# Thursday, March 18

## $\Spinc$ Structures and Invariance

:::{.remark}
Recall that given a Heegard diagram \( ( \Sigma, \alpha, \beta, z ) \) gives an equivalence relation
\[
x \sim y \iff \eps(x, y) = 0 \in H_1(M) \equalsbecause{\PD} H^2(M)
.\]
This yields a decomposition of $\hat{\CF}$ into a direct sum over equivalence classes of subcomplexes defined by $\Spinc$ structures.
Note that the differential will preserve each direct summand.
We defined $\Spinc(M)$ as the set of nowhere vanishing vector fields on $M$ modulo being homotopic outside finitely many 3-balls in $M$.
We had a map 
\[
\TT_{ \alpha} \intersect \TT_{\beta} \mapsvia{s_z} \Spinc(M)
,\]
recalling that the left-hand side are the generators of $\hat{\CF}$.
We took a self-indexing Morse function on $M$, took the inverse image of $3/2$ to get the Heegard surface, and each intersection point $x_i$ gave a flow line from an index 2 critical point to an index 1 critical point passing through $x_i$:

![Trajectories of negative gradient flow](figures/image_2021-03-18-11-19-00.png)

We proceeded by cancelling adjacent flow lines (at the level of vector fields), and then modifying $\gamma_z$ (the flow line passing through the basepoint $z$ connecting the index 0 to the index 3) to get a nowhere vanishing vector field.
We then took a trivialization $\tau: TM \to M \cross\RR^3$ defined a map
\[
\Spinc(M) &\mapsvia{\gamma^ \tau} H^2(M) \\
s = [v] &\mapsto f_v^*( \alpha)
.\]
where \( \alpha \) is the volume form of $S^2$ and 
\[
f_v: M &\to S^2 \\
x &\mapsto \hat{v_x} \da { v_x \over \norm{v_x} }
.\]
Note that \( \delta^\tau \) a priori depends on \( \tau \), but 
\[
\delta(s_1, s_2) = \delta^{ \tau}(s_1) - \delta^{ \tau}(s_2) \in H^2(M)
,\]
and the difference is independent of \( \tau \).
:::

:::{.lemma title="?"}
For $x, y\in \TT_{ \alpha} \intersect \TT_{ \beta}$, defining $s_1 - s_2 = \delta(s_1, s_2) \in H^2(M)$, we have
\[
s_z(y) - s_z(x) = \PD[\eps(x, y) ]
.\]

As corollaries,

1. If $x\sim y$ then $s_z(y) = s_z(x)$, and
2. If $x\not\sim y$ then the above equation holds.

:::

:::{.exercise title="?"}
Prove this!

*Hint, take the Poincaré dual of the link below to get the formula:*
\[
s_z(y) - s_z(x) = \PD[ \gamma_y \union (- \gamma_x)]
.\]
*This implies that the two vector fields are equal everywhere outside of a tubular neighborhood of the link.
Then show that \( [ \gamma_x \union (-\gamma_x) = [ \eps(x, y) ] \).*
:::

:::{.remark}
We thus have
\[
\hat{\CF}( \Sigma, \alpha, \beta, z)
= \bigoplus _{\mfs \in \Spinc(M)} 
\hat{\CF}( \Sigma, \alpha, \beta, z, \mfs)
.\]
:::

:::{.remark}
We have several properties of $\Spinc$ structures.
There is a map
\[
J: \Spinc(M) &\to \Spinc(M) \\
s = [v] &\mapsto \conj{s} \da [-v]
.\]
There is also a first Chern class
\[
c_1: \Spinc(M) &\to H^2(M) \\
s &\mapsto s - \conj{s}
,\]
i.e. $c_1(s) = \delta(s, \conj{s})$.
:::

:::{.theorem title="Topological Invariance"}
The association
\[
( \Sigma, \alpha, \beta, z), J \leadsto 
\hat{\HF}
( \Sigma, \alpha, \beta, z)
\]
does not depend on the choice of Heegard diagram or the almost complex structure $J$, so this yields a well-defined invariant of $M$ which we'll denote \( \hat{\HF}(M) \) for \( M\in \Mfd^3(\RR) \).
:::

:::{.remark}
There are few things to discuss:

1. The almost complex structure $J$:

2. Isotopies

3. Handle slides

4. Stabilization

Remarks on these:

