# Heegard diagrams for three-manifolds

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.

• 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.
– mme
Aug 27 '20 at 1:38
• I’ve taken the liberty of editing your title, because I don’t think anyone who studies 3-manifolds calls them “3-folds”.
– HJRW
Aug 27 '20 at 6:43
• You may also look at the first three chapters of beautiful paper of P. Ozsváth and Z. Szabó: math.mit.edu/~petero/Introduction.pdf Aug 27 '20 at 17:01
• @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 $S^2 \sqcup S^2$? Aug 28 '20 at 20:42
• 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.
– mme
Aug 28 '20 at 21:18

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".

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. 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. So the followings are Heegaard splittings of $$S^3$$ and $$S^1 \times S^2$$ respectively: The following is for a Heegaard diagram of lens space $$L(5,2)$$: And the last scheme is for the famous Poincaré homology sphere $$\Sigma(2,3,5)$$: 