# Thursday, March 25

:::{.remark}
Recall that we have several variants: namely \( \hat{\HF}, \HF^-, \HF^+, \HF^{\infty }\).
Let $M\in \Mfd^3$ and take a Heegard diagram \( (\Sigma, \alpha, \beta, z) \).
Note that $\CF^-( \Sigma, \alpha, \beta, z)$ is the free $\ZZ/2[u]\dash$module generated by $\TT_{ \alpha} \intersect \TT_{\beta}$ with differential given by ?.

\todo[inline]{Missing some stuff.}

:::

:::{.definition title="Nice Diagrams"}
A Heegaard diagram is called **nice** if every connected component of \( \Sigma\sm \alpha\union \beta \) that does not contain $z$ is either a bigon or a rectangle.
:::

:::{.remark}
For nice Heegaard diagrams, the Maslov index 1 holomorphic discs with $n_z( \varphi) = 0$ are embedded bigons and rectangles.
:::

:::{.lemma title="?"}
Any 3-manifold has a nice Heegaard diagram, so computing \( \hat{\HF} \) is combinatorial.
:::

:::{.remark}
Some properties:

1. $\Spinc$ structures: we have a decomposition 
  \[
  \CF^-(M) = \bigoplus _{s\in \Spinc(M) } \CF^-(M, s)
  \] 
  which induces 
  \[
  \HF^\star(M) = \bigoplus_{s\in \Spinc(M)} \HF^\star(M, s)
  \]
  where $\star = +, -, \infty$.

2. Maslov grading: For a $\QHS^3$, \( \HF^-(M) \) is relatively $\ZZ\dash$graded.
  The degree of $u$ is -2, and this grading can be lifted to an absolute $\QQ\dash$grading.

3. There is a SES
\[
0 \to \CF^-(M, s) \mapsvia{\cdot u} \CF^-(M, s) \to \hat{\CF}(M, s) \to 0
.\]

  This yields an exact triangle

\begin{tikzcd}
	{\HF^-(M, s)} && {\HF^-(M, s)} \\
	\\
	& {\hat{\HF}(M, s)}
	\arrow["{\cdot u}", from=1-1, to=1-3]
	\arrow[from=1-3, to=3-2]
	\arrow[from=3-2, to=1-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXEhGXi0oTSwgcykiXSxbMiwwLCJcXEhGXi0oTSwgcykiXSxbMSwyLCJcXGhhdHtcXEhGfShNLCBzKSJdLFswLDEsIlxcY2RvdCB1Il0sWzEsMl0sWzIsMF1d)

4. There is a short exact sequence

\[
0 \to \CF^-(M) \to \CF^{\infty }(M) \to \CF^+(M) \to 0
.\]

yielding an exact triangle

\begin{tikzcd}
	{\HF^-(M, s)} && {\HF^\infty(M, s)} \\
	\\
	& {{\HF^+}(M, s)}
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=3-2]
	\arrow[from=3-2, to=1-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXEhGXi0oTSwgcykiXSxbMiwwLCJcXEhGXlxcaW5mdHkoTSwgcykiXSxbMSwyLCJ7XFxIRl4rfShNLCBzKSJdLFswLDFdLFsxLDJdLFsyLDBdXQ==)

5. $\ZZ/2[u]$ is a PID, so by the structure theorem, any module over it will decompose and we have
\[
\HF^-(M, s) = \bigoplus_{i} \ZZ/2[u] \oplus \bigoplus_j {\ZZ/2[u] \over \gens{u^{n_j}} }
.\]

Supposing that $M \in \QHS^3$, then by Osvath-Szabo, for any $s\in \Spinc(M)$ there is exactly one free summand.
Let $d$ be the Maslov grading of the free generator, and $c_j$ be the grading of the torsion part.
We write the $u\dash$torsion part as $\HF_\red(M, s)$.
:::

:::{.definition title="$d\dash$invariant"}
The Maslov grading of the free summand $d = d(M, s)$ is referred to as the **$d\dash$invariant** or correction term, and
\[
d(M, s) = \max \ts{ \gr( \alpha) \st \alpha\in \HF^-(M, s),\, u^n \alpha \neq 0 \forall n  }
.\]

:::

:::{.definition title="Rational Homology Cobordism Group"}
The **rational homology cobordism group** is denoted
\[
\qty{\Theta_\QQ^3 \da \ts{ M \in \QHS^3 } /\sim, \connectsum}
\]
where \( M_1\sim M_2 \) if and only if they are *rationally homology cobordant*, i.e. 

1. There exists an $W \in \Mfd^4$ (connected, oriented) such that $\bd W = -M_1 \disjoint M_2$, i.e. $W$ is a cobordism from $M_1$ to $M_2$.

2. $H_i(W; \QQ) = 0$ for $i=1, 2$, so $W$ is a rational homology cylinder.

:::

:::{.remark}
Note that this is only a monoid without the equivalence relation, but this equivalence creates inverses.
:::

:::{.definition title="?"}
Define the **$d\dash$invariant** of $M$ as 
\[
d(M) = \sum_{s\in \Spinc(M) } d(M, s)
.\]
:::

:::{.remark}
This induces a homomorphism 
\[
d: \Theta_\QQ^3 \to \QQ
.\]
:::