Casson invariant

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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

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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 KL 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

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  • If K is the trefoil then
.
where is the coefficient of in the Alexander–Conway polynomial , and is congruent (mod 2) to the Arf invariant of K.
where

The Casson invariant as a count of representations

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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

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Rational homology 3-spheres

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Kevin Walker found an extension of the Casson invariant to rational homology 3-spheres. A Casson-Walker invariant is a surjective map λCW 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(MK, 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 MK, 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

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Christine Lescop defined an extension λCWL 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 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.

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. (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

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  • 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