Generalized 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 λ from oriented integral homology 3-spheres to Z satisfying the following properties:

• λ(S³) = 0.
• Let Σ be an integral homology 3-sphere. Then for any knot K and for any integer n, the difference

${\displaystyle \lambda \left(\Sigma +{\frac {1}{n+1}\cdot K\right)-\lambda \left(\Sigma +{\frac {1}{n}\cdot K\right)}$

is independent of n. Here ${\displaystyle \Sigma +{\frac {1}{m}\cdot K}$ denotes ${\displaystyle {\frac {1}{m}$ Dehn surgery on Σ by K.

• For any boundary link K ∪ L in Σ the following expression is zero:

${\displaystyle \lambda \left(\Sigma +{\frac {1}{m+1}\cdot K+{\frac {1}{n+1}\cdot L\right)-\lambda \left(\Sigma +{\frac {1}{m}\cdot K+{\frac {1}{n+1}\cdot L\right)-\lambda \left(\Sigma +{\frac {1}{m+1}\cdot K+{\frac {1}{n}\cdot L\right)+\lambda \left(\Sigma +{\frac {1}{m}\cdot K+{\frac {1}{n}\cdot L\right)}$

The Casson invariant is unique (with respect to the above properties) up to an overall multiplicative constant.

Properties

• If K is the trefoil then

${\displaystyle \lambda \left(\Sigma +{\frac {1}{n+1}\cdot K\right)-\lambda \left(\Sigma +{\frac {1}{n}\cdot K\right)=\pm 1}$.

• The Casson invariant is 1 (or −1) for the Poincaré homology sphere.
• The Casson invariant changes sign if the orientation of M is reversed.
• The Rokhlin invariant of M is equal to 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 integer n

${\displaystyle \lambda \left(M+{\frac {1}{n+1}\cdot K\right)-\lambda \left(M+{\frac {1}{n}\cdot K\right)=\phi _{1}(K),}$

where ${\displaystyle \phi _{1}(K)}$ is the coefficient of ${\displaystyle z^{2}$ in the Alexander–Conway polynomial ${\displaystyle \nabla _{K}(z)}$, and is congruent (mod 2) to the Arf invariant of K.

• The Casson invariant is the degree 1 part of the Le–Murakami–Ohtsuki invariant.
• The Casson invariant for the Seifert manifold ${\displaystyle \Sigma (p,q,r)}$ is given by the formula:

${\displaystyle \lambda (\Sigma (p,q,r))=-{\frac {1}{8}\left[1-{\frac {1}{3pqr}\left(1-p^{2}q^{2}r^{2}+p^{2}q^{2}+q^{2}r^{2}+p^{2}r^{2}\right)-d(p,qr)-d(q,pr)-d(r,pq)\right]}$

where

${\displaystyle d(a,b)=-{\frac {1}{a}\sum _{k=1}^{a-1}\cot \left({\frac {\pi k}{a}\right)\cot \left({\frac {\pi bk}{a}\right)}$

The Casson invariant as a count of representations

Informally speaking, the Casson invariant counts half 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 ${\displaystyle {\mathcal {R}(M)=R^{\mathrm {irr} }(M)/SU(2)}$ where ${\displaystyle R^{\mathrm {irr} }(M)}$ denotes the space of irreducible SU(2) representations of ${\displaystyle \pi _{1}(M)}$. For a Heegaard splitting ${\displaystyle \Sigma =M_{1}\cup _{F}M_{2}$ of ${\displaystyle M}$, the Casson invariant equals ${\displaystyle {\frac {(-1)^{g}{2}$ times the algebraic intersection of ${\displaystyle {\mathcal {R}(M_{1})}$ with ${\displaystyle {\mathcal {R}(M_{2})}$.

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 λ_CW from oriented rational homology 3-spheres to Q satisfying the following properties:

1. λ(S³) = 0.

2. For every 1-component Dehn surgery presentation (K, μ) of an oriented rational homology sphere M′ in an oriented rational homology sphere M:

${\displaystyle \lambda _{CW}(M^{\prime })=\lambda _{CW}(M)+{\frac {\langle m,\mu \rangle }{\langle m,\nu \rangle \langle \mu ,\nu \rangle }\Delta _{W}^{\prime \prime }(M-K)(1)+\tau _{W}(m,\mu ;\nu )}$

where:

