- First, one shows that every surface can be represented by a polygon with edge identifications,
- Then, one shows that there are several ``moves'' on fundamental polygons that modify the combinatorial data of the fundamental polygon but correspond to the same surface,
1: Heegaard Diagrams:combinatorial descriptions of 3 manifolds
definition 1.0.1
Let \(M\) be a closed 3-manifold. A genus \(g\) Heegaard splitting of \(M\) is a decomposition \[M=U_1\cup_{\Sigma_g} U_2\] where \(U_1, U_2\) are genus \(g\) handlebodies along with an identification of their boundaries (the surface \(\Sigma_g)\) with a diffeomorphism.example 1.0.2
Consider the 3-sphere \[M=S^3=\left\{(x_0, x_1, x_2, x_3)\st x_i\in \RR, \sum_{i=1}^3 x_i^2=1.\right\}\] Consider the decomposition of this into two halves along the \(z_0\) coordinate: \begin{align*} U_1=\{(x_0, x_1, x_2, x_3)\in S^3 \st x_0\leq 0\}&& U_2=\{(x_0, x_1, x_2, x_3)\in S^3 \st x_0\geq 0\} \end{align*} Then both \(U_1, U_2\) are diffeomorphic to the 3-ball, and are glued together by their common boundary \(S^2=\Sigma_0\). See figure 1.0.3.proposition 1.0.4
Let \(M\) be a closed 3-manifold. There exists a Heegaard splitting for \(M\).lemma 1.0.5
Let \(U\) be a 3-manifold with boundary. \(U\) is a handlebody if and only if there exists a Morse function \(f: U\to \RR\) whose- Gradient \(\nabla f\) points outward along the boundary of \(U\);
- \(f\) has only critical points of index \(0\) and \(1\).
definition 1.0.6
A Heegaard diagram is a triple \((\Sigma_g, \{\alpha_i\}_{i=1}^g, \{\beta_i\}_{i=1}^g\)). Here, \(\Sigma_g\) is a genus \(g\) surface and \(\{\alpha_i\}_{i=1}^g, \{\beta_i\}_{i=1}^g\) are collections of smooth 1-cycles in \(\Sigma_g\) such that- \(\Sigma_g\setminus \{\alpha_i\}_{i=1}^g, \Sigma_g\setminus \{\beta_i\}_{i=1}^g\) are connected and;
- \(\alpha_i\cap \alpha_j=\emptyset = \beta_i \cap \beta_j\) whenever \(i\neq j\).
example 1.0.7
Consider the 3-sphere \[S^3=\{(z_1, z_2)\st z_i\in \CC, |z_1|^2+|z_2|^2=1\}.\] We will first describe a new Heegaard diagram for \(S^3\). Take the function \(f=|z_1|^2-|z_2|^2\), and consider the sets \begin{align*} U_1=f^{-1}([0, 1]) && U_2=f^{-1}([-1, 0]) \end{align*} These sets are fillings of the boundary \[\Sigma_1:=f^{-1}(0)=\left\{(z_1, z_2)\st |z_1|^2=\frac{1}{2}, |z_2|^2=\frac{1}{2}\right\}=\{(e^{i\theta_1}, e^{i\theta_2}), \theta_1, \theta_2\in S^1\}.\] which is a torus. Observe that \(\grad f\) is transverse to the boundary of \(\Sigma_1\), and that the critical locus of \(f\) can be parameterized by the cycles \(\{(e^{i\theta_1}, 0)\}\sqcup \{(0, e^{i\theta_2})\}\). It follows the sets \(U_1, U_2\) are diffeomorphic to \(S^1\times D^2\) and \(D^2\times S^1\) respectively. These are handlebodies, giving us a Heegaard decomposition. We now Morsify \(f\) by taking a perturbation. Take \(\rho:[-1, 1]\to [0, \eps]\) satisfying the constraints: \begin{align*} \rho|_{[-1, -.5]}=\eps/10 && \rho|_{[0, 1]}=0 && |\rho'|<\eps \end{align*} The the function \(f+ \rho(f)\cos(\theta_1)+\rho(-f)\cos(\theta_2)\) has 4 critical points at \((\pm 1 , 0)\) and \((0, \pm 1)\). The attaching disks associated to the index 2 and index 1 critical points give the cycle \(\alpha_1=S^1\ times \{1\}\) and \(\beta_1=\{1\}\times S^1\) inside \(T^2\). See figure 1.0.8.definition 1.0.9
Let \((\Sigma_g, \{\alpha_i\}_{i=1}^g, \{\beta_i\}_{i=1}^g)\) be a Heegaard diagram. Let \(p\in \Sigma_g\) be a point avoiding the cycles \(\alpha_i, \beta_i\). The stabilization of \(\Sigma_g\) at \(p\) is the diagram \((\Sigma_g \#_p T^2, \{\alpha_i\}_{i=1}^{g+1}, \{\beta_i\}_{i=1}^{g+1})\) where \(\alpha_{g+1}, \beta_{g+1}\) are the meridional and longitudinal classes of \(T^2\).definition 1.0.11
Let \((\Sigma_g,\{\alpha_i\}_{i=1}^g, \{\beta_i\}_{i=1}^g)\) be a Heegaard diagram. We say that another diagram \((\Sigma_g,\{\alpha_i'\}_{i=1}^{g}, \{\beta_i'\}_{i=1}^{g})\) is related to \((\Sigma_g,\{\alpha_i\}_{i=1}^g, \{\beta_i\}_{i=1}^g)\) by an- isotopy if the sets \(\{\alpha_i\}_{i=1}^g, \{\alpha_i'\}^g\) are isotopic in \(\Sigma_g\), or \(\{\beta_i\}_{i=1}^g, \{\beta_i'\}^g\) are isotopic in \(\Sigma_g\).
