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 \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

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

\( \lambda\left(\Sigma+\frac{1}{m+1}\cdot K+\frac{1}{n+1}\cdot L\right) -\lambda\left(\Sigma+\frac{1}{m}\cdot K+\frac{1}{n+1}\cdot L\right) -\lambda\left(\Sigma+\frac{1}{m+1}\cdot K+\frac{1}{n}\cdot L\right) +\lambda\left(\Sigma+\frac{1}{m}\cdot K+\frac{1}{n}\cdot L\right) \) 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 \( \lambda\left(\Sigma+\frac{1}{n+1}\cdot K\right)-\lambda\left(\Sigma+\frac{1}{n}\cdot K\right)=\pm 1. \)
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\( M_{K_{n+1}} \) 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+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 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:

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

m is an oriented meridian of a knot K and 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.
\( \tau_{W}(m,\mu;\nu)= -\mathrm{sgn}\langle y,m\rangle s(\langle x,m\rangle,\langle y,m\rangle)+\mathrm{sgn}\langle y,\mu\rangle s(\langle x,\mu\rangle,\langle y,\mu\rangle)+\frac{(\delta^2-1)\langle m,\mu\rangle}{12\langle m,\nu\rangle\langle \mu,\nu\rangle} \)

where x, y are generators of \( H_1(\partial N(K);\mathbb{Z}) \) 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:

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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 \( \lambda_{CWL}(M_1\#M_2)=\left\vert H_1(M_2)\right\vert\lambda_{CWL}(M_1)+\left\vert H_1(M_1)\right\vert\lambda_{CWL}(M_2) \)

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

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