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 $lambda$ from oriented integral homology 3-spheres to $mathbb\{Z\}$ satisfying the following properties:

• $lambda(S^3)=0$.
• Let $Sigma$ be an integral homology 3-sphere. Then for any knot K and for any $ninmathbb\{Z\}$, the difference

$lambdaleft(Sigma+frac\{1\}\{n+1\}cdot\; Kright)-lambdaleft(Sigma+frac\{1\}\{n\}cdot\; Kright)$ is independent of n. Here $Sigma+frac\{1\}\{m\}cdot\; K$ denotes $frac\{1\}\{m\}$ Dehn surgery on $Sigma$ by K. * $lambdaleft(Sigma+frac\{1\}\{m+1\}cdot\; K+frac\{1\}\{n+1\}cdot\; Lright)$ $-lambdaleft(Sigma+frac\{1\}\{m\}cdot\; K+frac\{1\}\{n+1\}cdot\; Lright)$ $-lambdaleft(Sigma+frac\{1\}\{m+1\}cdot\; K+frac\{1\}\{n\}cdot\; Lright)$ $+lambdaleft(Sigma+frac\{1\}\{m\}cdot\; K+frac\{1\}\{n\}cdot\; Lright)$ is equal to zero for any boundary link $Kcup\; L$ in $Sigma$.

The Casson invariant is unique up to sign.

Properties

• If K is the trefoil then $lambdaleft(Sigma+frac\{1\}\{n+1\}cdot\; Kright)-lambdaleft(Sigma+frac\{1\}\{n\}cdot\; Kright)=pm\; 1$.
• The Casson invariant is 2 (or − 2) for the Poincaré homology sphere.
• The Casson invariant changes sign if the orientation of M is reversed.
• The Rokhlin invariant of M is equal 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 $nin\; mathbb\{Z\}$ let $M\_\{K\_n\}$ be the result of $frac\{1\}\{n\}$ Dehn surgery on M along K. Then the Casson invariant of $M\_\{K\_\{n+1\}\}$ minus the Casson invariant of $M\_\{K\_n\}$

is the Arf invariant of $K$.

• The Casson invariant is the degree 1 part of the LMO invariant.
• The Casson invariant for the Seifert manifold $Sigma(p,q,r)$ is given by the formula:

$lambda(Sigma(p,q,r))=-frac\{1\}\{8\}left[1-frac\{1\}\{3pqr\}left(1-p^2q^2r^2+p^2q^2+q^2r^2+p^2r^2right)\; -d(p,qr)-d(q,pr)-d(r,pq)right]$ where $d(a,b)=-frac\{1\}\{a\}sum\_\{k=1\}^\{a-1\}cotleft(frac\{pi\; k\}\{a\}right)cotleft(frac\{pi\; bk\}\{a\}right)$

The Casson Invariant as a count of representations

Informally speaking, the Casson invariant counts 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 $mathcal\{R\}(M)=R^\{mathrm\{irr\}\}(M)/SO(3)$ 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 $Sigma$, 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 $lambda\_\{CW\}$ from oriented rational homology 3-spheres to $mathbb\{Q\}$ satisfying the following properties:

• $lambda(S^3)=0$.
• For every 1-component Dehn surgery presentation $(K,mu)$ of an oriented rational homology sphere $M^prime$ in an oriented rational homology sphere M:

$lambda\_\{CW\}(M^prime)=lambda\_\{CW\}(M)+frac\{langle\; m,murangle\}\{langle\; m,nuranglelangle\; mu,nurangle\}Delta\_\{W\}^\{primeprime\}(M-K)(1)+tau\_\{W\}(m,mu;nu)$ where:

• m is an oriented meridian of a knot K and $mu$ is the characteristic curve of the surgery.
• $nu$ is a generator the kernel of the natural map from $H\_1(partial\; N(K),mathbb\{Z\})$ to $H\_1(M-K,mathbb\{Z\})$.
• $langlecdot,cdotrangle$ is the intersection form on the tubular neighbourhood of the knot, N(K).
• $Delta$ 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.
• $tau\_\{W\}(m,mu;nu)=\; -mathrm\{sgn\}langle\; y,mrangle\; s(langle\; x,mrangle,langle\; y,mrangle)+mathrm\{sgn\}langle\; y,murangle\; s(langle\; x,murangle,langle\; y,murangle)+frac\{(delta^2-1)langle\; m,murangle\}\{12langle\; m,nuranglelangle\; mu,nurangle\}$

where x, y are generators of $H\_1(partial\; N(K);mathbb\{Z\})$ such that $langle\; x,yrangle=1$, and $v=delta\; y$ for an integer $delta$. $s(p,q)$ is the Dedekind sum.

Compact oriented 3-manifolds

Christine Lescop defined an extension $lambda\_\{CWL\}$ 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, $lambda\_\{CWL\}(M)=frac\{leftvert\; H\_1(M)rightvertlambda\_\{CW\}(M)\}\{2\}$.
• If the first Betti number of M is one, $lambda\_\{CWL\}(M)=frac\{Delta^\{primeprime\}\_M(1)\}\{2\}-frac\{mathrm\{torsion\}(H\_1(M,mathbb\{Z\}))\}\{12\}$ where $Delta$ is the Alexander polynomial normalized to be symmetric and take a positive value at 1.
• If the first Betti number of M is two, $lambda\_\{CWL\}(M)=leftvertmathrm\{torsion\}(H\_1(M))rightvertmathrm\{Link\}\_M\; (gamma,gamma^prime)$ where $gamma$ 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 $gamma$ induced by the trivialization of the tubular neighbourhood of $gamma$ 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 $lambda\_\{CWL\}(M)=leftvertmathrm\{torsion\}(H\_1(M;mathbb\{Z\}))rightvertleft((acup\; bcup\; c)([M])right)^2$.
• If the first Betti number of M is greater than three, $lambda\_\{CWL\}(M)=0$.

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

• $$
• 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 $lambda\_\{CWL\}(M\_1\#M\_2)=leftvert\; H\_1(M\_2)rightvertlambda\_\{CWL\}(M\_1)+leftvert\; H\_1(M\_1)rightvertlambda\_\{CWL\}(M\_2)$

SU(N)

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 0691021325
• 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
• 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