We will associate to the data of a Heegaard diagram \((\Sigma_g, \{\alpha_i\}_{i=1}^g, \{\beta_i\}_{i=1}^g)\) a pair of Lagrangian submanifolds in the \(g\)-th symmetric product of \(\Sigma_g\), and compute a version of their Lagrangian intersection Floer cohomology.

1: Symmetric products in symplectic geometry

definition 1.0.1

Let \(X\) be a topological space. The \(k\)-symmetric product is the space \[\Sym^k(X):=X^k/S_k\] where \(S_n\) is the symmetric group on \(k\)-generators which acts by permutation on the coordinates of \(\Sym^k(X)\). In general, the symmetric product of a manifold does not have the structure of a manifold (as the symmetric group does not act freely on the diagonal). However, when \(X\) is a complex curve (real surface) we are able to equip \(\Sym^k(X)\) with the structure of a smooth complex manifold.

theorem 1.0.2

Let \((\Sigma, \jmath)\) be a complex curve. Then \(\Sym^n(\Sigma)\) is a complex manifold, and the map \(\Sigma^k\to \Sym^k(\Sigma)\) is holomorphic. We prove this in the setting where \(\Sigma=\CC\) (this will serve as a local model for the general setting). Consider the space of polynomials of degree \(n\) with leading coefficient 1, \[\CC[z]_n:=\{z^n+a_{n-1}z^{n-1}+\cdots + a_1z+a_0\st a_i\in \CC\}\] There is a map from \(\CC[z]_n\to \Sym^n(\CC)\) which sends each polynomial to its (unordered) set of zeros. This map is a bijection as every set of \(n\) points (with possible repetition) in \(\CC\) uniquely determines a degree \(n\) polynomial with leading coefficient 1. The complex structure comes from identifying \(\CC[z]_n = \CC^n\).

example 1.0.3

We identify the symmetric product \(\Sym^2(\CP^1)\) with \(\CP^2\). To each point in \(\CP^2\) we can associate a degree 2 homogenous polynomial in 2-variables: \[(z_0:z_1:z_2)\mapsto z_0 s^2 + z_1 st + z_2 t^2\] We then factor the polynomial as \[( z_0 s^2 + z_1 st + z_2 t^2)=(y_0 s + y_1 t)\cdot (x_0 s + x_1 t)\] This gives us a bijection between the points of \(\CP^2\) and \(\Sym^2(\CP^1)\). \[(z_0:z_1:z_2)\mapsto [(x_0:x_1), (y_0,y_1)]\] In figure 1.0.4 the moment polytope of \(\CP^1\times \CP^1\) is the square given by \[\text{Convex Hull}((0,0), (0,1), (1, 0), (1, 1)),\] while the moment polytope of \(\CP^2\) is given by the convex hull of \[\text{Convex Hull}((0,0), (0,1), (1,1)).\] There is map from the first moment polytope to the second (given by ``folding'' along the diagonal) which is 2-to-1 away from the diagonal, allowing us to see the symmetric product on the level of moment polytopes. Along the diagonal, we cannot define a relation between the symplectic form on \(\CP^1\times \CP^1\) and \(\CP^2\).

figure 1.0.4:\(\CP^2\) can be identified with the 2-fold symmetric product of \(\CP^1\). Away from the diagonal, we have symplectic maps.

While \(\Sym^k(\Sigma)\) is a complex manifold it does not come with a canonical choice of symplectic structure, so we are leaving some of the tools that we use to study Lagrangian intersection Floer cohomology behind.

2: 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 2.0.1

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 2.0.2: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 2.0.3: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 2.0.2 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 2.0.1 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 2.0.4

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.

Gromov-Compactness: Observe that this is defined over \(\ZZ/2\ZZ\) coefficients. In general, when we look at the strips connecting two points \(x, y\), we are only guaranteed compactness of the moduli space of pseudoholomorphic strips which have bounded energy. To get around this problem --- when strips from \(x\) to \(y\) have possibly unbounded energy --- we would employ Novikov coefficients in Lagrangian intersection Floer cohomology. Here, we do not use such a coefficient ring.
If we treat \(X\) as a symplectic manifold, the Lagrangian submanifolds \(L_{\underline \alpha}, L_{\underline \beta}\) are not tautologically unobstructed (i.e. \(\omega(\pi_2(X, L))\neq 0\)). We therefore must rule out disk and sphere bubbling to show that the differential squares to zero.

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 2.0.5

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

lemma 2.0.6

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.

3: admissibility and convergence

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 3.0.1

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 3.0.2

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 3.0.3

A Heegaard domain (or simply domain) is a formal linear combination of the connected components of \(\Sigma\setminus (\underline \alpha\cup \underline \beta)\).

definition 3.0.4

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 3.0.5

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 3.0.6 [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 3.0.7 [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.
[Per08]	Timothy Perutz. Hamiltonian handleslides for Heegaard Floer homology, 2008.