proposition 0.0.1
Let \(\underline \alpha = \{\alpha_i\}_{i=1}^g, \underline \beta = \{\beta_i\}_{i=1}^g\) be \(g\)-tuples of disjoint curves in \(\Sigma_g\), so that \(\alpha_i\cap \beta_j\) intersect transversely. Then there is a bijection between \[L_{\underline \alpha}\cap L_{\underline \beta} = \{(x_1, \ldots, x_g)\st x_i\neq x_j, x_i \in \underline \alpha \cap \underline \beta.\}\]theorem 0.0.2
Given a holomorphic strip \(u\in \mathcal M(x, y)\) there exists a \(g\)-branched cover \(\pi: \hat D\to D\) and a holomorphic map from \(\hat u: \hat D \to \Sigma_g\) so that for all \(z\in D\), \[u(z)=([\hat u(z_1), \hat u(z_2), \cdots , \hat u(z_g)])\] where \(\{z_1, \ldots, z_g\}\in \pi^{-1}(z)\).definition 0.0.3
A Heegaard domain (or simply domain) is a formal linear combination of the connected components of \(\Sigma\setminus (\underline \alpha\cup \underline \beta)\).definition 0.0.4
A domain is periodic if its boundary can be written as a sum of the cycles in \(\underline \alpha, \underline \beta\) and it has no intersection with the marked point \(z\).definition 0.0.5
A pointed Heegaard diagram is weakly admissible for a spin-c structure \(s\) if for every non-trivial periodic domain \(\mathcal D\) with \[\langle c_1(s), H(\mathcal D)\rangle = 0\] \(\mathcal D\) has both positive and negative coefficients.lemma 0.0.6 [lemma 4.12 of OS04a]
The following are equivalent:- \((\Sigma_g, \underline \alpha, \underline \beta)\) is admissible for all \(\Spinc\) structures
- There exists a symplectic form on \(\Sigma\) so that every periodic domain as total signed area 0.
lemma 0.0.7 [Lemma 4.14 of OS04a]
Suppose that \((\Sigma, \alpha, \beta, z)\) is a weakly admissible Heegaard diagram. There are only finite many \(\phi\in \pi_2(x, y)\) with \(\mu(\phi)-j, n_z(\phi)=k, \mathcal D(\phi)\geq 0\).References
[OS04a] | Peter Ozsváth and Zoltán Szabó. Holomorphic disks and three-manifold invariants: properties and applications. Annals of Mathematics, pages 1159--1245, 2004. |