# Tuesday, February 23

## Whitney Discs 

:::{.remark}
For $x,y \in \TT_{ \alpha} \intersect \TT_{ \beta}$, recall that we had the following situation:

![Whiteney Disc](figures/image_2021-02-23-11-14-22.png)

Then $\pi_2(x, y)$ was defined to be the homotopy classes of discs connecting $x$ to $y$.
The obstruction to the existence of such discs was denoted $\eps(x, y) \in H_1(M)$ for $M\in \Mfd^3$.
We're checking if there exist two paths connecting $x$ to $y$,
\[
a: I &\to \TT_{ \alpha} \\
b: I &\to \TT_{ \beta}
\]
such that $a-b$ is nullhomotopic.
In this case, $\pi_2(x, y) \neq \emptyset$.

![image_2021-02-23-11-17-30](figures/image_2021-02-23-11-17-30.png)

We had a theorem that $\pi_1(\Sym^g \Sigma) \cong H_1( \Sym^g \Sigma)$, so we can replace nullhomotopic with nullhomologous above.
We can also use the fact that \( H_1( \Sym^g \Sigma) \cong H_1 \Sigma \).
Note that $[a-b]$ isn't well-defined, since we can append any loop to $a$ for example, but the following is well-defined:
\[
\eps(x, y) 
\da [a-b] \in { H_1 \Sym^g \Sigma \over H_1 \TT_{ \alpha} \oplus H_1 \TT_{ \beta} }
\cong {H_1 \Sigma \over \gens{ [ \alpha_1], \cdots, [ \beta_1 ], \cdots }} 
\cong H_1 M 
.\]

How can we compute $\eps$ using the Heegard diagrams?
Recall that a path in $\Sym^g \Sigma$ was a union of $g$ paths in \( \Sigma \).
So choose arcs \( a_1 \union \cdots \union a_g \) on \( \Sigma \) such that \( a_i \subseteq \alpha_i \) is sub-arc and \( \bd( a_1 \union \cdots \union a_g ) = y_1 + \cdots + y_g - x_1 - \cdots - x_g \), and similarly choose \( b_1 \union \cdots \union b_g \).
Note that if $\eps(x, y) \neq 0$ then $\pi_2(x, y) = \emptyset$.


:::

:::{.example title="$L(2, 3)$"}
The following is a Heegard diagram for $L(2, 3)$ of minimal genus, where we take \( \alpha \) to be the horizontal line and \( \beta \) will be a line of slope $2/3$:

![image_2021-02-23-11-28-10](figures/image_2021-02-23-11-28-10.png)

Then \( \TT_{ \alpha} \intersect \TT_{ \beta} = \ts{ A, B } \).
Now draw arcs connecting $A$ and $B$, e.g. the ones in orange and green here:

![image_2021-02-23-11-29-54](figures/image_2021-02-23-11-29-54.png)

Note that we have two generators of homology for the torus, say $x,y$, and we can write

![image_2021-02-23-11-30-37](figures/image_2021-02-23-11-30-37.png)

Then the union of the two arcs is exactly $x+y$, so we can write 
\[
H_1( L(2, 3)) 
= { \ZZ \gens{ x, y } \over \gens{ y, 2x + 3y }}  
.\]
Moreover, $\eps(A, B) = x + y \neq 0$ in this quotient, so there is not Whitney disc connecting $A$ to $B$ and $\pi_2(A, B) = \emptyset$.
:::

:::{.remark}
We'll define $x\sim y \iff \eps(x, y) = 0$, and this turns out to be an equivalence relation which partitions the set of paths.

- $\eps(x, y) = 0 \implies \eps(y, x) = 0$, which follows from \( \eps(x, y) = [a-b] = [b-a] = \eps(y, x) \)

- $\eps(x, x) = 0$ by picking $a,b$ constant.

:::

:::{.exercise title="?"}
Show that \( \eps(x, y) + \eps(y, z) = \eps(x, z) \).
:::

:::{.corollary title="?"}
If $x\sim y$ and $y\sim z$, so $\eps(x, y) = \eps(y, z) = 0$, we have $\eps(x, z) = 0 \implies x \sim z$.
:::

:::{.exercise title="?"}
Find the equivalence classes under $\sim$ for the Poincaré homology sphere using the genus 2 Heegard diagram.
:::

