Casson invariant explained

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 mapλ 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

λ \left(\Sigma + (1)/(n +1) ⋅ K\right) -λ \left(\Sigma + (1)/(n) ⋅ K\right)

is independent of n. Here

\Sigma + (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:

λ \left(\Sigma + (1)/(m +1) ⋅ K + (1)/(n +1) ⋅ L\right) -λ \left(\Sigma + (1)/(m) ⋅ K + (1)/(n +1) ⋅ L\right) -λ \left(\Sigma + (1)/(m +1) ⋅ K + (1)/(n) ⋅ L\right) +λ \left(\Sigma + (1)/(m) ⋅ K + (1)/(n) ⋅ 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

λ \left(\Sigma + (1)/(n +1) ⋅ K\right) -λ \left(\Sigma + (1)/(n) ⋅ 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

λ \left ( M + (1)/(n +1) ⋅ K\right ) - λ \left ( M + (1)/(n) ⋅ 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.

\Sigma(p,q,r) is given by the formula:λ (\Sigma(p,q,r)) =- (1)/(\left[ 1- 8) (1)/(3 pqr) \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) =- (1)/(a) \sum ^{a -1} _{k =1} \cot\left( (\pi k)/(\right)\cot\left( a) (\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

l{R}(M)=R ^{irr} (M)/SU( 2 ) where R ^{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 (( -1 ) ^{g})/(2) times the algebraic intersection of l{R}(M _{1)} with l{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 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 _{CW} (M ^{\prime) =λ} _{CW} (M) + (\langle m,\mu\rangle)/(\langle m,\nu\rangle\langle \mu,\nu\rangle) \Delta ^{\prime\prime} _{W} (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 ⋅ , ⋅ \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)/Torsion} in the infinite cyclic cover of M−K, and is symmetric and evaluates to 1 at 1.\tau _{W} (m,\mu;\nu) = - sgn \langle y,m\rangle s(\langle x,m\rangle,\langle y,m\rangle) + sgn \langle y,\mu\rangle s(\langle x,\mu\rangle,\langle y,\mu\rangle) + ((\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:

λ _{CW} (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,

λ _{CWL} (M) = \tfrac{ 1 }{ 2 }\left\vert H _{1(M)\right\vertλ} _{CW} (M) .

• If the first Betti number of M is one,

λ _{CWL} (M) = (\Delta ^{\prime\prime} _{M( 1 )})/(- 2) (torsion (H _{1(M,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,

λ _{CWL} (M) = \left\vert torsion (H _{1(M))\right\vertLink} _{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;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;Z)} , then λ _{CWL} (M) = \left\vert torsion (H _{1(M;Z))\right\vert\left((a\cup} b\cup c)([M])\right) ^{2} .

• If the first Betti number of M is greater than three,

λ _{CWL} (M) =0 .

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

• When the orientation of M changes the behavior of

λ _{CWL} (M) depends on the first Betti number b _{1(M)} = \operatorname{rank} H _{1(M;Z)} of M: if \overline{M} is M with the opposite orientation, thenλ _{CWL} (\overline{M}) = ( -1 ) ^{b _{1(M)+1}} λ _{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

λ _{CWL} (M _{1\#M} _{2)=\left\vert} H _{1(M} _{2)\right\vertλ} _{CWL} (M _{1)+\left\vert} H _{1(M} _{1)\right\vertλ} _{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

l{A}/ l{G} , where l{A} is the space of SU(2) connections on M and l{G} is the group of gauge transformations. He regarded the Chern–Simons invariant as a S ^{1} -valued Morse function on l{A}/ l{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.

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.
• 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,
• Nikolai Saveliev, Lectures on the topology of 3-manifolds: An introduction to the Casson Invariant. de Gruyter, Berlin, 1999.
  • Kevin Walker, An extension of Casson's invariant. Annals of Mathematics Studies, 126. Princeton University Press, Princeton, NJ, 1992.