# 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.

## Definition

A Casson invariant is a surjective map λ from oriented integral homology 3-spheres to Z satisfying the following properties:
* λ(S$^3$) = 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.

## Properties

* 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

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 ${\mathcal {R}}(M_{2})$.

## 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 λ$_{CW}$ from oriented rational homology 3-spheres to Q satisfying the following properties:
1. λ(S$^3$) = 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:
* m is an oriented meridian of a knot K and μ is the characteristic curve of the surgery.
* ν is a generator the kernel of the natural map H$_1$(∂N(K), Z) → H$_1$(M−K, Z).
* $\langle \cdot ,\cdot \rangle$ is the intersection form on the tubular neighbourhood of the knot, N(K).
* Δ is the Alexander polynomial normalized so that the action of t corresponds to an action of the generator of $H_{1}(M-K)/{\text{Torsion}}$ in the infinite cyclic cover of M−K, and is symmetric and evaluates to 1 at 1.
* $\tau _{W}(m,\mu ;\nu )=-\mathrm {sgn} \langle y,m\rangle s(\langle x,m\rangle ,\langle y,m\rangle )+\mathrm {sgn} \langle y,\mu \rangle s(\langle x,\mu \rangle ,\langle y,\mu \rangle )+{\frac {(\delta ^{2}-1)\langle m,\mu \rangle }{12\langle m,\nu \rangle \langle \mu ,\nu \rangle }}$
    where x, y are generators of H$_1$(∂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

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:
* When the orientation of M changes the behavior of $\lambda _{CWL}(M)$ depends on the first Betti number $b_{1}(M)=\operatorname {rank} H_{1}(M;\mathbb {Z} )$of M: if ${\overline {M}}$ is M with the opposite orientation, then
       $\lambda _{CWL}({\overline {M}})=(-1)^{b_{1}(M)+1}\lambda _{CWL}(M).$
    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
       $\lambda _{CWL}(M_{1}\#M_{2})=\left\vert H_{1}(M_{2})\right\vert \lambda _{CWL}(M_{1})+\left\vert H_{1}(M_{1})\right\vert \lambda _{CWL}(M_{2})$

### 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 ${\mathcal {A}}/{\mathcal {G}}$, 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.