curve given by the intersection of two generators $$S_1,S_2$$ of $$H_2(M;\mathbb{Z})$$ and $$\gamma^\prime$$ 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 $$H_1(M;\mathbb{Z})$$, then
 * $$\lambda_{CWL}(M)=\left\vert\mathrm{torsion}(H_1(M;\mathbb{Z}))\right\vert\left((a\cup b\cup c)([M])\right)^2$$.


 * If the first Betti number of M is greater than three, $$\lambda_{CWL}(M)=0$$.

The Casson–Walker–Lescop invariant has the following properties:
 * When the orientation of M changes the behavior of $$\lambda_{CWL}(M)$$ depends on the first Betti number $$b_1(M) = \operatorname{rank} H_1(M;\mathbb{Z})$$of M: if $$\overline{M}$$ is M with the opposite orientation, then
 * $$\lambda_{CWL}(\overline{M}) = (-1)^{b_1(M)+1}\lambda_{CWL}(M).$$
 * That is, if the first Betti number of M is odd the Casson–Walker–Lescop invariant is unchanged, while if it is even it changes sign.


 * For connect-sums of manifolds
 * $$\lambda_{CWL}(M_1\#M_2)=\left\vert H_1(M_2)\right\vert\lambda_{CWL}(M_1)+\left\vert H_1(M_1)\right\vert\lambda_{CWL}(M_2)$$

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 $$\mathcal{A}/\mathcal{G}$$, 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 $$\mathcal{A}/\mathcal{G}$$ 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.