\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip: admissibility and convergence

admissibility and convergence

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In order to obtain a replacement for the symplectic form providing a bound on the energy of holomorphic strips, we need to work a little bit harder. The key insight is that there is a dictionary between holomorphic strips in the symmetric product and maps from more-complicated domains to the surface.

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 \[L_{\underline \alpha}\cap L_{\underline \beta} = \{(x_1, \ldots, x_g)\st x_i\neq x_j, x_i \in \underline \alpha \cap \underline \beta.\}\] This means that we can understand the intersections between our Lagrangians \(L_{\underline \alpha}, L_{\underline \beta}\) in terms of the intersections between the collection of cycles. Remarkably, we can also understand the holomorphic strips between the intersections points by looking at data on \(\Sigma_g\).

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 \(\{z_1, \ldots, z_g\}\in \pi^{-1}(z)\). With this relation, we can ``by hand'' rule out the kinds of problematic disks which would interfere with the arguments of Gromov-compactness.

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 \(z\). Every periodic domain can be represented by a surface with boundary \(\hat D\to \Sigma\), thus every periodic domain \(\mathcal D\) gives a homology class \(H_2(M; \ZZ)\) by gluing the attaching disks associated to each of the \(\alpha_i, \beta_j\) to the appropriate boundaries in \(\hat D\). We call this homology class \(H(\mathcal D)\).

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. The following lemma may give us some intuition for where the admissibility condition enters into the definition of Heegaard-Floer cohomology.

lemma 0.0.6 [lemma 4.12 of OS04a]

The following are equivalent:

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 \(\mu(\phi)-j, n_z(\phi)=k, \mathcal D(\phi)\geq 0\). This shows that the symplectic energy of the holomorphic strips that we consider in the definition of the differential \(d_z\) is bounded.

References

[OS04a]Peter Ozsváth and Zoltán Szabó. Holomorphic disks and three-manifold invariants: properties and applications. Annals of Mathematics, pages 1159--1245, 2004.