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I have a basic question about the Heegaard diagrams involved in providing a framework for calculation of Floer-Homology of three-manifolds.

Typically such diagrams look like Figure 1 and Figure 2 here or these two (Image1 or Image2) from researchgate network.

And I would thankful if anybody could try to explain how to "read" this diagrams to a non-topologist.

We see a compact surface, which is probably assumed to be the boundary of certain three-manifold, and theory of compact surfaces states that this surface is up to homeomorphism always a connect sum of $g$ tori for $g \ge 1$. Looking at images in Image2 in last two links we find two sets of $g$ disjoint curves ("team red" $\alpha_0,..., \alpha_g$ and "team blue" $\beta_0,..., \beta_g$).

Now how does this information provide instructions to build a three-manifold?

My non-expert guess is that this data tells us: Start with two identical disjoint three-manifolds which have two $g$-tori as surfaces and the data provided by these Heegaard diagrams is nothing else than instructions how to glue the two three-manifolds along the surfaces. The instruction says probably that the curve $\alpha_i$ of one surface has to be glued homeomorphically with $\beta_i$ for other surface. And seemingly if we know all pairs of curves $\alpha_i$ and $\beta_i$ are glued together, then the gluing of the two surfaces is already uniquely determined up to homeomorphism and therefore we know how to glue the two disjoint three-manifold also the boundary.

Is this exactly the correct way to read a Heegaard diagram? Does there exist a more conventional way? Sorry, if the question is too elementary, I'm not an algebraic topologist and the motivation of this question is pure curiosity.

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    $\begingroup$ Start with $\Sigma \times I$. For each $\alpha$ curve (which you ought to thicken into an annulus, which you can do in a way which is unique up too isotopy), you glue on a copy of $D^2 \times I$ by pasting $S^1 \times I$ along the thickened $\alpha$ curve inside of $\Sigma \times \{0\}$. Do the same thing for the $\beta$ curves inside of $\Sigma \times \{1\}$. What you have produced is a compact 3-manifold with boundary $S^2 \sqcup S^2$; glue on 3-balls to make this a closed manifold. Gompf and Stipsicz have a good discussion. $\endgroup$
    – mme
    Aug 27, 2020 at 1:38
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    $\begingroup$ I’ve taken the liberty of editing your title, because I don’t think anyone who studies 3-manifolds calls them “3-folds”. $\endgroup$
    – HJRW
    Aug 27, 2020 at 6:43
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    $\begingroup$ You may also look at the first three chapters of beautiful paper of P. Ozsváth and Z. Szabó: math.mit.edu/~petero/Introduction.pdf $\endgroup$
    – Oğuz Şavk
    Aug 27, 2020 at 17:01
  • $\begingroup$ @MikeMiller: You mean it in the sense that you glue along each thickened tubulus of every $\alpha_i \subset \Sigma \times {0}$ a $2$-handle and them the same game for $\beta_j \subset \Sigma \times {1}$. What I still not understand is why the $3$-manifold obtained by this construction has boundary $S^2 \sqcup S^2$? $\endgroup$
    – user267839
    Aug 28, 2020 at 20:42
  • $\begingroup$ You are at each stage performing a handle attachment, which changes the boundary by a surgery along the attaching sphere. Check that this process reduces the genus by 1 every time you do a handle attachment. You do it g times. Again, G&S is a good source. $\endgroup$
    – mme
    Aug 28, 2020 at 21:18

2 Answers 2

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Chapter four of "Knots, Links, Braids and 3-Manifolds" by Prasolov and Sossinsky gives a highly readable (and nicely illustrated) introduction to three-manifolds via Heegaard splittings. Another, more classical, reference is chapter two of "Three-manifolds" by Hempel. Note that Hempel calls handlebodies "cubes with handles".

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You are probably familiar with definitions and theorems. But I prefer to write those for completeness. And also excuse for a paint-like drawing. I hope that they will be useful.

A handlebody of genus $g$ is a $3$-manifold constructed from the standart $3$-ball $B^3$ by adding $g$ copies of $1$-handles $B^2 \times B^1$. It is denoted by $H$ and $\partial H \approx \Sigma_g$ where $\Sigma_g$ is a genus $g$ surface, see the following figures.enter image description here

Let $Y$ be a $3$-manifold. A Heegaard splitting of $Y$ is a decomposition of $Y$ such that

  • $Y=H_0 \cup H_1$ where $H_0$ and $H_1$ are handlebodies,
  • $\partial H_0 = \partial H_1 = \Sigma_g$.

Theorem(Singer, 1933): Any closed oriented 3-manifold $Y$ admits a Heegaard splitting.

The genus $g$-surface $\Sigma_g$ is constructed from $S^2 = \mathbb{R}^2 \cup \{ \infty \}$ by attaching $g$ copies of $1$-handles, where we draw attaching spheres as pairs of matching disks.

enter image description here

So the followings are Heegaard splittings of $S^3$ and $S^1 \times S^2$ respectively:

enter image description here

The following is for a Heegaard diagram of lens space $L(5,2)$:

enter image description here

And the last scheme is for the famous Poincaré homology sphere $\Sigma(2,3,5)$:

enter image description here

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