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Casson invariant - Wikipedia

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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.

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:

λ(S³) = 0.
Let Σ be an integral homology 3-sphere. Then for any knot K and for any integer n, the difference

$\lambda \left(\Sigma +{\frac {1}{n+1}}\cdot K\right)-\lambda \left(\Sigma +{\frac {1}{n}}\cdot K\right)$

is independent of n. Here $\Sigma +{\frac {1}{m}}\cdot K$ denotes ${\frac {1}{m}}$ Dehn surgery on Σ by K.

For any boundary link K ∪ L in Σ the following expression is zero:

$\lambda \left(\Sigma +{\frac {1}{m+1}}\cdot K+{\frac {1}{n+1}}\cdot L\right)-\lambda \left(\Sigma +{\frac {1}{m}}\cdot K+{\frac {1}{n+1}}\cdot L\right)-\lambda \left(\Sigma +{\frac {1}{m+1}}\cdot K+{\frac {1}{n}}\cdot L\right)+\lambda \left(\Sigma +{\frac {1}{m}}\cdot K+{\frac {1}{n}}\cdot L\right)$

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

If K is the trefoil then

$\lambda \left(\Sigma +{\frac {1}{n+1}}\cdot K\right)-\lambda \left(\Sigma +{\frac {1}{n}}\cdot K\right)=\pm 1$ .

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

$\lambda \left(M+{\frac {1}{n+1}}\cdot K\right)-\lambda \left(M+{\frac {1}{n}}\cdot K\right)=\phi _{1}(K),$

where $\phi _{1}(K)$ is the coefficient of $z^{2}$ in the Alexander–Conway polynomial $\nabla _{K}(z)$ , and is congruent (mod 2) to the Arf invariant of K.

The Casson invariant is the degree 1 part of the Le–Murakami–Ohtsuki invariant.
The Casson invariant for the Seifert manifold $\Sigma (p,q,r)$ is given by the formula:

$\lambda (\Sigma (p,q,r))=-{\frac {1}{8}}\left[1-{\frac {1}{3pqr}}\left(1-p^{2}q^{2}r^{2}+p^{2}q^{2}+q^{2}r^{2}+p^{2}r^{2}\right)-d(p,qr)-d(q,pr)-d(r,pq)\right]$

where

$d(a,b)=-{\frac {1}{a}}\sum _{k=1}^{a-1}\cot \left({\frac {\pi k}{a}}\right)\cot \left({\frac {\pi bk}{a}}\right)$

The Casson invariant as a count of representations

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

Rational homology 3-spheres

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Kevin Walker found an extension of the Casson invariant to rational homology 3-spheres. A Casson-Walker invariant is a surjective map λ_CW 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:

$\lambda _{CW}(M^{\prime })=\lambda _{CW}(M)+{\frac {\langle m,\mu \rangle }{\langle m,\nu \rangle \langle \mu ,\nu \rangle }}\Delta _{W}^{\prime \prime }(M-K)(1)+\tau _{W}(m,\mu ;\nu )$

where:

where x, y are generators of H₁(∂N(K), Z) such that $\langle x,y\rangle =1$ , 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: $\lambda _{CW}(M)=2\lambda (M)$ .

Compact oriented 3-manifolds

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Christine Lescop defined an extension λ_CWL 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,

$\lambda _{CWL}(M)={\tfrac {1}{2}}\left\vert H_{1}(M)\right\vert \lambda _{CW}(M)$ .

If the first Betti number of M is one,

$\lambda _{CWL}(M)={\frac {\Delta _{M}^{\prime \prime }(1)}{2}}-{\frac {\mathrm {torsion} (H_{1}(M,\mathbb {Z} ))}{12}}$

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,

$\lambda _{CWL}(M)=\left\vert \mathrm {torsion} (H_{1}(M))\right\vert \mathrm {Link} _{M}(\gamma ,\gamma ^{\prime })$

where γ is the oriented curve given by the intersection of two generators $S_{1},S_{2}$ of $H_{2}(M;\mathbb {Z} )$ and $\gamma ^{\prime }$ is the parallel curve to γ induced by the trivialization of the tubular neighbourhood of γ determined by $S_{1},S_{2}$ .

If the first Betti number of M is three, then for a,b,c a basis for $H_{1}(M;\mathbb {Z} )$ , then

$\lambda _{CWL}(M)=\left\vert \mathrm {torsion} (H_{1}(M;\mathbb {Z} ))\right\vert \left((a\cup b\cup c)([M])\right)^{2}$ .

If the first Betti number of M is greater than three, $\lambda _{CWL}(M)=0$ .

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

$\lambda _{CWL}({\overline {M}})=(-1)^{b_{1}(M)+1}\lambda _{CWL}(M).$

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.

For connect-sums of manifolds

$\lambda _{CWL}(M_{1}\#M_{2})=\left\vert H_{1}(M_{2})\right\vert \lambda _{CWL}(M_{1})+\left\vert H_{1}(M_{1})\right\vert \lambda _{CWL}(M_{2})$

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 ${\mathcal {A}}/{\mathcal {G}}$ , where ${\mathcal {A}}$ is the space of SU(2) connections on M and ${\mathcal {G}}$ is the group of gauge transformations. He regarded the Chern–Simons invariant as a $S^{1}$ -valued Morse function on ${\mathcal {A}}/{\mathcal {G}}$ 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.

Selman Akbulut and John 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
Michael 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.
Hans Boden and Christopher Herald, The SU(3) Casson invariant for integral homology 3-spheres. Journal of Differential Geometry 50 (1998), 147–206.
Christine Lescop, Global Surgery Formula for the Casson-Walker Invariant. 1995, ISBN 0-691-02132-5
Nikolai 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
Taubes, Clifford Henry (1990), "Casson's invariant and gauge theory.", Journal of Differential Geometry, 31: 547–599
Kevin 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