Heegaard Floer homology

Definition

Let $Y$ be a closed oriented 3-manifold. Let $(\Sigma, \alpha, \beta, z)$ be a pointed Heegaard diagram of $Y$ coming from a Heegaard splitting. Here $\Sigma$ is a closed oriented surface of genus $g$. The families $\alpha = (\alpha_1, \ldots, \alpha_g)$ and $\beta = (\beta_1, \ldots, \beta_g)$ consist of disjoint simple closed curves on $\Sigma$. The base point $z \in \Sigma$ lies off all of these curves. Assume the diagram is admissible, and fix a generic path of almost complex structures on the symmetric product $\mathrm{Sym}^g(\Sigma)$.

1. The tori $\mathbb{T}_\alpha = \alpha_1 \times \cdots \times \alpha_g$ and $\mathbb{T}_\beta = \beta_1 \times \cdots \times \beta_g$ are submanifolds of $\mathrm{Sym}^g(\Sigma)$ that meet transversally in finitely many points.
2. The chain complex $\widehat{CF}(\Sigma, \alpha, \beta, z)$ is the vector space over $\F_2$ with basis $\mathbb{T}_\alpha \cap \mathbb{T}_\beta$.
3. Let $x$ and $y$ be basis elements. Consider the holomorphic disks from $x$ to $y$ with Maslov index $1$ whose domain avoids $z$. The coefficient of $y$ in $\partial x$ is their number modulo $2$. Disks that differ by a translation of the parameter are counted once.

The homology $\widehat{HF}(Y)$ of this complex is the hat version of Heegaard Floer homology of $Y$. The versions $HF^-$, $HF^\infty$ and $HF^+$ are defined in the same way over $\F_2[U]$ or $\F_2[U, U^{-1}]$. In them the disks may cover $z$, and the power of $U$ records how many times they do.