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

[math]\displaystyle{ \lambda\left(\Sigma+\frac{1}{n+1}\cdot K\right)-\lambda\left(\Sigma+\frac{1}{n}\cdot K\right) }[/math]

is independent of n. Here [math]\displaystyle{ \Sigma+\frac{1}{m}\cdot K }[/math] denotes [math]\displaystyle{ \frac{1}{m} }[/math] Dehn surgery on Σ by K.

• For any boundary link K ∪ L in Σ the following expression is zero:

[math]\displaystyle{ \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) }[/math]

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

Properties

• If K is the trefoil then

[math]\displaystyle{ \lambda\left(\Sigma+\frac{1}{n+1}\cdot K\right)-\lambda\left(\Sigma+\frac{1}{n}\cdot K\right)=\pm 1 }[/math].

• 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

[math]\displaystyle{ \lambda \left ( M + \frac{1}{n+1}\cdot K\right ) - \lambda \left ( M + \frac{1}{n}\cdot K\right ) = \phi_1 (K), }[/math]

where [math]\displaystyle{ \phi_1 (K) }[/math] is the coefficient of [math]\displaystyle{ z^2 }[/math] in the Alexander–Conway polynomial [math]\displaystyle{ \nabla_K(z) }[/math], 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 [math]\displaystyle{ \Sigma(p,q,r) }[/math] is given by the formula:

[math]\displaystyle{ \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] }[/math]

where

[math]\displaystyle{ 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) }[/math]

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

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(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:

[math]\displaystyle{ \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) }[/math]

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).
• [math]\displaystyle{ \langle\cdot,\cdot\rangle }[/math] 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 [math]\displaystyle{ H_1(M-K)/\text{Torsion} }[/math] in the infinite cyclic cover of M−K, and is symmetric and evaluates to 1 at 1.
• [math]\displaystyle{ \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} }[/math]

where x, y are generators of H₁(∂N(K), Z) such that [math]\displaystyle{ \langle x,y\rangle=1 }[/math], 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: [math]\displaystyle{ \lambda_{CW}(M) = 2 \lambda(M) }[/math].

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,

[math]\displaystyle{ \lambda_{CWL}(M)=\tfrac{1}{2}\left\vert H_1(M)\right\vert\lambda_{CW}(M) }[/math].

• If the first Betti number of M is one,

[math]\displaystyle{ \lambda_{CWL}(M)=\frac{\Delta^{\prime\prime}_M(1)}{2}-\frac{\mathrm{torsion}(H_1(M,\mathbb{Z}))}{12} }[/math]

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,

[math]\displaystyle{ \lambda_{CWL}(M)=\left\vert\mathrm{torsion}(H_1(M))\right\vert\mathrm{Link}_M (\gamma,\gamma^\prime) }[/math]

where γ is the oriented curve given by the intersection of two generators [math]\displaystyle{ S_1,S_2 }[/math] of [math]\displaystyle{ H_2(M;\mathbb{Z}) }[/math] and [math]\displaystyle{ \gamma^\prime }[/math] is the parallel curve to γ induced by the trivialization of the tubular neighbourhood of γ determined by [math]\displaystyle{ S_1, S_2 }[/math].

• If the first Betti number of M is three, then for a,b,c a basis for [math]\displaystyle{ H_1(M;\mathbb{Z}) }[/math], then

[math]\displaystyle{ \lambda_{CWL}(M)=\left\vert\mathrm{torsion}(H_1(M;\mathbb{Z}))\right\vert\left((a\cup b\cup c)([M])\right)^2 }[/math].

• If the first Betti number of M is greater than three, [math]\displaystyle{ \lambda_{CWL}(M)=0 }[/math].

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

• When the orientation of M changes the behavior of [math]\displaystyle{ \lambda_{CWL}(M) }[/math] depends on the first Betti number [math]\displaystyle{ b_1(M) = \operatorname{rank} H_1(M;\mathbb{Z}) }[/math]of M: if [math]\displaystyle{ \overline{M} }[/math] is M with the opposite orientation, then

[math]\displaystyle{ \lambda_{CWL}(\overline{M}) = (-1)^{b_1(M)+1}\lambda_{CWL}(M). }[/math]

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

[math]\displaystyle{ \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) }[/math]

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 [math]\displaystyle{ \mathcal{A}/\mathcal{G} }[/math], where [math]\displaystyle{ \mathcal{A} }[/math] is the space of SU(2) connections on M and [math]\displaystyle{ \mathcal{G} }[/math] is the group of gauge transformations. He regarded the Chern–Simons invariant as a [math]\displaystyle{ S^1 }[/math]-valued Morse function on [math]\displaystyle{ \mathcal{A}/\mathcal{G} }[/math] and used invariance under perturbations to define an invariant which he equated with the SU(2) Casson invariant. ((Taubes 1990))

H. Boden and C. Herald (1998) used a similar approach to define an SU(3) Casson invariant for integral homology 3-spheres.

References

• Selman Akbulut and John 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
• Michael 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.
• Hans Boden and Christopher Herald, The SU(3) Casson invariant for integral homology 3-spheres. Journal of Differential Geometry 50 (1998), 147–206.
• Christine Lescop, Global Surgery Formula for the Casson-Walker Invariant. 1995, ISBN:0-691-02132-5
• Nikolai 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
• Taubes, Clifford Henry (1990), "Casson’s invariant and gauge theory.", Journal of Differential Geometry 31: 547–599 
• Kevin 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