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.

1 Definition
2 Properties
3 The Casson invariant as a count of representations
4 Generalizations
5 References

Definition

A Casson invariant is a surjective map $\lambda$ from oriented integral homology 3-spheres to $\mathbb{Z}$ satisfying the following properties:

$\lambda(S^3)=0$ .
Let $\Sigma$ be an integral homology 3-sphere. Then for any knot K and for any $n\in\mathbb{Z}$ , the difference

$\lambda\left(\Sigma%2B\frac{1}{n%2B1}\cdot K\right)-\lambda\left(\Sigma%2B\frac{1}{n}\cdot K\right)$ is independent of n. Here $\Sigma%2B\frac{1}{m}\cdot K$ denotes $\frac{1}{m}$ Dehn surgery on $\Sigma$ by K.

$\lambda\left(\Sigma%2B\frac{1}{m%2B1}\cdot K%2B\frac{1}{n%2B1}\cdot L\right)$ $-\lambda\left(\Sigma%2B\frac{1}{m}\cdot K%2B\frac{1}{n%2B1}\cdot L\right)$ $-\lambda\left(\Sigma%2B\frac{1}{m%2B1}\cdot K%2B\frac{1}{n}\cdot L\right)$ $%2B\lambda\left(\Sigma%2B\frac{1}{m}\cdot K%2B\frac{1}{n}\cdot L\right)$ is equal to zero for any boundary link $K\cup L$ in $\Sigma$ .

The Casson invariant is unique up to sign.

Properties

If K is the trefoil then $\lambda\left(\Sigma%2B\frac{1}{n%2B1}\cdot K\right)-\lambda\left(\Sigma%2B\frac{1}{n}\cdot K\right)=\pm 1$ .
The Casson invariant is 2 (or − 2) 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 $n\in \mathbb{Z}$ let $M_{K_n}$ be the result of $\frac{1}{n}$ Dehn surgery on M along K. Then the Casson invariant of $M_{K_{n%2B1}}$ minus the Casson invariant of $M_{K_n}$

is the Arf invariant of $K$ .

The Casson invariant is the degree 1 part of the LMO invariant.
The Casson invariant for the Seifert manifold $\Sigma(p,q,r)$ is given by the formula:

$\lambda(\Sigma(p,q,r))=-\frac{1}{8}\left[1-\frac{1}{3pqr}\left(1-p^2q^2r^2%2Bp^2q^2%2Bq^2r^2%2Bp^2r^2\right) -d(p,qr)-d(q,pr)-d(r,pq)\right]$ 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 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 $\mathcal{R}(M)=R^{\mathrm{irr}}(M)/SO(3)$ 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 $\Sigma$ , the Casson invariant equals $\frac{(-1)^g}{2}$ times the algebraic intersection of $\mathcal{R}(M_1)$ with $\mathcal{R}(M_2)$ .

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 $\lambda_{CW}$ from oriented rational homology 3-spheres to $\mathbb{Q}$ satisfying the following properties:

$\lambda(S^3)=0$ .
For every 1-component Dehn surgery presentation $(K,\mu)$ of an oriented rational homology sphere $M^\prime$ in an oriented rational homology sphere M:

$\lambda_{CW}(M^\prime)=\lambda_{CW}(M)%2B\frac{\langle m,\mu\rangle}{\langle m,\nu\rangle\langle \mu,\nu\rangle}\Delta_{W}^{\prime\prime}(M-K)(1)%2B\tau_{W}(m,\mu;\nu)$ where:

- m is an oriented meridian of a knot K and $mu$ is the characteristic curve of the surgery.
- $\nu$ is a generator the kernel of the natural map from $H_1(\partial N(K),\mathbb{Z})$ to $H_1(M-K,\mathbb{Z})$ .
- $\langle\cdot,\cdot\rangle$ is the intersection form on the tubular neighbourhood of the knot, N(K).
- $\Delta$ 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.
- $\tau_{W}(m,\mu;\nu)= -\mathrm{sgn}\langle y,m\rangle s(\langle x,m\rangle,\langle y,m\rangle)%2B\mathrm{sgn}\langle y,\mu\rangle s(\langle x,\mu\rangle,\langle y,\mu\rangle)%2B\frac{(\delta^2-1)\langle m,\mu\rangle}{12\langle m,\nu\rangle\langle \mu,\nu\rangle}$

where x, y are generators of $H_1(\partial N(K);\mathbb{Z})$ such that $\langle x,y\rangle=1$ , and $v=\delta y$ for an integer $\delta$ . $s(p,q)$ is the Dedekind sum.

Compact oriented 3-manifolds

Christine Lescop defined an extension $\lambda_{CWL}$ 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, $\lambda_{CWL}(M)=\frac{\left\vert H_1(M)\right\vert\lambda_{CW}(M)}{2}$ .
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 $\Delta$ 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 $\gamma$ 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 $\gamma$ induced by the trivialization of the tubular neighbourhood of $\gamma$ 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:

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 $\lambda_{CWL}(M_1\#M_2)=\left\vert H_1(M_2)\right\vert\lambda_{CWL}(M_1)%2B\left\vert H_1(M_1)\right\vert\lambda_{CWL}(M_2)$

SU(N)

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

References

S. Akbulut and J. McCarthy, Casson's invariant for oriented homology 3-spheres— an exposition. Mathematical Notes, 36. Princeton University Press, Princeton, NJ, 1990. ISBN 0-691-08563-3
M. Atiyah, New invariants of 3- and 4-dimensional manifolds. The mathematical heritage of Hermann Weyl (Durham, NC, 1987), 285-299, Proc. Sympos. Pure Math., 48, Amer. Math. Soc., Providence, RI, 1988.
H. Boden and C. Herald, The SU(3) Casson invariant for integral homology 3-spheres. J. Differential Geom. 50 (1998), 147–206.
C. Lescop, Global Surgery Formula for the Casson-Walker Invariant. 1995, ISBN 0691021325
N. Saveliev, Lectures on the topology of 3-manifolds: An introduction to the Casson Invariant. de Gruyter, Berlin, 1999. ISBN 3-11-016271-7 ISBN 3-11-016272-5
K. Walker, An extension of Casson's invariant. Annals of Mathematics Studies, 126. Princeton University Press, Princeton, NJ, 1992. ISBN 0-691-08766-0 ISBN 0-691-02532-0