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

:

is independent of n. Here denotes Dehn surgery on Σ by K.

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

:

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

Properties

• If K is the trefoil then

:.

• 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

:

where is the coefficient of in the Alexander–Conway polynomial , 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 integer Homology Sphere is given by the formula:

:

where

:

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 where denotes the space of irreducible SU(2) representations of . For a Heegaard splitting of , the Casson invariant equals times the algebraic intersection of with .

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:

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).
• 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 in the infinite cyclic cover of M−K, and is symmetric and evaluates to 1 at 1.

where x, y are generators of H₁(∂N(K), Z) such that , 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: .

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,

:.

• If the first Betti number of M is one,

:

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,

:

where γ is the oriented curve given by the intersection of two generators of and is the parallel curve to γ induced by the trivialization of the tubular neighbourhood of γ determined by .

• If the first Betti number of M is three, then for a,b,c a basis for , then

:.

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

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

• When the orientation of M changes the behavior of depends on the first Betti number of M: if is M with the opposite orientation, then

:

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

:

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 , where is the space of SU(2) connections on M and is the group of gauge transformations. He regarded the Chern–Simons invariant as a -valued Morse function on 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.