# Lecture 2 (Tuesday, January 19)

\todo[inline]{Copy in references recommended by Akram!}

## Constructing Heegard Floer

:::{.remark}
For Morse Theory, there are some good exercises in Audin's book -- essentially anything other than the existence questions.
The first 8 look good on p. 18.
\

Today: 

1. Overview of the construction of $\HF$, and

2. A discussion of Morse Theory.

First goal: discuss how the name "Heegard" fits in.
:::

:::{.definition title="Genus $g$ handlebody"}
A **genus $g$ handlebody** $H_g$ is a compact oriented 3-manifold with boundary obtained from $\B^3$ by attaching $g$ solid handles (a neighborhood of an arc).

:::

:::{.example title="Attaching $g=2$ handles to a sphere"}
For $g=2$ attached to a sphere, we glue $\DD^2 \cross I$ by its boundary to $S^2$.

![image_2021-01-19-00-35-48](figures/image_2021-01-19-00-35-48.png)

In general, $\bd H_g = \Sigma_g$ is a genus $g$ surface, and $H_g \sm \disjoint_{i=1}^g D_i = \Ball^3$.
We can keep track of the data by specifying $(\Sigma, \alpha_1, \alpha_2, \cdots, \alpha_g)$ where $\bd D_i = \alpha_i$.

![Attaching a handlebody](figures/image_2021-01-19-11-26-35.png)

:::

:::{.definition title="Heegard Decomposition"}
A **Heegard diagram** is $M = H_1 \union_\bd H_2$ where $H_i$ are genus $g$ handlebodies and there is a diffeomorphism $\bd H_1 \to \bd H_2$.
:::

:::{.theorem title="?"}
Every closed 3-manifold has a Heegard decomposition, although it is not unique.
:::

:::{.definition title="Heegard Diagram"}
A **Heegard diagram** is the data 
\[
(\Sigma_g, \alpha = \ts{ \alpha_1, \cdots, \alpha_g}, \beta = \ts{ \beta_1, \cdots, \beta_g})
\]
where the $\alpha$ correspond to $H_1$ and \( \beta \) to $H_2$ and \( \Sigma_g = \bd H_1 = \bd H_2 \).
:::

## Lagrangian Floer Homology

:::{.remark}
This is essentially an infinite-dimensional version of Morse homology.
:::

:::{.definition title="Symplectic Manifold"}
A **symplectic manifold** is a pair $(M^{2n}, \omega)$ such that

- \( \omega \) is *closed*, i.e. \( d \omega = 0 \), and 
- \( \omega \) is *nondegenerate*, i.e. \( \wedge^n \omega \neq 0 \).
:::

:::{.definition title="Lagrangian"}
A **Lagrangian submanifold** is an $L^n \subseteq M$ such that \( \ro{\omega}{L} = 0 \).
:::

:::{.remark}
If $L_1 \intersect L_2$ is finitely many points, case we can define a chain complex 
\[
CF(M^{2n}, L_1, L_2) \da \ZZ_2[L_1 \intersect L_2]
,\]
the $\ZZ_2\dash$vector space generated by the intersection points of the Lagrangian submanifolds.
We'll define a differential by essentially counting discs between intersection points:

![Two intersection points](figures/image_2021-01-19-11-46-38.png)

We'll want to write $\bd x = c_y y + \cdots$ where $c_y$ is some coefficient. 
How do we compute it?
In this case, we have half of the boundary on $L_1$ and half is on $L_2$

![Interior region is the continuous image of a disc](figures/image_2021-01-19-00-40-08.png)

So we can the number of *holomorphic* discs from $x$ to $y$.
We'll get $\del^2 = 0 \iff \im \del \subset \ker \del$, and $\HF$ will be kernels modulo images.
In more detail, we'll have
\[
\bd x = \sum_y \sum_{\mu(\varphi) = 1} \# \hat{\mathcal{M}} (\varphi)y &&  \hat{\mathcal{M}}(\varphi) = \mathcal{M}(\varphi) / \RR  
\]
where \( \hat {\mathcal{M}}  \) will (in good cases) be a 1-dimensional manifold with finitely many points.


:::{.warnings}
$\CF$ does not necessarily have a grading!
:::


Given a 3-manifold $M^3$, we'll associate a Heegard diagram \( \Sigma, \alpha, \beta \).
Note the $g\dash$element symmetric group acts on \( \prod_{i=1}^g \Sigma \) by permuting the $g$ coordinates, so we can define \( \Sym^g(\Sigma) \da \prod_{i=1}^g \Sigma / S_g\).

:::

:::{.theorem title="?"}
The space \( X\da \Sym^g(\Sigma) \) is a smooth complex manifold of $\dim_\RR X = 2g$.
:::

:::{.remark}
Write $\TT _{\alpha} \da \prod_{i=1}^g \alpha_i \subseteq \prod_{i=1}^g \Sigma$ for a $g\dash$dimensional torus; this admits a quotient map to $\Sym^g(\Sigma)$.
We can repeat this to obtain $\TT _{\beta}$.
Then $\HF^\wait(M)$ will be a variation of Lagrangian Floer Homology for \( (\Sym^g(\Sigma), \TT _{\alpha}, \TT _{\beta} ) \).

:::

:::{.example title="?"}
Consider constructing a genus $g=1$ Heegard diagram.
Recall that $S^3$ can be constructed by gluing two solid torii.

![image_2021-01-19-12-20-16](figures/image_2021-01-19-12-20-16.png)

Here $(T, \alpha, \beta)$ will be a Heegard diagram for $S^3$.
:::

:::{.exercise title="?"}
Show that the following diagram with \( \beta \) defined as some perturbation of \( \alpha \) is a Heegard diagram for \( S^1 \cross S^2 \).

![image_2021-01-19-12-21-56](figures/image_2021-01-19-12-21-56.png)

:::

:::{.definition title="Dehn Surgery"}
Consider $M$ a 3-manifold containing a knot $K$, we can construct a new 3-manifold by first removing a neighborhood of $K$ to yield $M\sm N(K)$:

![image_2021-01-19-12-23-16](figures/image_2021-01-19-12-23-16.png)

Taking a new solid torus $S \da \DD^2 \cross S^1$ and a diffeomorphism $i: \bd S \to \bd (M \sm N(K))$, this yields a new manifold $M _{\varphi} (K)$, a **surgery** along $K$.

![image_2021-01-19-12-25-25](figures/image_2021-01-19-12-25-25.png)

:::

:::{.remark}
Note that the diffeomorphism is entirely determined by the image of the curve \( \alpha \) .
The Knot Floer chain complex of $K$ will allow us to compute any flavor $\HF^\wait M _{\varphi} (K)$ of Floer homology.
Why is this important: any closed 3-manifold is surgery on a link in $S^3$.
However there are many more computational tools available here and not in the other theories: combinatorial approaches to compute, exact sequences, bordered Floer homology.

\

Next time: we'll talk about "integer surgeries".
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