• Images
• Videos
• Documents
• Books
• Articles

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

λ ( Σ + 1 n + 1 ⋅ K ) − λ ( Σ + 1 n ⋅ K ) {\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 Σ + 1 m ⋅ K {\displaystyle \Sigma +{\frac {1}{m}}\cdot K} denotes 1 m {\displaystyle {\frac {1}{m}}} Dehn surgery on Σ by K.

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

λ ( Σ + 1 m + 1 ⋅ K + 1 n + 1 ⋅ L ) − λ ( Σ + 1 m ⋅ K + 1 n + 1 ⋅ L ) − λ ( Σ + 1 m + 1 ⋅ K + 1 n ⋅ L ) + λ ( Σ + 1 m ⋅ K + 1 n ⋅ L ) {\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

λ ( Σ + 1 n + 1 ⋅ K ) − λ ( Σ + 1 n ⋅ K ) = ± 1 {\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

λ ( M + 1 n + 1 ⋅ K ) − λ ( M + 1 n ⋅ K ) = ϕ 1 ( K ) , {\displaystyle \lambda \left(M+{\frac {1}{n+1}}\cdot K\right)-\lambda \left(M+{\frac {1}{n}}\cdot K\right)=\phi _{1}(K),} where ϕ 1 ( K ) {\displaystyle \phi _{1}(K)} is the coefficient of z 2 {\displaystyle z^{2}} in the Alexander–Conway polynomial ∇ K ( z ) {\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 Σ ( p , q , r ) {\displaystyle \Sigma (p,q,r)} is given by the formula:

λ ( Σ ( p , q , r ) ) = − 1 8 [ 1 − 1 3 p q r ( 1 − p 2 q 2 r 2 + p 2 q 2 + q 2 r 2 + p 2 r 2 ) − d ( p , q r ) − d ( q , p r ) − d ( r , p q ) ] {\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 d ( a , b ) = − 1 a ∑ k = 1 a − 1 cot ⁡ ( π k a ) cot ⁡ ( π b k a ) {\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 R ( M ) = R i r r ( M ) / S U ( 2 ) {\displaystyle {\mathcal {R}}(M)=R^{\mathrm {irr} }(M)/SU(2)} where R i r r ( M ) {\displaystyle R^{\mathrm {irr} }(M)} denotes the space of irreducible SU(2) representations of π 1 ( M ) {\displaystyle \pi _{1}(M)} . For a Heegaard splitting Σ = M 1 ∪ F M 2 {\displaystyle \Sigma =M_{1}\cup _{F}M_{2}} of M {\displaystyle M} , the Casson invariant equals ( − 1 ) g 2 {\displaystyle {\frac {(-1)^{g}}{2}}} times the algebraic intersection of R ( M 1 ) {\displaystyle {\mathcal {R}}(M_{1})} with R ( M 2 ) {\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 λCW 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:

λ C W ( M ′ ) = λ C W ( M ) + ⟨ m , μ ⟩ ⟨ m , ν ⟩ ⟨ μ , ν ⟩ Δ W ′ ′ ( M − K ) ( 1 ) + τ W ( m , μ ; ν ) {\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 H 1 ( M − K ) / Torsion {\displaystyle H_{1}(M-K)/{\text{Torsion}}} in the infinite cyclic cover of M−K, and is symmetric and evaluates to 1 at 1.
• τ W ( m , μ ; ν ) = − s g n ⟨ y , m ⟩ s ( ⟨ x , m ⟩ , ⟨ y , m ⟩ ) + s g n ⟨ y , μ ⟩ s ( ⟨ x , μ ⟩ , ⟨ y , μ ⟩ ) + ( δ 2 − 1 ) ⟨ m , μ ⟩ 12 ⟨ m , ν ⟩ ⟨ μ , ν ⟩ {\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 ⟨ x , y ⟩ = 1 {\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: λ C W ( M ) = 2 λ ( M ) {\displaystyle \lambda _{CW}(M)=2\lambda (M)} .

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,

λ C W L ( M ) = 1 2 | H 1 ( M ) | λ C W ( M ) {\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,

λ C W L ( M ) = Δ M ′ ′ ( 1 ) 2 − t o r s i o n ( H 1 ( M , Z ) ) 12 {\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,

λ C W L ( M ) = | t o r s i o n ( H 1 ( M ) ) | L i n k M ( γ , γ ′ ) {\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 S 1 , S 2 {\displaystyle S_{1},S_{2}} of H 2 ( M ; Z ) {\displaystyle H_{2}(M;\mathbb {Z} )} and γ ′ {\displaystyle \gamma ^{\prime }} is the parallel curve to γ induced by the trivialization of the tubular neighbourhood of γ determined by S 1 , S 2 {\displaystyle S_{1},S_{2}} .

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

λ C W L ( M ) = | t o r s i o n ( H 1 ( M ; Z ) ) | ( ( a ∪ b ∪ c ) ( [ M ] ) ) 2 {\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, λ C W L ( M ) = 0 {\displaystyle \lambda _{CWL}(M)=0} .

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

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

λ C W L ( M ¯ ) = ( − 1 ) b 1 ( M ) + 1 λ C W L ( M ) . {\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

λ C W L ( M 1 # M 2 ) = | H 1 ( M 2 ) | λ C W L ( M 1 ) + | H 1 ( M 1 ) | λ C W L ( M 2 ) {\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 A / G {\displaystyle {\mathcal {A}}/{\mathcal {G}}} , where A {\displaystyle {\mathcal {A}}} is the space of SU(2) connections on M and G {\displaystyle {\mathcal {G}}} is the group of gauge transformations. He regarded the Chern–Simons invariant as a S 1 {\displaystyle S^{1}} -valued Morse function on A / G {\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.

• 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