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.
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.
- 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 LMO invariant.
- The Casson invariant for the Seifert manifold is given by the formula:
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 .
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 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:
- 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 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 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:
- 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
Boden and Herald (1998) defined an SU(3) Casson invariant.
- 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 0-691-02132-5
- 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