Definitions
Nearby Words

# 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 $mathbb\left\{Z\right\}$ satisfying the following properties:

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

$lambdaleft\left(Sigma+frac\left\{1\right\}\left\{n+1\right\}cdot Kright\right)-lambdaleft\left(Sigma+frac\left\{1\right\}\left\{n\right\}cdot Kright\right)$ is independent of n. Here $Sigma+frac\left\{1\right\}\left\{m\right\}cdot K$ denotes $frac\left\{1\right\}\left\{m\right\}$ Dehn surgery on $Sigma$ by K. * $lambdaleft\left(Sigma+frac\left\{1\right\}\left\{m+1\right\}cdot K+frac\left\{1\right\}\left\{n+1\right\}cdot Lright\right)$ $-lambdaleft\left(Sigma+frac\left\{1\right\}\left\{m\right\}cdot K+frac\left\{1\right\}\left\{n+1\right\}cdot Lright\right)$ $-lambdaleft\left(Sigma+frac\left\{1\right\}\left\{m+1\right\}cdot K+frac\left\{1\right\}\left\{n\right\}cdot Lright\right)$ $+lambdaleft\left(Sigma+frac\left\{1\right\}\left\{m\right\}cdot K+frac\left\{1\right\}\left\{n\right\}cdot Lright\right)$ 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 $lambdaleft\left(Sigma+frac\left\{1\right\}\left\{n+1\right\}cdot Kright\right)-lambdaleft\left(Sigma+frac\left\{1\right\}\left\{n\right\}cdot Kright\right)=pm 1$.
• 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 $nin mathbb\left\{Z\right\}$ let $M_\left\{K_n\right\}$ be the result of $frac\left\{1\right\}\left\{n\right\}$ Dehn surgery on M along K. Then the Casson invariant of $M_\left\{K_\left\{n+1\right\}\right\}$ minus the Casson invariant of $M_\left\{K_n\right\}$

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 $Sigma\left(p,q,r\right)$ is given by the formula:

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

## 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 $mathcal\left\{R\right\}\left(M\right)=R^\left\{mathrm\left\{irr\right\}\right\}\left(M\right)/SO\left(3\right)$ where $R^\left\{mathrm\left\{irr\right\}\right\}\left(M\right)$ denotes the space of irreducible SU(2) representations of $pi_1 \left(M\right)$. For a Heegaard splitting $Sigma=M_1 cup_F M_2$ of $Sigma$, the Casson invariant equals $frac\left\{\left(-1\right)^g\right\}\left\{2\right\}$ times the algebraic intersection of $mathcal\left\{R\right\}\left(M_1\right)$ with $mathcal\left\{R\right\}\left(M_2\right)$.

## 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 $lambda_\left\{CW\right\}$ from oriented rational homology 3-spheres to $mathbb\left\{Q\right\}$ satisfying the following properties:

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

$lambda_\left\{CW\right\}\left(M^prime\right)=lambda_\left\{CW\right\}\left(M\right)+frac\left\{langle m,murangle\right\}\left\{langle m,nuranglelangle mu,nurangle\right\}Delta_\left\{W\right\}^\left\{primeprime\right\}\left(M-K\right)\left(1\right)+tau_\left\{W\right\}\left(m,mu;nu\right)$ 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 $H_1\left(partial N\left(K\right),mathbb\left\{Z\right\}\right)$ to $H_1\left(M-K,mathbb\left\{Z\right\}\right)$.
• $langlecdot,cdotrangle$ 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 $H_1\left(M-K\right)/text\left\{Torsion\right\}$ in the infinite cyclic cover of M-K, and is symmetric and evaluates to 1 at 1.
• $tau_\left\{W\right\}\left(m,mu;nu\right)= -mathrm\left\{sgn\right\}langle y,mrangle s\left(langle x,mrangle,langle y,mrangle\right)+mathrm\left\{sgn\right\}langle y,murangle s\left(langle x,murangle,langle y,murangle\right)+frac\left\{\left(delta^2-1\right)langle m,murangle\right\}\left\{12langle m,nuranglelangle mu,nurangle\right\}$

where x, y are generators of $H_1\left(partial N\left(K\right);mathbb\left\{Z\right\}\right)$ such that $langle x,yrangle=1$, and $v=delta y$ for an integer $delta$. $s\left(p,q\right)$ is the Dedekind sum.

### Compact oriented 3-manifolds

Christine Lescop defined an extension $lambda_\left\{CWL\right\}$ 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, $lambda_\left\{CWL\right\}\left(M\right)=frac\left\{leftvert H_1\left(M\right)rightvertlambda_\left\{CW\right\}\left(M\right)\right\}\left\{2\right\}$.
• If the first Betti number of M is one, $lambda_\left\{CWL\right\}\left(M\right)=frac\left\{Delta^\left\{primeprime\right\}_M\left(1\right)\right\}\left\{2\right\}-frac\left\{mathrm\left\{torsion\right\}\left(H_1\left(M,mathbb\left\{Z\right\}\right)\right)\right\}\left\{12\right\}$ 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, $lambda_\left\{CWL\right\}\left(M\right)=leftvertmathrm\left\{torsion\right\}\left(H_1\left(M\right)\right)rightvertmathrm\left\{Link\right\}_M \left(gamma,gamma^prime\right)$ where $gamma$ is the oriented curve given by the intersection of two generators $S_1,S_2$ of $H_2\left(M;mathbb\left\{Z\right\}\right)$ 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 $H_1\left(M;mathbb\left\{Z\right\}\right)$, then $lambda_\left\{CWL\right\}\left(M\right)=leftvertmathrm\left\{torsion\right\}\left(H_1\left(M;mathbb\left\{Z\right\}\right)\right)rightvertleft\left(\left(acup bcup c\right)\left(\left[M\right]\right)right\right)^2$.
• If the first Betti number of M is greater than three, $lambda_\left\{CWL\right\}\left(M\right)=0$.

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 $lambda_\left\{CWL\right\}\left(M_1#M_2\right)=leftvert H_1\left(M_2\right)rightvertlambda_\left\{CWL\right\}\left(M_1\right)+leftvert H_1\left(M_1\right)rightvertlambda_\left\{CWL\right\}\left(M_2\right)$

### 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
Search another word or see Casson invarianton Dictionary | Thesaurus |Spanish