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 homology 3-spheres, introduced by Andrew Casson.
Kevin Walker (1992) found an extension to rational homology 3-spheres, called the Casson-Walker invariant, and Lescop (1995) extended the invariant to all 3-manifolds
[edit] Definition
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 description should not be taken literally: the number of conjugacy classes is often infinite, and the Casson invariant can be negative. However we can interpret it formally as follows.
Take a Heegaard decomposition of M as the union of two handlebodies A and B with intersection a surface S. Write R(X) for the space of conjugacy classes of representations of the fundamental group of a space X into the group SU(2). Then R(M) is the intersection of the two subspaces R(A) and R(B) of R(S). So we can formally define the number of points of R(M) to be the intersection number of R(A) and R(B) in R(S) (which may be negative, as points of intersection are counted with an orientation which can be +1 or −1).
There are several technical problems with this definition: the subspaces R(A) and R(B) have singularities, and it is not obvious that the definition of R(M) is independent of the choice of Heegaard decomposition. However these problems can be solved, and we get a well defined integer R(M) called the Casson invariant of M, and denoted by λ(M). (Sometimes the Casson invariant is defined to be half of this.)
[edit] Properties
- The Casson invariant is 0 on the 3-sphere, and 2 (or − 2) for the Poincaré sphere.
- The Casson invariant changes sign if the orientation of M is reversed.
- The Rokhlin invariant of M is equal to half the Casson invariant mod 2.
- The Casson invariant is additive on connected sums of homology 3-spheres.
- The Casson invariant is a sort of Euler characteristic for Floer homology.
[edit] References
- 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
- Atiyah, Michael 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.
- Christine Lescop Global Surgery Formula for the Casson-Walker Invariant. 1995, ISBN 0691021325
- Walker, Kevin 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
