Casson invariant

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 λ from oriented integral homology 3-spheres to Z satisfying the following properties:

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

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 ${\mathcal {R}}(M)=R^{\mathrm {irr} }(M)/SO(3)$ ${\mathcal {R}}(M)=R^{{{\mathrm {irr}}}}(M)/SO(3)$ where $R^{\mathrm {irr} }(M)$ $R^{{{\mathrm {irr}}}}(M)$ denotes the space of irreducible SU(2) representations of $\pi _{1}(M)$ $\pi _{1}(M)$ . For a Heegaard splitting $\Sigma =M_{1}\cup _{F}M_{2}$ $\Sigma =M_{1}\cup _{F}M_{2}$ of $M$ $M$ , the Casson invariant equals ${\frac {(-1)^{g}}{2}}$ ${\frac {(-1)^{g}}{2}}$ times the algebraic intersection of ${\mathcal {R}}(M_{1})$ ${\mathcal {R}}(M_{1})$ with ${\mathcal {R}}(M_{2})$ ${\mathcal {R}}(M_{2})$ .

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