Heegaard Splitting

In this page, $Y$ is a closed oriented three-manifold.

Motivation

One of the effective way to study the object is decompose until the object is realizable. We can classify the closed surfaces through the number of genus.

Okay then , there would be a natural question.

Could we decompose 3-manifold using 2-manifolds which are well known?

The answer is yes and through the Heegaard Splitting.

Definition

handlebody

Heegaard diagrams

Review of the original Heegaard diagram

Let $Y^3$ be a closed orientable connected 3-manifold, and ${\Sigma }_g$ (or $\Sigma$) be the genus $g$ surface.

Definition.
A complete set of attaching circles is a set $\{{\alpha }_1,\dots ,{\alpha }_g\}\subset {\Sigma }_g$ where the ${\alpha }_i$ are simple closed curves such that $[{\alpha }_i]\in H_1({\Sigma }_g;\mathbb{Z})$ are linearly independent, and ${\alpha }_i\cap {\alpha }_j=\varnothing$ for $i\ne j$.

A complete set of attaching circles determines a handlebody as follows:

1. Thicken ${\Sigma }_g$ to ${\Sigma }_g\times I$.
2. Fill the boundary $S^2$ by a ${\mathbb{D}}^3$ (i.e. attach 2-handles).

Definition.
A Heegaard diagram $H$ is a triple $({\Sigma }_g,\mathbf{\alpha },\mathbf{\beta })$ where ${\alpha }_i⋔{\beta }_j$. It is generic if each ${\alpha }_i$ intersects each ${\beta }_j$ transversally.

Using a Heegaard diagram, we can specify a 3-manifold. Examples: $S^3,L(3,1),T^3$.

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Existence of Heegaard diagrams

Theorem. Every 3-manifold has a Heegaard diagram.

Two approaches:

1. Triangulation.
   Every 3-manifold has a triangulation (Moise, 1952).
   A handlebody can be seen as a thickened 1-skeleton.
   The complement is also a handlebody (via dual graph).

2. Morse function.
   Every smooth manifold has a self-indexing Morse function.
   The Heegaard surface is ${\Sigma }_g=f^{-1}(3/2)$.

   • $\alpha$ circles: ascending manifolds of index 1 critical points.
   • $\beta$ circles: descending manifolds of index 2 critical points.

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Heegaard Moves

Definition.
Heegaard moves are transformations of diagrams:

1. Isotopies of $\alpha$ and $\beta$ curves (keeping them disjoint).
2. Handleslides of $\alpha$ (resp. $\beta$) curves over others (basis change in $H_1({\Sigma }_g;\mathbb{Z})$).
3. Stabilization: connect sum with genus 1 diagram of $S^3$.

Theorem.
If $H_1$ and $H_2$ are Heegaard diagrams of the same 3-manifold $Y$,
then they are equivalent after finitely many Heegaard moves.

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Pointed and doubly pointed diagrams

Definition.
A pointed Heegaard diagram is $({\Sigma }_g,\alpha ,\beta ,z)$ with $z\in {\Sigma }_g-\alpha -\beta$.

Moves must avoid $z$.

Definition.
A doubly pointed Heegaard diagram has two basepoints $w,z$.
Moves must avoid both.

References

• Moise (1952): Affine Structures in 3-Manifolds.