Heegaard splittings of Brieskorn spheres

Question

The genus $g$ handlebodies are building blocks of $3$-manifolds. They are constructed from $3$-ball $B^3$ by adding $g$-copies of $1$-handles $B^2 \times B^1$. Their boundaries are homeomorphic to the genus $g$ surface $\Sigma_g$.

It turns out that any closed orientable $3$-manifold $Y$ can be obtained by gluing together two handlebodies $H_1$ and $H_2$ (such a decomposition is called Heegaard splitting):

• $Y= H_1 \cup H_2$,
• $\partial H_1 = \partial H_2 = \Sigma_g$.

The basic examples of such $3$-manifolds are

• $S^3$,
• $S^1 \times S^2$,
• $S^1 \times S^1 \times S^1$,
• Lens spaces $L(p,q) = S^3 / \mathbb Z_p$,
• Brieskorn spheres $\Sigma(p,q,r) = \{ x^p + y^q +z^r = 0 \} \cap S^5 \subset \mathbb C^3$.

There are many references for Heegaard splittings of the first four of these examples, for example chapter 1 of Saveliev's book: Lectures on the Topology of 3-Manifolds.

How about the Brieskorn spheres? Is there an easy way to think about their Heegaard splittings? How can we draw them? Is there any good reference for this?

Answer 1

This is a late reply but it should be still helpful. It is from Zoltan Szabó's PCMI lecture notes.

Consider the following Heegaard splitting:

It is the genus $2$ Heegaard splitting for the $3$-manifold $W_2$.

To generalize this example to the family $W_n$, let us focus the $\beta_2$ cycle winding the right circle twice. Instead of twisting around the right circle two times, twist $n$-times to obtain the Heegaard splitting of $W_n$.

Let $K$ be the right-handed trefoil in $S^3$. Then show that

1. Let $Y$ be the $3$-manifold whose Heegaard diagram obtained by the Heegaard diagram of $W_n$ by omitting the $\beta_2$ curve. Then $Y$ is homeomorphic to $S^3 \setminus K$.

2. In general, $W_n$ is homeomorphic to $S^3_{n-4}(K)$, which is the $3$-manifold obtained by $n-4$-surgery along $K$ in $S^3$.

3. In particular, $W_3$ is homeomorphic to Poincaré homology sphere $\Sigma(2,3,5)$.

4. Further, $W_2$ is homeomorphic to $\Sigma(2,3,4)$ (Manolescu's example) and $W_1$ is homeomorphic to $\Sigma(2,3,3)$. The latter two are the boundaries of the plumbing graphs $E_7$ and $E_6$ respectively.

Answer 2

All Brieskorn spheres are small Seifert fibred spaces (small SFS, in brief), i.e. they admit a fibration $S^1 \to \Sigma(p,q,r) \to S^2$ with three multiple fibres. This is easier to see when $p,q,r$ are pairwise coprime: the fibration come from the action of $S^1\subset \mathbb{C}$ on $\Sigma(p,q,r)$ given by $\theta\cdot(x,y,z) = (\theta^{qr}x, \theta^{rp}y, \theta^{pq}z)$

Each small SFS $M$ admit a genus-2 Heegaard splitting; for instance, take two singular fibres of $M$ and an arc in $M$ that lifts a simple arc connecting the images of the two singular fibres. A neighbourhood of the two fibres and the arc is a 2-handlebody, whose complement is also a 2-handlebody, so we have a Heegaard decomposition of $M$. This is called a vertical Heegaard splitting of $M$.

Actually, genus-2 Heegaard splittings have been classified independently by Boileau, Collins, and Zieschang (Ann. Inst. Fourier 41 no. 4, 1991) and by Moriah (Invent. Math. 91, 1988).

Answer 3

This is not an answer, just a comment. It is from Manolescu's website. It seems to be related to your way of thinking Brieskorn spheres, but $2,3$ and $4$ are not pairwise coprime.