• m is an oriented meridian of a knot K and μ is the characteristic curve of the surgery.
• ν is a generator the kernel of the natural map H₁(∂N(K), Z) → H₁(M−K, Z).
• ${\displaystyle \langle \cdot ,\cdot \rangle }$ 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 ${\displaystyle H_{1}(M-K)/{\text{Torsion}$ in the infinite cyclic cover of M−K, and is symmetric and evaluates to 1 at 1.
• ${\displaystyle \tau _{W}(m,\mu ;\nu )=-\mathrm {sgn} \langle y,m\rangle s(\langle x,m\rangle ,\langle y,m\rangle )+\mathrm {sgn} \langle y,\mu \rangle s(\langle x,\mu \rangle ,\langle y,\mu \rangle )+{\frac {(\delta ^{2}-1)\langle m,\mu \rangle }{12\langle m,\nu \rangle \langle \mu ,\nu \rangle }$

where x, y are generators of H₁(∂N(K), Z) such that ${\displaystyle \langle x,y\rangle =1}$, v = δy for an integer δ and s(p, q) is the Dedekind sum.

Note that for integer homology spheres, the Walker's normalization is twice that of Casson's: ${\displaystyle \lambda _{CW}(M)=2\lambda (M)}$.

Compact oriented 3-manifolds

Christine Lescop defined an extension λ_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,

${\displaystyle \lambda _{CWL}(M)={\tfrac {1}{2}\left\vert H_{1}(M)\right\vert \lambda _{CW}(M)}$.

• If the first Betti number of M is one,

${\displaystyle \lambda _{CWL}(M)={\frac {\Delta _{M}^{\prime \prime }(1)}{2}-{\frac {\mathrm {torsion} (H_{1}(M,\mathbb {Z} ))}{12}$

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,

${\displaystyle \lambda _{CWL}(M)=\left\vert \mathrm {torsion} (H_{1}(M))\right\vert \mathrm {Link} _{M}(\gamma ,\gamma ^{\prime })}$

where γ is the oriented curve given by the intersection of two generators ${\displaystyle S_{1},S_{2}$ of ${\displaystyle H_{2}(M;\mathbb {Z} )}$ and ${\displaystyle \gamma ^{\prime }$ is the parallel curve to γ induced by the trivialization of the tubular neighbourhood of γ determined by ${\displaystyle S_{1},S_{2}$.

• If the first Betti number of M is three, then for a,b,c a basis for ${\displaystyle H_{1}(M;\mathbb {Z} )}$, then

${\displaystyle \lambda _{CWL}(M)=\left\vert \mathrm {torsion} (H_{1}(M;\mathbb {Z} ))\right\vert \left((a\cup b\cup c)([M])\right)^{2}$.

• If the first Betti number of M is greater than three, ${\displaystyle \lambda _{CWL}(M)=0}$.

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

• When the orientation of M changes the behavior of ${\displaystyle \lambda _{CWL}(M)}$ depends on the first Betti number ${\displaystyle b_{1}(M)=\operatorname {rank} H_{1}(M;\mathbb {Z} )}$of M: if ${\displaystyle {\overline {M}$ is M with the opposite orientation, then

${\displaystyle \lambda _{CWL}({\overline {M})=(-1)^{b_{1}(M)+1}\lambda _{CWL}(M).}$

That is, if the first Betti number of M is odd the Casson–Walker–Lescop invariant is unchanged, while if it is even it changes sign.

• For connect-sums of manifolds

${\displaystyle \lambda _{CWL}(M_{1}\#M_{2})=\left\vert H_{1}(M_{2})\right\vert \lambda _{CWL}(M_{1})+\left\vert H_{1}(M_{1})\right\vert \lambda _{CWL}(M_{2})}$

SU(N)

In 1990, C. Taubes showed that the SU(2) Casson invariant of a 3-homology sphere M has a gauge theoretic interpretation as the Euler characteristic of ${\displaystyle {\mathcal {A}/{\mathcal {G}$, where ${\displaystyle {\mathcal {A}$ is the space of SU(2) connections on M and ${\displaystyle {\mathcal {G}$ is the group of gauge transformations. He regarded the Chern–Simons invariant as a ${\displaystyle S^{1}$-valued Morse function on ${\displaystyle {\mathcal {A}/{\mathcal {G}$ and used invariance under perturbations to define an invariant which he equated with the SU(2) Casson invariant. (Taubes (1990))

H. Boden and C. Herald (1998) used a similar approach to define an SU(3) Casson invariant for integral homology 3-spheres.

References

• Selman Akbulut and John 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
• Michael 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.
• Hans Boden and Christopher Herald, The SU(3) Casson invariant for integral homology 3-spheres. Journal of Differential Geometry 50 (1998), 147–206.
• Christine Lescop, Global Surgery Formula for the Casson-Walker Invariant. 1995, ISBN 0-691-02132-5
• Nikolai 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
• Taubes, Clifford Henry (1990), "Casson's invariant and gauge theory.", Journal of Differential Geometry, 31: 547–599
• Kevin 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