Casson invariant

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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.

Contents

[edit] Definition

A Casson invariant is a surjective map λ from oriented integral homology 3-spheres to \mathbb{Z} satisfying the following properties:

  • λ(S3) = 0.
  • Let Σ be an integral homology 3-sphere. Then for any knot K and for any n\in\mathbb{Z}, the difference

\lambda\left(\Sigma+\frac{1}{n+1}\cdot K\right)-\lambda\left(\Sigma+\frac{1}{n}\cdot K\right) is independent of n. Here \Sigma+\frac{1}{m}\cdot K denotes \frac{1}{m} Dehn surgery on Σ by K.

\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) is equal to zero for any boundary link K\cup L in Σ.

The Casson invariant is unique up to sign.

[edit] Properties

  • If K is the trefoil then \lambda\left(\Sigma+\frac{1}{n+1}\cdot K\right)-\lambda\left(\Sigma+\frac{1}{n}\cdot K\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 n\in \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 Σ(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\right) -d(p,qr)-d(q,pr)-d(r,pq)\right] where 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)

[edit] 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 Rirr(M) denotes the space of irreducible SU(2) representations of π1(M). For a Heegaard splitting \Sigma=M_1 \cup_F M_2 of Σ, the Casson invariant equals \frac{(-1)^g}{2} times the algebraic intersection of \mathcal{R}(M_1) with \mathcal{R}(M_2).

[edit] Generalizations

[edit] 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 \mathbb{Q} satisfying the following properties:

  • λ(S3) = 0.
  • For every 1-component Dehn surgery presentation (K,μ) 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\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 mu is the characteristic curve of the surgery.
    • ν is a generator the kernel of the natural map from H_1(\partial N(K),\mathbb{Z}) to H_1(M-K,\mathbb{Z}).
    • \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 H1(M − K) / 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,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 H_1(\partial N(K);\mathbb{Z}) such that \langle x,y\rangle=1, and v = δy for an integer δ. s(p,q) is the Dedekind sum.

[edit] 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:

  • If the first Betti number of M is zero, \lambda_{CWL}(M)=\frac{\left\vert H_1(M)\right\vert\lambda_{CW}(M)}{2}.
  • If the first Betti number of M is one, \lambda_{CWL}(M)=\frac{\Delta^{\prime\prime}_M(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, \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 S1,S2 of H_2(M;\mathbb{Z}) and \gamma^\prime is the parallel curve to γ induced by the trivialization of the tubular neighbourhood of γ determined by S1,S2.
  • 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)=\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, λCWL(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)=\left\vert H_1(M_2)\right\vert\lambda_{CWL}(M_1)+\left\vert H_1(M_1)\right\vert\lambda_{CWL}(M_2)

[edit] SU(N)

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

[edit] 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