:::{.remark}
For \( \varphi\in \pi_2(x, y) \), the **shadow** is the 2-chain \( D( \varphi ) \) on \( \Sigma \) defined in the following way: remove the \( \alpha, \beta \) arcs to obtain
\[
\Sigma \sm (\alpha\union \sigma) = \Disjoint_{i=1}^m D_i
,\]

where ${}^{o}$ denotes that the set is open.
Then \( D( \varphi) = \sum_{i=1}^m a_i D_i \).
:::

:::{.definition title="?"}
Given $z\in \Sigma\sm (\alpha\union \beta)$, define a hyperplane 
\[
L_z = \ts{ \vector{w} \in \Sym^g( \Sigma) \st z\in \vector{w} } 
.\]
Note that this will be codimension 2.
Then for a disc \( \varphi\in \pi_2(x, y) \), define
\[
n_z( \varphi ) = \# \qty{ \im( \varphi) \intersect L_z }
.\]
which is an algebraic (signed) count of how many entries in a tuple contain the point $z$.
We can then define $a_i \da n_{z_i}( \varphi)$ and define
\[
D( \varphi) = \sum_{i=1}^m a_i D_i, && z_i \in {}^{o} D_i
.\]

:::

:::{.remark}
The following comes from "Introduction to Heegard Floer Homology" (Osvath-Szabo), which we've been following relatively closely so far.
:::

:::{.exercise title="?"}
Let $D$ be a domain of a disc connecting \( \ts{ x_1, x_2 } \)  to \( \ts{ y_1, y_2 } \) in the following way:

![image_2021-02-23-11-54-09](figures/image_2021-02-23-11-54-09.png)

Attach 1-handles in the following way to obtain \( \beta \) curves:

![image_2021-02-23-11-55-13](figures/image_2021-02-23-11-55-13.png)

Use these handles to add curves running through the handles:

![image_2021-02-23-11-56-26](figures/image_2021-02-23-11-56-26.png)


:::{.exercise title="?"}
What is a Heegard diagram for?
:::

Pick a point in the center of the rectangle and connect it to the 4 vertices, noting that it includes in \( \Sigma \) :

![image_2021-02-23-11-58-15](figures/image_2021-02-23-11-58-15.png)

Applying a rotation by $\pi$ and taking the quotient, we get a 2-fold branched cover of $S^1$:

![image_2021-02-23-11-59-29](figures/image_2021-02-23-11-59-29.png)

Here $x_1, x_2 \mapsto -i$ and $y_1, y_2 \mapsto +i$.
We can now get a map \( \varphi \) to \( \Sym^2( \Sigma) \):

![image_2021-02-23-12-01-02](figures/image_2021-02-23-12-01-02.png)

In the image we get 2 points with multiplicity on \( \Sigma \), and thus an element of \( \Sym^2 \Sigma \).
We know \( \varphi(-i) = \ts{ x_1, x_2 } \)  and \( \varphi(+i) = \ts{ y_1, y_2 } \).

:::

:::{.example title="?"}
Show that $D'$ is the domain of a disc from \( \ts{ x_1, x_2 } \to \ts{ y_1, y_2 } \):

![image_2021-02-23-12-18-15](figures/image_2021-02-23-12-18-15.png)

We want to make a similar 2-fold cover like in the previous example, so we'll take the two rectangles bounding the arcs, then taking the rotation by $\pi$ yields the cover:

![image_2021-02-23-12-20-27](figures/image_2021-02-23-12-20-27.png)

As before, we get a map to $\Sym^2 \Sigma$:

![image_2021-02-23-12-22-36](figures/image_2021-02-23-12-22-36.png)

As a result, we again get \( \varphi(-i) = \ts{ x_1, x_2 } \)  and \( \varphi(+i) = \ts{ y_1, y_2 } \).

:::

:::{.exercise title="?"}
. Suppose \( x = \ts{ x_1, \cdots, x_g } \)  and \( y = \ts{ y_1, \cdots, y_g } \) such that \( x_i \in \alpha_i \intersect \beta_i \) and \( y_i \in \alpha_i \intersect \beta_{ \sigma\inv(i)} \) for some permutation \( \sigma\in S_g \).
  Then for any \( \varphi\in \pi_2(x, y) \),  show that
  \[
  \bd \qty{\bd D( \varphi) \intersect \alpha_i } = y_i - x_i
  ,\]
  where the inner term is a 1-chain in \( \alpha_i \), and 
  \[
  \bd \qty{ \bd D( \varphi) \intersect \beta_i } = x_i - y_{ \sigma(i) }
  .\]

:::

:::{.remark}
This will characterize the coefficients $a_i$ for which discs exist.
Next time we'll talk about holomorphic discs.
:::