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.
A Casson invariant is a surjective map
from oriented integral homology 3-spheres to
satisfying the following properties:
- Let be an integral homology 3-sphere. Then for any knot K and for any , the difference
is independent of n. Here denotes Dehn surgery on by K.
is equal to zero for any boundary link in .
The Casson invariant is unique up to sign.
- If K is the trefoil then .
- The Casson invariant is 2 (or − 2) for the Poincaré homology sphere.
- The Casson invariant changes sign if the orientation of M is reversed.
- The Rokhlin invariant of M is equal 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 let be the result of Dehn surgery on M along K. Then the Casson invariant of minus the Casson invariant of
is the Arf invariant of .
- 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 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
from oriented rational homology 3-spheres to
satisfying the following properties:
- For every 1-component Dehn surgery presentation of an oriented rational homology sphere in an oriented rational homology sphere M:
- m is an oriented meridian of a knot K and is the characteristic curve of the surgery.
- is a generator the kernel of the natural map from to .
- is the intersection form on the tubular neighbourhood of the knot, N(K).
- 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 such that , and for an integer . is the Dedekind sum.
Compact oriented 3-manifolds
Christine Lescop defined an extension
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 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 is the oriented curve given by the intersection of two generators of and is the parallel curve to induced by the trivialization of the tubular neighbourhood of 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)
- 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 0691021325
- 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