- handle slide if \(\alpha_i=\alpha_{i'}\) for \(i\neq g\), and the curves \(\alpha_{g-1}, \alpha_g, \alpha_g'\) bound a pair of pants in \(\Sigma_g\) disjoint from \(\{\alpha_i\}_{i=1}^{g-2}\) (or similarly for the \(\beta_i\)). See figure 1.0.12
theorem 1.0.13 [Sin33]
Suppose that \((\Sigma_g,\underline \alpha, \underline \beta)\) and \((\Sigma_{g'},\underline \alpha', \underline \beta')\) are Heegaard diagrams for \(M\). There exist a sequence of Heegaard moves and stabilizations taking one to the other.2: Construction of Heegaard Floer cohomology
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.2.1: Symmetric products in symplectic geometry
definition 2.1.1
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 \(\Sym^k(X)\).theorem 2.1.2
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.example 2.1.3
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 2.1.4 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 \(\CP^2\).2.2: 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 2.2.1
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. If the diagram is chosen so that \(\alpha, \beta\) are disjoint, then the Lagrangian intersection Floer cohomology \(\HF(\alpha, \beta)\) vanishes. 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 2.2.2 will be exact unless the areas of the two strips agree --- that is, the Lagrangians \(\alpha, \beta\) are Hamiltonian isotopic.definition 2.2.4
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 \(\underline \alpha, \underline \beta\).- 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.
lemma 2.2.5
Suppose that \(g>2\). Then \(\pi_2(\Sym^g(\Sigma_g))=\ZZ\). The generator of this group intersects \(Y_z\).lemma 2.2.6
Suppose that \(u: D^2\to L_\alpha\) is a holomorphic disk. Then the image of \(u\) intersects \(Y_z\).2.3: 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 2.3.1
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 2.3.2
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 2.3.3
A Heegaard domain (or simply domain) is a formal linear combination of the connected components of \(\Sigma\setminus (\underline \alpha\cup \underline \beta)\).definition 2.3.4
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 2.3.5
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 2.3.6 [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 2.3.7 [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 \(\mu(\phi)-j, n_z(\phi)=k, \mathcal D(\phi)\geq 0\).3: invariance of Heegaard-Floer cohomology
theorem 3.0.1
Suppose that \((\Sigma_g,\underline \alpha, \underline \beta,z)\) and \((\Sigma_{g'},\underline \alpha', \underline \beta',z')\) are pointed admissible Heegaard diagrams for \(M\). Then \[\HHeF(\Sigma, \underline{\alpha}, \underline{\beta},z)\cong\HHeF(\Sigma, \underline{\alpha}, \underline{\beta},z).\]- Show that a similar theorem to theorem 1.0.13 holds for pointed Heegaard diagrams.
- Invariance of \(\HHeF(\Sigma_g, \underline \alpha, \underline \beta, z)\) under isotopies of almost complex structure.
- Invariance of \(\HHeF(\Sigma_g,\underline \alpha, \underline \beta,z)\) under isotopies which do not create/destroy critical points. This can be reduced to the previous step by pulling back the almost complex structure along the isotopy.
- Invariance of \(\HHeF(\Sigma_g,\underline \alpha, \underline \beta,z)\) under isotopies which do not create/destroy critical points: these isotopies can always be taken to be Hamiltonian isotopies of the Lagrangian \(L_{\underline \alpha}\).
