# Heegaard splittings of Brieskorn spheres

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?

• The construction of the Heegaard splitting of Seifert manifolds is described in Saveliev's book, see page 27, 28, 188 and 189. Feb 7, 2020 at 12:25

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.

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).

• Hi @Marco Golla, could you please provide a reference for showing that Brieskorn sphere are small seifert fibered spaces? I would like to try to understand what the Kirby diagram for $\Sigma(2,3,4)$ might look like (as a SFS, rather than $E_{7}$). Thanks! Jun 17, 2020 at 0:01
• I guess that this is (among other places) in Saveliev's book, or in Eisenbud and Neumann's book. A way of seeing it is to show that they're branched covers of $S^3$ branched over a torus knot/link and lift the Seifert fibration of $S^3$. (Also, I guess I was pretty loose with terminology: I think that technically a Brieskorn sphere has $\gcd(p,q) = \gcd(q,r) = \gcd(r,p) = 1$, so it's a homology sphere, and then the proof is at the beginning of my answer.) Jun 17, 2020 at 10:14

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.