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$lambda$ 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$lambdaleft(Sigma+frac\{1\}\{n+1\}cdot\; K\; ight)-lambdaleft(Sigma+frac\{1\}\{n\}cdot\; K\; ight)$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\; L\; ight)$$-lambdaleft(Sigma+frac\{1\}\{m\}cdot\; K+frac\{1\}\{n+1\}cdot\; L\; ight)$$-lambdaleft(Sigma+frac\{1\}\{m+1\}cdot\; K+frac\{1\}\{n\}cdot\; L\; ight)$$+lambdaleft(Sigma+frac\{1\}\{m\}cdot\; K+frac\{1\}\{n\}cdot\; L\; ight)$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 $lambdaleft(Sigma+frac\{1\}\{n+1\}cdot\; K\; ight)-lambdaleft(Sigma+frac\{1\}\{n\}cdot\; K\; ight)=pm\; 1$.
*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:$lambda(Sigma(p,q,r))=-frac\{1\}\{8\}left\; [1-frac\{1\}\{3pqr\}left(1-p^2q^2r^2+p^2q^2+q^2r^2+p^2r^2\; ight)-d(p,qr)-d(q,pr)-d(r,pq)\; ight]$where$d(a,b)=-frac\{1\}\{a\}sum\_\{k=1\}^\{a-1\}cotleft(frac\{pi\; k\}\{a\}\; ight)cotleft(frac\{pi\; bk\}\{a\}\; ight)$

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 spaceof 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":$lambda\_\{CW\}(M^prime)=lambda\_\{CW\}(M)+frac\{langle\; m,mu\; angle\}\{langle\; m,\; u\; anglelangle\; mu,\; u\; angle\}Delta\_\{W\}^\{primeprime\}(M-K)(1)+\; au\_\{W\}(m,mu;\; u)$ where:
**"m" is an oriented meridian of a knot "K" and $mu$ is the characteristic curve of the surgery.
**$u$ is a generator the kernel of the natural map from $H\_1(partial\; N(K),mathbb\{Z\})$ to $H\_1(M-K,mathbb\{Z\})$.
**$langlecdot,cdot\; angle$ 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 $H\_1(M-K)/\; ext\{Torsion\}$ in the infinite cyclic cover of "M-K", and is symmetric and evaluates to 1 at 1.
**$au\_\{W\}(m,mu;\; u)=\; -mathrm\{sgn\}langle\; y,m\; angle\; s(langle\; x,m\; angle,langle\; y,m\; angle)+mathrm\{sgn\}langle\; y,mu\; angle\; s(langle\; x,mu\; angle,langle\; y,mu\; angle)+frac\{(delta^2-1)langle\; m,mu\; angle\}\{12langle\; m,\; u\; anglelangle\; mu,\; u\; angle\}$where "x, y" are generators of $H\_1(partial\; N(K);mathbb\{Z\})$ such that $langle\; x,y\; angle=1$, 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, $lambda\_\{CWL\}(M)=frac\{leftvert\; H\_1(M)\; ightvertlambda\_\{CW\}(M)\}\{2\}$.
*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, $lambda\_\{CWL\}(M)=leftvertmathrm\{torsion\}(H\_1(M))\; ightvertmathrm\{Link\}\_M\; (gamma,gamma^prime)$ 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 $lambda\_\{CWL\}(M)=leftvertmathrm\{torsion\}(H\_1(M;mathbb\{Z\}))\; ightvertleft((acup\; bcup\; c)(\; [M]\; )\; ight)^2$.
*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 $lambda\_\{CWL\}(M\_1\#M\_2)=leftvert\; H\_1(M\_2)\; ightvertlambda\_\{CWL\}(M\_1)+leftvert\; H\_1(M\_1)\; ightvertlambda\_\{CWL\}(M\_2)$

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