# Heegaard splittings of graph manifolds

Castellón, 12/12/2017

# Before starting

• Thank the organizers (applause).
• Joint work with E. Artal-Bartolo and S. Isaza.
• Warning: topology inside. Might contain traces of singularities.

# Motivation

Take an isolated surface singularity.

By Milnor fibration theorem, the germ is the topological cone over the link.

• Take a resolution of singularities, with exceptional divisor $$E$$.
• The link becomes the boundary of a regular neighborhood of $$E$$.
• It can be decomposed in pieces corresponding to the components of $$E$$.
• Each piecse fibers over a component of $$E$$, with fiber $$\mathbb{S}^1$$
• At each normal crossing, two pieces must be glued:
• The fiber of one piece corresponds to a section of the second.

# $$\mathbb{S}^1$$-bundles over (closed,oriented) surfaces.

Let $$\pi: M \rightarrow \Sigma$$ be a $$\mathbb{S}^1$$ bundle

• Take a disk $$D$$ on the surface $$\Sigma$$.
• Over $$D$$, the bundle is a product, take a section $$s_1$$.
• Over $$\Sigma\setminus D$$, it is also a product, with section $$s_2$$.
• $$s_1(\partial D)$$ and $$s_2(\partial (\Sigma \setminus D))$$ are two curves in a torus $$\mathbb{S}^1 \times \mathbb{S}^1$$
• The intersection number of these two curves is called the Euler number of the bundle

# The Euler number of an $$\mathbb{S}^1$$ bundle

• Is well defined
• Classifies the bundle
• In the case of regular neighborhoods of curves in a surface, coincides with the self intersection

# Plumbing

Given two $$\mathbb{S}^1$$ bundles $$\pi_1: M_1 \rightarrow \Sigma_1$$, $$\pi_2: M_2 \rightarrow \Sigma_2$$

• Consider a disk $$D_i$$ in each $$\Sigma_i$$ and the trivialization $$\pi_i^{-1}(D_i)=D_i\times \mathbb{S}^1$$
• In $$\partial D_i \times \mathbb{S}^1$$ we have a fiber $$F_i$$ and a section $$S_i$$
• Take $$\pi_i^{-1}(\Sigma_i\setminus D_i)$$, they have a torus as boundary.
• Glue the tori, in such a way that $$F_1\cong S_2$$, $$F_2\cong S_1$$.
• We get a new manifold.
• This operation is called plumbing.

# Graph manifolds

Are obtained by several plumbings

• They can be described by a decorated graph:
• Each vertex is a bundle.
• Each edge is a plumbing.
• The links of a surface singularity are graph manifolds, described by the resolution graph.

# Handle bodies

Are the result of adding handles to closed balls (equivalently, identifying pairs of disks in the boundary)

There are special curves that identify the handles (cutting curves).

# Heegaard splittings

Two handle bodies of the same genus can be glued along the boundary, obtaining a closed $$3$$-manifold.

Every $$3$$-manifold can be obtained in this way (although not in a unique way).

The gluing surface, together with the two systems of cutting curves is called the Heegaard diagram of the splitting.

Heegaard-Floer homology is defined after such diagrams.

# Previous lemmas

Let $$\mathbb{T}$$ be a solid torus, $$c$$ a closed curve in $$\partial \mathbb{T}$$ isotopic to its core. $$M$$ a $$3$$-manifold with boundary, and $$s$$ a closed curve in $$\partial M$$. Consider $$A_c$$, $$A_s$$ small neighborhoods in the boundaries of $$c$$ and $$s$$.

Lemma: The result of gluing $$\mathbb{T}$$ with $$M$$ through an orientation reversing homeomorphism $$A_c\to A_s$$ is homeomorphic to $$M$$.

This operation is called a float gluing

# Heegaard diagrams of $$\mathbb{S}^1$$ bundles

Lemma: Let $$\Sigma'$$ be an oriented surface of genus $$g$$ with one boundary component. Then $$\Sigma'\times I$$ is a handle body of genus $$2g$$.