1. This involves a standard argument from Lagrangian Floer homology.
2. There are two cases:
    - If the isotopy doesn't create a new intersection, we have a 1-to-1 correspondence between generators for any two choices, and changing $J$ to $J'$ will give a correspondence between the differentials.
    This just involves picking a diffeomorphism that maps \( \alpha \) circles to \( \alpha' \) circles, and so on.
    So this reduces to showing 1.
    - If is *does* create new intersection points, there are again standard arguments in Lagrangian Floer homology for this.
3. This involves the following situation, which induces a map

  ![image_2021-03-18-11-51-38](figures/image_2021-03-18-11-51-38.png)
  
  For an appropriate choice of $J$ on \( \Sigma \connectsum T^2 \), the map $f$ above will induce a chain homotopy equivalence
\[
\tilde f: 
\hat{\CF}
(\Sigma, \alpha, \beta, z)
\mapsvia{\sim} 
\hat{\CF}
(\Sigma \connectsum T^2, \alpha', \beta', z)
.\]

4. What's the picture? 

![image_2021-03-18-11-56-59](figures/image_2021-03-18-11-56-59.png)

  This will yield a map
  \[
  (\Sigma, \alpha, {\color{blue} \beta}, z)
  \leadsto
  (\Sigma, \alpha, {\color{green} \gamma}, z)
  .\]
  For \( i = 1, \cdots, g-1 \), we'll have \( \gamma_i \) isotopic to \( \beta_i \), and for \( i=g \), \( \gamma_g \) is obtained by sliding \( \beta_g \) over \( \beta_{g-1} \).
  We'll combine these into the same diagram with different colors to compare them, yielding a **Heegard triple**:
  \[
  (\Sigma, {\color{red} \alpha}, {\color{blue} \beta}, { \color{green} \gamma}, z)
  .\]
  We can think of this as three separate diagrams:
  \[
  (\Sigma, {\color{red} \alpha}, {\color{blue} \beta}, z)
  &\leadsto M\\
  (\Sigma, {\color{blue} \beta}, { \color{green} \gamma}, z)
  &\leadsto ? \\
  (\Sigma, {\color{red} \alpha}, { \color{green} \gamma}, z)
  &\leadsto M
  .\]
  What does the middle one represent?

  ![Heegard diagram](figures/image_2021-03-18-12-02-39.png)

  Here this is a diagram for \( (S^1 \cross S^2)^{\connectsum 2} \).

  Note that we draw \( \gamma_i \) such that it intersects \( \beta_i \) in two transverse intersection points to make sure the diagram is admissible.
:::

:::{.remark}
Give a Heegard triple
\( ( \Sigma, \alpha, \beta, \gamma, z \), pick three intersection points
\[
x &\in \TT_{ \alpha} \intersect \TT_{ \beta} \\
y &\in \TT_{ \beta} \intersect \TT_{ \gamma} \\
w &\in \TT_{ \gamma} \intersect \TT_{ \alpha} 
.\]
We can use **Whitney triangles** to connect $x,y,w$:

![image_2021-03-18-12-23-07](figures/image_2021-03-18-12-23-07.png)

We then define \( \pi_2(x,y,z) \) to be the homotopy class of Whitney triangles connecting $x,y,w$.
We can similarly define \( \mcm( \psi )  \) to be the moduli space of $J\dash$holomorphic representatives of \( \psi\in \pi_2(x,y,w) \), along with a chain map
\[
f_{\alpha \beta \gamma}: 
\hat{\CF}(\Sigma, \alpha, \beta, z)
\tensor
\hat{\CF}(\Sigma, \beta, \gamma z)
&\to
\hat{\CF}(\Sigma, \alpha, \gamma, z) \\
x\tensor y 
&\mapsto 
\sum_{ w \in \TT_{ \alpha} \intersect\TT_{ \beta}} 
\sum_{\substack{ \psi\in \pi_2(x,y,w) \\ \mu( \psi) = 0 \\ n_z( \psi) = 0 }}
\# \mcm( \psi) \cdot w
.\]
:::

:::{.theorem title="?"}
\( f_{ \alpha \beta \gamma} \) is a chain map.
:::

:::{.remark}
Next time: we'll show how to get a chain homotopy equivalence from the first tensor term above to the codomain.
We'll also see surgery exact triangles.
:::