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.

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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:

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

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

Properties

.
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

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 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:

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:

.
where Δ is the Alexander polynomial normalized to be symmetric and take a positive value at 1.
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 .
.

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

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.

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. (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