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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?

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    $\begingroup$ The construction of the Heegaard splitting of Seifert manifolds is described in Saveliev's book, see page 27, 28, 188 and 189. $\endgroup$ – Oğuz Şavk Feb 7 '20 at 12:25
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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:

enter image description here

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.

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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).

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  • $\begingroup$ 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! $\endgroup$ – no_idea Jun 17 '20 at 0:01
  • $\begingroup$ 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.) $\endgroup$ – Marco Golla Jun 17 '20 at 10:14
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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.

enter image description here

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