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.
 A: 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".
A: 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 $\Sigma(2,3,5)$:

