# 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(S3) = 0.
• Let Σ be an integral homology 3-sphere. Then for any knot K and for any integer n, the difference
${\displaystyle \lambda \left(\Sigma +{\frac {1}{n+1}}\cdot K\right)-\lambda \left(\Sigma +{\frac {1}{n}}\cdot K\right)}$
is independent of n. Here ${\displaystyle \Sigma +{\frac {1}{m}}\cdot K}$ denotes ${\displaystyle {\frac {1}{m}}}$ Dehn surgery on Σ by K.
• For any boundary link KL in Σ the following expression is zero:
${\displaystyle \lambda \left(\Sigma +{\frac {1}{m+1}}\cdot K+{\frac {1}{n+1}}\cdot L\right)-\lambda \left(\Sigma +{\frac {1}{m}}\cdot K+{\frac {1}{n+1}}\cdot L\right)-\lambda \left(\Sigma +{\frac {1}{m+1}}\cdot K+{\frac {1}{n}}\cdot L\right)+\lambda \left(\Sigma +{\frac {1}{m}}\cdot K+{\frac {1}{n}}\cdot L\right)}$

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

## Properties

• If K is the trefoil then
${\displaystyle \lambda \left(\Sigma +{\frac {1}{n+1}}\cdot K\right)-\lambda \left(\Sigma +{\frac {1}{n}}\cdot K\right)=\pm 1}$.
${\displaystyle \lambda \left(M+{\frac {1}{n+1}}\cdot K\right)-\lambda \left(M+{\frac {1}{n}}\cdot K\right)=\phi _{1}(K),}$
where ${\displaystyle \phi _{1}(K)}$ is the coefficient of ${\displaystyle z^{2}}$ in the Alexander-Conway polynomial ${\displaystyle \nabla _{K}(z)}$, and is congruent (mod 2) to the Arf invariant of K.
• The Casson invariant is the degree 1 part of the LMO invariant.
• The Casson invariant for the Seifert manifold ${\displaystyle \Sigma (p,q,r)}$ is given by the formula:
${\displaystyle \lambda (\Sigma (p,q,r))=-{\frac {1}{8}}\left[1-{\frac {1}{3pqr}}\left(1-p^{2}q^{2}r^{2}+p^{2}q^{2}+q^{2}r^{2}+p^{2}r^{2}\right)-d(p,qr)-d(q,pr)-d(r,pq)\right]}$
where
${\displaystyle d(a,b)=-{\frac {1}{a}}\sum _{k=1}^{a-1}\cot \left({\frac {\pi k}{a}}\right)\cot \left({\frac {\pi bk}{a}}\right)}$

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

## 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 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:

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

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 H1(∂N(K), Z) → H1(MK, Z).
• ${\displaystyle \langle \cdot ,\cdot \rangle }$ 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 ${\displaystyle H_{1}(M-K)/{\text{Torsion}}}$ in the infinite cyclic cover of MK, and is symmetric and evaluates to 1 at 1.
• ${\displaystyle \tau _{W}(m,\mu ;\nu )=-\mathrm {sgn} \langle y,m\rangle s(\langle x,m\rangle ,\langle y,m\rangle )+\mathrm {sgn} \langle y,\mu \rangle s(\langle x,\mu \rangle ,\langle y,\mu \rangle )+{\frac {(\delta ^{2}-1)\langle m,\mu \rangle }{12\langle m,\nu \rangle \langle \mu ,\nu \rangle }}}$
where x, y are generators of H1(∂N(K), Z) such that ${\displaystyle \langle x,y\rangle =1}$, 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: ${\displaystyle \lambda _{CW}(M)=2\lambda (M)}$.

### 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:

${\displaystyle \lambda _{CWL}(M)={\tfrac {1}{2}}\left\vert H_{1}(M)\right\vert \lambda _{CW}(M)}$.
• If the first Betti number of M is one,
${\displaystyle \lambda _{CWL}(M)={\frac {\Delta _{M}^{\prime \prime }(1)}{2}}-{\frac {\mathrm {torsion} (H_{1}(M,\mathbb {Z} ))}{12}}}$
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,
${\displaystyle \lambda _{CWL}(M)=\left\vert \mathrm {torsion} (H_{1}(M))\right\vert \mathrm {Link} _{M}(\gamma ,\gamma ^{\prime })}$
where γ is the oriented curve given by the intersection of two generators ${\displaystyle S_{1},S_{2}}$ of ${\displaystyle H_{2}(M;\mathbb {Z} )}$ and ${\displaystyle \gamma ^{\prime }}$ is the parallel curve to γ induced by the trivialization of the tubular neighbourhood of γ determined by ${\displaystyle S_{1},S_{2}}$.
• If the first Betti number of M is three, then for a,b,c a basis for ${\displaystyle H_{1}(M;\mathbb {Z} )}$, then
${\displaystyle \lambda _{CWL}(M)=\left\vert \mathrm {torsion} (H_{1}(M;\mathbb {Z} ))\right\vert \left((a\cup b\cup c)([M])\right)^{2}}$.
• If the first Betti number of M is greater than three, ${\displaystyle \lambda _{CWL}(M)=0}$.

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

• If the orientation of M, then if the first Betti number of M is odd the Casson–Walker–Lescop invariant is unchanged, otherwise it changes sign.
• For connect-sums of manifolds
${\displaystyle \lambda _{CWL}(M_{1}\#M_{2})=\left\vert H_{1}(M_{2})\right\vert \lambda _{CWL}(M_{1})+\left\vert H_{1}(M_{1})\right\vert \lambda _{CWL}(M_{2})}$

### SU(N)

In 1990, C. Taubes showed that the SU(2) Casson invariant of a 3-homology sphere M has gauge theoretic interpretation as the Euler characteristic of ${\displaystyle {\mathcal {A}}/{\mathcal {G}}}$, where ${\displaystyle {\mathcal {A}}}$ is the space of SU(2) connections on M and ${\displaystyle {\mathcal {G}}}$ is the group of gauge transformations. He led Chern–Simons invariant as a ${\displaystyle S^{1}}$-valued Morse function on ${\displaystyle {\mathcal {A}}/{\mathcal {G}}}$ and pointed out that the SU(3) Casson invariant is important to make the invariants independent on perturbations. (Taubes (1990))

Boden and Herald (1998) defined an SU(3) Casson invariant.

## 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
• M. 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.
• H. Boden and C. Herald, The SU(3) Casson invariant for integral homology 3-spheres. J. Differential Geom. 50 (1998), 147–206.
• C. Lescop, Global Surgery Formula for the Casson-Walker Invariant. 1995, ISBN 0-691-02132-5
• N. 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.", J. Differential Geom., 31: 547–599
• K. 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