# Heegaard diagrams of $$\mathbb{S}^1$$ bundles

Lemma: Let $$\Sigma'$$ be an oriented surface of genus $$g$$ with $$n$$ boundary components. Then $$\Sigma'\times I$$ is a handle body of genus $$2g+n-1$$.

# Heegaard diagrams of $$\mathbb{S}^1$$ bundles

Lemma: Let $$\Sigma'$$ be an oriented surface of genus $$g$$ with $$n$$ boundary components. Then $$\Sigma'\times I$$ is a handle body of genus $$2g+n-1$$.

# Heegaard splittings of $$\mathbb{S}^1$$ bundles

Let's start with a $$(g,e)$$ $$\mathbb{S}^1$$ bundle $$\pi$$.

• Take one or more disjoint disks $$D_1,\ldots, D_n$$ in $$\Sigma_g$$.
• Assign a $$\pm 1$$ to each one, such that the total sum is $$e$$
• Choose one of them to be called the main disk.
• Over $$\Sigma'$$, take two parallel sections $$s_1,s_2$$
• $$s_1(\Sigma')$$ and $$s_2(\Sigma')$$ divide $$\pi^{-1}(\Sigma')$$ in two parts: $$M_s$$ and $$M_b$$.
• $$M_s\cong M_b \cong \Sigma'\times I$$

# Heegaard splittings of $$\mathbb{S}^1$$ bundles

Over the $$D_i$$, choose trivializations such that local Euler numbers are the $$\pm 1$$ chosen before.

• The boundaries of the sections $$s_i$$ correspond to the local curve $$-s_d\pm f_d$$, so they are isotopic to the core of the solid torus.
• So we can perform float gluings of the solid tori.

Our Heegaard splitting is $$M_s$$, $$M_b\cup \pi^{-1}(D_1\cup\cdots\cup D_n)$$

# Heegaard diagrams of $$\mathbb{S}^1$$ bundles

Our Heegaard surface is formed by two copies of a genus $$g$$ surface, joined by $$n$$ tubes.

# Heegaard diagrams of $$\mathbb{S}^1$$ bundles

One system of cutting curves:

# Heegaard diagrams of $$\mathbb{S}^1$$ bundles

The other system goes parallel except in the boundary of the floats:

# Heegaard diagrams of $$\mathbb{S}^1$$ bundles

The other system goes parallel except in the boundary of the floats:

# Plumbing: previous lemmas

Lemma: Let $$M_1,M_2$$ be handle bodies of geni $$g_1,g_2$$. $$c_i$$ is a curve in $$\partial M_i$$ that meets only une cutting curve, transversally, and only at one point. $$A_i$$ is an annulus in $$\partial M_i$$ that is a regular neighborhood of $$c_i$$. The gluing of $$M_1$$ and $$M_2$$ along an orientation-reversing homeomorphism $$A_1\to M_2$$ is a handle body of genus $$g_1+g_2-1$$.

# Plumbing of two handle bodies

If we have chosen the same sign in the two tubes, the pieces $$M_s,M_b$$ match, so we just have to move the cutting curves in the common part until they match.

# Plumbing of two handle bodies

So we have take away the cylinders of the tori being glued, and glue the cutting curves accordingly.

# Plumbing of two handle bodies

One of them are just glued, the other are glued in one end, and make a turn in the other one.

# Closing loops

Theorem: Let $$M$$ be a handle body of genus $$g$$. $$c_1,c_2$$ two disjoint curves in $$\partial M$$ that intersect transversally two different cutting curves in only one point each, and touch no other one. Consider $$A_1,A_2$$ annuli in $$\partial M$$ that are regular neighborhoods of them. The result of gluing along an orientation reversing homeomorphism $$A_1\to A_2$$ is again a handle body of genus $$g$$.

# Closing loops

The resulting cutting curves are:

• One obtained by gluing the two preexisting ones
• Another one that is the commutator of a preexisting and the meridian of the gluing cylinder.

# Closing loops

The resulting cutting curves are:

• One obtained by gluing the two preexisting ones
• Another one that is the commutator of a preexisting and the meridian of the gluing cylinder.

# Another examples

The Poincare sphere