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

construction of Heegaard Floer complex

Let \((\Sigma_g, \underline{\alpha}, \underline{\beta})\) be a Heegaard diagram. Let \(X=\Sym^g(\Sigma_g)\). Consider the submanifolds of \(X\) \begin{align*} L_{\underline \alpha}:=\{[(z_1, \ldots, z_g)]\in X\st z_i\in \alpha_i\} && L_{\underline \beta}:=\{[(z_1, \ldots, z_g)]\in X\st z_i\in \beta_i\} \end{align*} Since the \(\alpha_i\) are disjoint from one another, this gives a smooth submanifold whose topology is \(T^g\). Furthermore, since each of the \(\alpha_i\) is a real subspace of \(\Sigma_g\), the submanifold \(L_\alpha\) is a real submanifold of \(\Sigma_g\). A similar statement holds for \(L_\beta\). [Per08] shows that \(\Sym^g(\Sigma_g)\) can be equipped with a symplectic form which makes \(L_{\underline \alpha},L_{\underline \beta}\) Lagrangian submanifolds. We wish to define the Heegaard-Floer cohomology as the Lagrangian intersection Floer cohomology of \(L_{\underline \alpha},L_{\underline \beta}\). However, this is not an invariant of Heegaard diagrams --- note that isotopies of Heegaard diagrams give isotopies of Lagrangian submanifolds, while Lagrangian intersection Floer cohomology is only invariant under Hamiltonian isotopies of Lagrangian submanifolds. We given an example exhibiting some of the problems with this preliminary approach.

example 0.0.1

We look at the example of \(M=S^2\times S^1\). Observe that \(S^2=D^2\cup_{S^1} D^2\), so we can write \(M=D^2\times S^1 \cup_{\Sigma_1} D^2\times S^1\). The Heegaard diagram \((\Sigma_1, \alpha, \beta)\) consists of a torus with two meridional cycles.
figure 0.0.2:A non-admissible Heegaard diagram
If the diagram is chosen so that \(\alpha, \beta\) are disjoint, then the Lagrangian intersection Floer cohomology \(\HF(\alpha, \beta)\) vanishes.
figure 0.0.3:An admissible Heegaard diagram
However, if the diagram is chosen so that \(\alpha, \beta'\) intersect transversely, the Lagrangian intersection Floer cohomology (with \(\ZZ/2\ZZ\) coefficients) is \(\ZZ/2\ZZ\oplus \ZZ/2\ZZ\). Note that \(\beta'\) can be chosen so that it is Hamiltonian isotopic to \(\beta\). The discrepancy between these two answers comes from the non-convergence of the homotopy between the composition of continuation maps \[f\circ g:\CF(\alpha, \beta')\to \CF(\alpha, \beta) \to \CF(\alpha, \beta')\] \[\id: \CF(\alpha, \beta')\to \CF(\alpha, \beta')\] over \(\ZZ/2\ZZ\) coefficients. The presence of an annulus between \(\alpha, \beta\) is the culprit for the non-convergence. One can make the quantities converge by using Novikov coefficients instead of \(\ZZ/2\ZZ\) coefficients. In that setting, the differential in figure 0.0.2 will be exact unless the areas of the two strips agree --- that is, the Lagrangians \(\alpha, \beta\) are Hamiltonian isotopic.
We desire the best of both worlds: invariance under Lagrangian isotopies, and convergence. In example 0.0.1 this can be achieved by only looking at strips which avoid the marked point \(z\). When this marked point is chosen well, we will obtain a criterion (admissible Heegaard diagrams) which precludes the existence of annuli disjoint from the marked point \(z\). In general, the admissibility criterion will limit us to configurations of cycles for which the number of holomorphic strips contributing to the Floer differential is finite. The marked point is introduced by slightly generalizing our definition of a Heegaard diagram.

definition 0.0.4

A pointed Heegaard diagram \((\Sigma_g, \underline \alpha, \underline \beta, z)\) is a Heegaard diagram \((\Sigma_g, \underline \alpha, \underline \beta)\) along with a choice of point \(z\) disjoint from the cycles \(\underline \alpha, \underline \beta\).
As a vector space, the Heegaard Floer cohomology of the pointed Heegaard diagram \((\Sigma,\underline{\alpha}, \underline{\beta},z)\) is \[\HeF(\Sigma, \underline{\alpha}, \underline{\beta},z):=\bigoplus_{x\in L_{\underline \alpha}\cap L_{\underline{\beta}}} \ZZ/2\ZZ\langle x\rangle.\] The differential is defined in a manner similar to the Lagrangian intersection Floer cohomology, but we will need to overcome the following difficulties. The choice of base point defines a subset \(Y_z\subset X\) given by \[ Y_z:=\{[(z, z_2, \ldots, z_g)] \st z_i\in \Sigma_g\} \subset X\] which is disjoint from \(L_{\underline \alpha}\cup L_{\underline \beta}\). Both issues above can be avoided by only looking at strips which are disjoint from \(Y_z\). For \(x, y\in L_{\underline \alpha}\cap L_{\underline \beta}\) let \(\mathcal M(x, y)\) denote the moduli space of holomorphic strips with boundary on \(L_{\underline \alpha}\cup L_{\underline \beta}\), and ends limiting to \(x\) and \(y\). Let \(\mathcal M(x, y)_{z=0}^0\) be the zero-dimensional component of \(\mathcal M(x, y)\) consisting of strips which are disjoint from \(Y_z\). The differential of Heegaard Floer cohomology is defined by the structure coefficients \[\langle d_zx , y \rangle = \#\mathcal M(x, y)_{z=0}^0.\] The following lemmas rule out the presence of disk and sphere bubbling.

lemma 0.0.5

Suppose that \(g>2\). Then \(\pi_2(\Sym^g(\Sigma_g))=\ZZ\). The generator of this group intersects \(Y_z\).

lemma 0.0.6

Suppose that \(u: D^2\to L_\alpha\) is a holomorphic disk. Then the image of \(u\) intersects \(Y_z\).
Provided that we can show that \((d_z)^2=0\), the Heegaard-Floer cohomology is defined to be the homology of the chain complex \[\HHeF(M):= H^\bullet(\HeF(\Sigma, \underline \alpha, \underline \beta ,z), d_z).\] As a result of these two lemmas, our restriction to counting disks which do not pass through \(Y_z\) gives us moduli spaces whose compactifications, if defined, would be free from disk and sphere bubbling. However, it remains to show that we have Gromov compactness.


[Per08]Timothy Perutz. Hamiltonian handleslides for Heegaard Floer homology, 2008.