Casson invariant

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 maplambda 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 differencelambdaleft(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] whered(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 asmathcal{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 equalsfrac{(-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 maplambda_{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


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