- Invariance of \(\HHeF(\Sigma_g,\underline \alpha, \underline \beta,z)\) under stabilizations: see exercise 4.0.8.
- Invariance of \(\HHeF(\Sigma_g,\underline \alpha, \underline \beta,z)\): by far, the trickiest portion of the proof. This requires looking at the product structure on Lagrangian intersection Floer cohomology.
4: Exercises
These exercises were curated by Fraser Binns.exercise 4.0.1
What is \(\Sym^2(S^1)\)?exercise 4.0.2
Let \((\Sigma, \{\alpha_i\}_{i=1}^g, \{\beta_i\}_{i=1}^g)\) be a Heegaard diagram for \(M\). Let \(A\) be the \(g\times g\) matrix whose entries over \(\ZZ/2\ZZ\) are given by \[a_{ij}=\alpha_{i}\cap \beta_j \text{\;\;mod \;\;} 2.\] Given the matrix \(A\), compute \(H^\bullet(M, \ZZ/2\ZZ)\).exercise 4.0.3
- Let \(\Sigma\) be a surface. Show that \(H_1(\Sym^g(\Sigma))\cong H_1(\Sigma)\).
- Show that \(H_1(\Sym^g(\Sigma))/(H_1(T_\alpha)\oplus H_1(T_\beta)\cong H_1(\Sigma))/\langle\{\alpha_i,\beta_j\}_{i,j}\rangle\)
- What is the homology class of the boundary of a psuedoholomorphic disk?
exercise 4.0.4
A Lens space \(L(p,q)\) can be defined as the \(3\)-manifold with a Heegaard splitting of genus 1, with an \(\alpha\)-curve given by a longitude of the the torus, and the \(\beta\)-curve in the class \((p, q)\). Use the above obstruction to compute \(\widehat{HHF}^\bullet(L(p,q))\).exercise 4.0.6
Compute the Heegaard Floer homology of the manifold with Heegaard splitting as shown, from the diagram. What is the manifold?exercise 4.0.7
- Let \(Y\) be a \(3\)-manifold with Heegaard splitting \((\Sigma,\mathbf{\alpha},\mathbf{\beta})\). Pick orientations on \(\Sigma,\alpha,\beta\). Explain how this induces a \(\ZZ/2\) grading on \(\widehat{HHF}^\bullet(Y)\).
- Let \(Y\) be a rational homology sphere. Show that \[\chi(\widehat{HHF}^\bullet(Y)):=\operatorname{rank}(\widehat{HHF}^\bullet_0(Y))-\operatorname{rank}(\widehat{HHF}^\bullet_1(Y))=|H_1(Y;\ZZ)|.\]
- A rational homology sphere (i.e. a 3 manifold with \(H_*(Y)\cong H_*(S^3)\)) is called an L-space, if \(\operatorname{rank}(\widehat{HHF}^\bullet(Y))\) is minimal in the sense that \(\operatorname{rank}(\widehat{HHF}^\bullet(Y))=|H_1(Y)|\). Explain why Lens spaces are \(L\)-spaces
exercise 4.0.8
- Consider the ``decatigorification" of \(\widehat{HHF}^\bullet\) given by the Euler characteristic as described in the previous question. Show that it is invariant under Heegaard moves. How might the proof of invariance of \(\widehat{HHF}^\bullet\) differ?
- Suppose \(Y_1,Y_2\) are \(3\)-manifolds. How are \(\widehat{HHF}^\bullet(Y_1\# Y_2),\widehat{HHF}^\bullet(Y_1),\) and \(\widehat{HHF}^\bullet(Y_2)\) related?
- Show that \(\widehat{HHF}^\bullet\) is invariant under stabilization and Hamiltonian isotopy in the sense that the resulting groups are isomorphic.
References
[OS04a] | Peter Ozsváth and Zoltán Szabó. Holomorphic disks and three-manifold invariants: properties and applications. Annals of Mathematics, pages 1159--1245, 2004. |
[OS04b] | Peter Ozsváth and Zoltán Szabó. An introduction to Heegaard Floer homology. Floer homology, gauge theory, and low-dimensional topology, 5:3--27, 2004. |
[Per08] | Timothy Perutz. Hamiltonian handleslides for Heegaard Floer homology, 2008. |
[Sin33] | James Singer. Three-dimensional manifolds and their Heegaard diagrams. Transactions of the American Mathematical Society, 35(1):88--111, 1933. |