History of Lorentz transformations

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

$\lambda (\mathrm{\Sigma }+\frac{1}{n+1}\cdot K)-\lambda (\mathrm{\Sigma }+\frac{1}{n}\cdot K)$

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

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

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

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

Properties

• If K is the trefoil then

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

• 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

$\lambda (M+\frac{1}{n+1}\cdot K)-\lambda (M+\frac{1}{n}\cdot K)=\alpha (K),$

where α(K) 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 $\mathrm{\Sigma }(p,q,r)$ is given by the formula:

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

where

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

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 $\mathcal{ℛ}(M)=R^{\mathrm{i}\mathrm{r}\mathrm{r}}(M)/SO(3)$ where $R^{\mathrm{i}\mathrm{r}\mathrm{r}}(M)$ denotes the space of irreducible SU(2) representations of ${\pi }_1(M)$. For a Heegaard splitting $\mathrm{\Sigma }=M_1{\cup }_F{M}_2$ of $M$, the Casson invariant equals $\frac{(-1{)}^g}{2}$ 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 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:

${\lambda }_{CW}(M^{\prime })={\lambda }_{CW}(M)+\frac{⟨m,\mu ⟩}{⟨m,\nu ⟩⟨\mu ,\nu ⟩}{\mathrm{\Delta }}_W^{\prime \prime }(M-K)(1)+{\tau }_W(m,\mu ;\nu )$

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).
• $⟨\cdot ,\cdot ⟩$ 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 $H_1(M-K)/\text{Torsion}$ in the infinite cyclic cover of M−K, and is symmetric and evaluates to 1 at 1.
• ${\tau }_W(m,\mu ;\nu )=-\mathrm{s}\mathrm{g}\mathrm{n}⟨y,m⟩s(⟨x,m⟩,⟨y,m⟩)+\mathrm{s}\mathrm{g}\mathrm{n}⟨y,\mu ⟩s(⟨x,\mu ⟩,⟨y,\mu ⟩)+\frac{({\delta }^2-1)⟨m,\mu ⟩}{12⟨m,\nu ⟩⟨\mu ,\nu ⟩}$

where x, y are generators of H₁(∂N(K), Z) such that $⟨x,y⟩=1$, 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: ${\lambda }_{CW}(M)=2\lambda (M)$.

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,

${\lambda }_{CWL}(M)=\frac{1}{2}|H_1(M)|{\lambda }_{CW}(M)$.

• If the first Betti number of M is one,

${\lambda }_{CWL}(M)=\frac{{\mathrm{\Delta }}_M^{\prime \prime }(1)}{2}-\frac{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}(H_1(M,\mathbb{\mathbb{Z}}))}{12}$

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,

${\lambda }_{CWL}(M)=|\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}(H_1(M))|{\mathrm{L}\mathrm{i}\mathrm{n}\mathrm{k}}_M(\gamma ,{\gamma }^{\prime })$

where γ is the oriented curve given by the intersection of two generators $S_1,S_2$ of $H_2(M;\mathbb{\mathbb{Z}})$ and ${\gamma }^{\prime }$ is the parallel curve to γ induced by the trivialization of the tubular neighbourhood of γ determined by $S_1,S_2$.

• If the first Betti number of M is three, then for a,b,c a basis for $H_1(M;\mathbb{\mathbb{Z}})$, then

${\lambda }_{CWL}(M)=|\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}(H_1(M;\mathbb{\mathbb{Z}}))|{((a\cup b\cup c)([M]))}^2$.

• 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

${\lambda }_{CWL}(M_1\#M_2)=|H_1(M_2)|{\lambda }_{CWL}(M_1)+|H_1(M_1)|{\lambda }_{CWL}(M_2)$

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 0-691-02132-5
• 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