# Heegaard splitting of a 3-manifold with boundary

A Heegaard splitting of a closed orientable 3-manifold $M$ is $M=H \cup H'$, where $H$ and $H'$ are handlebodies.

Is there any similar concept for orientable 3-manifolds with boundaries?

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The answer is yes. Decomposing $3$-manifolds using compression bodies instead of handlebodies results in "generalized Heegaard splittings."

If you follow the link to the compression body Wikipedia page, you'll see two dual definitions of "compression body." For the definition of generalized Heegaard splitting, I'm going to use this: a compression body $C$ is defined as follows. Take a closed surface $\Sigma$ and cross it with the interval $[0,1]$. This has two boundary components: $\partial_+ = \Sigma\times\{0\}$ and $\partial_- = \Sigma\times\{1\}$. Now add some number (possibly zero) of $2$-handles to $\partial_-$ and then add some (possibly no) $3$-handles to $\partial_-$ (that is, the component of the boundary of $\Sigma\times[0,1]\cup(\mbox{ 2-handles }$ which is not $\partial_+$). What remains is a compression body $C$. We take $\partial_+$ to be $\Sigma\times\{0\}$ and $\partial_-$ to be the complement of $\partial_+$ in $\partial C$. (Note that we do not require $\Sigma$ to be connected.)

(For useful pictures, see p. 5 of the notes by Saito-Scharlemann-Schultens below.)

A generalized Heegaard splitting of a $3$-manifold $M$ is a decomposition of $M$ into compression bodies $C_0,C_1,C_2,\ldots,C_n$ so that $\partial_- C_{2i} = \partial_- C_{2i-i}$ (and possibly some boundary components of $M$), and $\partial_+ C_{2i} = \partial_+ C_{2i+1}$.

If we have $n=1$ and $\partial M = \emptyset = \partial_- C_0 = \partial_- C_1$, then the generalized Heegaard splitting is a Heegaard splitting.

As you probably know, Heegaard splittings can arise by examining a Morse function $f$ on a closed $3$-manifold $M$ with all index $0$ and $1$ critical values less than all the index $2$ and $3$ critical values (e.g. a self-indexing Morse function). Take some regular value $x_0$ such that $x_0$ is greater than the index $0$ and $1$ critical values and less than the index $2$ and $3$ critical values. Then $f^{-1}(x_0)$ is a Heegaard surface for $M$, with the sublevel set of $x_0$ one handlebody and the superlevel set the other.

A generalized Heegaard splitting arises by taking any Morse function on $M$ (if $\partial M \neq\emptyset$, require that $f$ be constant on each boundary component) and grouping the index $0$ and $1$ points and index $2$ and $3$ points. Then take regular values $r_i$ (so that $r_0$ is between the first group of $0$,$1$-points and $2$,$3$-points, $r_1$ is between the first group of $2$,$3$-points and the second group of $0$,$1$-points, and so forth). Finally, define $C_0 = f^{-1}(-\infty,r_0]$, $C_1 = f^{-1}[r_0,r_1]$, etc.

Generalized Heegaard splittings are useful in analyzing the "complexity" of $3$-manifolds and can be related to the Cheeger constant of the $3$-manifold (c.f. Lackenby, "Heegaard splittings, the virtually Haken conjecture, and property $\tau$", Inventiones Mathematicae). There is also a neat way to construct a generalized Heegaard splitting of a link complement given some presentation of the link in $\mathbb{S}^3$.

For more information, read this Mathoverflow answer by Ian Agol, these notes by Martin Scharlemann, Toshio Saito, and Jennifer Schultens, or these notes by Jesse Johnson. I found all three very helpful as I learned about generalized Heegaard splittings.

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I'm just going to blantly quote Wikipedia since it is not something I know much (anything) about

Heegaard splittings can also be defined for compact 3-manifolds with boundary by replacing handlebodies with compression bodies. The gluing map is between the positive boundaries of the compression bodies.

where a compression body is a generalisation of a handlebody.

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