Let $M^3=H\bigcup_F H$ be a Heegaard diagram of an oriented 3-dimensional homology 3-sphere $M^3$, where $H$'s are handlebodies (of genus $g$) and $F=\partial H$ is the common boundary surface. The meridians of both sides become the $\alpha$-curves ($\alpha_1,\alpha_2,\dots,\alpha_g$) and the $\beta$-curves ($\beta_1,\beta_2,\dots,\beta_g$) on $F$. Assume that they intersect transversely. Fix a complex structure of $F$ and it induces a symplectic structure on the $g$-fold symmetric product ${\rm Sym}^g(F)$. The tori $T_\alpha =\alpha_1\times\alpha_2\times\cdots\times\alpha_g$ and $T_\beta=\beta_1\times\beta_2\times\cdots\times\beta_g$ are Lagrangians of ${\rm Sym}^g(T)$. Peter Ozsv\'ath and Zolt\'an Szaob\'o have considered the Lagrangian intersection homology theory \'a la Floer. It turns out that the chain complexes may depend on various choices but the homology groups are the same up to isomorphisms. These are the O-Z invariants of $M^3$. For the standard $S^3$, it is isomorphic to ${\bf Z}$. (We say that O-Z $=1$.)

In a very preliminary joint work with T.J. Li, we try to describe O-Z directly from the $\alpha$-curves and $\beta$-curves on $F$. Particularly, we are experimenting the the following question:

{\bf Question:} How do we describe the $\alpha$-curves and $\beta$-curves when O-Z $=1$ (and $g=2$)?

We also speculate the relative O-Z invariants. We hope that the relative invariants may be used to study the general case (i.e. $g>2$) for O-Z $=1$.