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
::$\lambda\left(\Sigma+\frac\cdot K\right)-\lambda\left(\Sigma+\frac\cdot K\right)$
:is independent of ''n''. Here $\Sigma+\frac\cdot K$ denotes $\frac$ Dehn surgery on Σ by ''K''.
*For any boundary link ''K'' ∪ ''L'' in Σ the following expression is zero:
::$\lambda\left(\Sigma+\frac\cdot K+\frac\cdot L\right) -\lambda\left(\Sigma+\frac\cdot K+\frac\cdot L\right)-\lambda\left(\Sigma+\frac\cdot K+\frac\cdot L\right) +\lambda\left(\Sigma+\frac\cdot K+\frac\cdot L\right)$

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

Properties

*If K is the trefoil then
::$\lambda\left(\Sigma+\frac\cdot K\right)-\lambda\left(\Sigma+\frac\cdot K\right)=\pm 1$.
*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''
::$\lambda \left ( M + \frac\cdot K\right ) - \lambda \left ( M + \frac\cdot K\right ) = \phi_1 (K),$
:where $\phi_1 (K)$ is the coefficient of $z^2$ in the Alexander–Conway polynomial $\nabla_K(z)$, 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 $\Sigma(p,q,r)$ is given by the formula:
:: \lambda(\Sigma(p,q,r))=-\frac\left -\frac\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/math>
:where
::$d(a,b)=-\frac\sum_^\cot\left(\frac\right)\cot\left(\frac\right)$

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 $\mathcal(M)=R^(M)/SU(2)$ where $R^(M)$ denotes the space of irreducible SU(2) representations of $\pi_1 (M)$. For a Heegaard splitting $\Sigma=M_1 \cup_F M_2$ of $M$, the Casson invariant equals $\frac$ times the algebraic intersection of $\mathcal(M_1)$ with $\mathcal(M_2)$.

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'':
:$\lambda_(M^\prime)=\lambda_(M)+\frac\Delta_^(M-K)(1)+\tau_(m,\mu;\nu)$
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 ''H''₁(∂''N''(''K''), Z) → ''H''₁(''M''−''K'', 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 $H_1(M-K)/\text$ in the infinite cyclic cover of ''M''−''K'', and is symmetric and evaluates to 1 at 1.
*$\tau_(m,\mu;\nu)= -\mathrm\langle y,m\rangle s(\langle x,m\rangle,\langle y,m\rangle)+\mathrm\langle y,\mu\rangle s(\langle x,\mu\rangle,\langle y,\mu\rangle)+\frac$
:where ''x'', ''y'' are generators of ''H''₁(∂''N''(''K''), Z) such that $\langle x,y\rangle=1$, ''v'' = δ''y'' 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: $\lambda_(M) = 2 \lambda(M)$.

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,
::$\lambda_(M)=\tfrac\left\vert H_1(M)\right\vert\lambda_(M)$.
*If the first Betti number of ''M'' is one,
::$\lambda_(M)=\frac-\frac$
: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,
::$\lambda_(M)=\left\vert\mathrm(H_1(M))\right\vert\mathrm_M (\gamma,\gamma^\prime)$
:where γ is the oriented curve given by the intersection of two generators $S_1,S_2$ of $H_2(M;\mathbb)$ 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 $H_1(M;\mathbb)$, then
::\lambda_(M)=\left\vert\mathrm(H_1(M;\mathbb))\right\vert\left((a\cup b\cup c)( \right)^2.
*If the first Betti number of ''M'' is greater than three, $\lambda_(M)=0$.

The Casson–Walker–Lescop invariant has the following properties:
*When the orientation of ''M'' changes the behavior of $\lambda_(M)$ depends on the first Betti number $b_1(M) = \operatorname H_1(M;\mathbb)$of ''M'': if $\overline$ is ''M'' with the opposite orientation, then
::$\lambda_(\overline) = (-1)^\lambda_(M).$
: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
::$\lambda_(M_1\#M_2)=\left\vert H_1(M_2)\right\vert\lambda_(M_1)+\left\vert H_1(M_1)\right\vert\lambda_(M_2)$

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 $\mathcal/\mathcal$, where $\mathcal$ is the space of SU(2) connections on ''M'' and $\mathcal$ is the group of gauge transformations. He regarded the Chern–Simons invariant as a $S^1$-valued Morse function on $\mathcal/\mathcal$ and used invariance under perturbations to define an invariant which he equated with the SU(2) Casson invariant. ()

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.
* 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,
*Nikolai Saveliev, ''Lectures on the topology of 3-manifolds: An introduction to the Casson Invariant.'' de Gruyter, Berlin, 1999.
*
*Kevin Walker, ''An extension of Casson's invariant.'' Annals of Mathematics Studies, 126. Princeton University Press, Princeton, NJ, 1992. {{ISBN, 0-691-02532-0

Geometric topology