\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip: Heegaard Diagrams:combinatorial descriptions of 3 manifolds

Heegaard Diagrams:combinatorial descriptions of 3 manifolds

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definition 0.0.1

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. A handlebody is a 3-manifold \(U\) with boundary with a collection disks \(\{f_i: D^2\to U\}_{i=1}^g\) which are embedded and disjoint, so that \(U\setminus \bigcup_{i=1}^g \Im(f_i)=D^3\).

example 0.0.2

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 0.0.3.
figure 0.0.3:After identifying \(S^3\setminus \{(1, 0, 0, 0)\}\) with \(\RR^3\) via stereographic projection, the Heegaard splitting of \(S^3\) is given by taking the unit sphere, which decomposes the sphere into two \(3\)-balls. We also draw the meridinal line \((\cos(\theta), \sin(\theta), 0, 0)\).

proposition 0.0.4

Let \(M\) be a closed 3-manifold. There exists a Heegaard splitting for \(M\). We first need this useful lemma.

lemma 0.0.5

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 Let \(f: M\to \RR\) be a self indexing Morse function. Consider the level set \(f^{-1}(1.5)\). Because \(1.5\) is not a critical value of \(f\), this is a smooth surface \(\Sigma_g\subset M\). Let \(U_1=f^{-1}([0, 1.5])\) and let \(U_2=f^{-1}([1.5, 3])\). By lemma 0.0.5, these are handlebodies. Seeing that we can decompose each 3-manifold into two handlebodies, we ask what extra do we need to equip \(U_1, U_2\) and \(\Sigma\) with in order to remember \(M\). We again turn to Morse theory. When we construct \(U_1, U_2\) and \(\Sigma\) by a self-indexing Morse function \(f: M\to \RR\), we can use the critical points of \(f\) to determine the fillings \(U_1\) and \(U_2\) of \(\Sigma_g\). The topology of the filling \(U_1\) is recovered by considering how the downward flow spaces of the critical points of \(f|_{U_1}\) attach to the \(\Sigma_g\). We may additionally assume that \(f\) has a unique minimum. The topological information of the filling is recovered by recording how the upward flow spaces \(W^\uparrow(p)\) intersect \(\Sigma_g\) for \(\{p\in \Crit(f), \ind(p)=1\}\). There are \(g\) such critical points which we enumerate as \(\{p_i\}_{i=1}^g\). These upward flow spaces are 2-dimensional, so \(\{\Sigma_g\cap W^\uparrow(p_i)\}_{i=1}^g\) gives \(g\) disjoint cycles in \(\Sigma_g\). We call these cycles \[\{\alpha_i\}_{i=1}^g:=\{ W^\downarrow(p_i)\cap \Sigma_g\}_{p_i\in \Crit(f), \deg(p_i)=1}.\] Similarly, the filling \(U_2\) is recovered by the attaching data for the downward flow spaces of index 2 critical points of \(f\), \[\{\beta\}_{i=1}^g:=\{ W^\downarrow(p_i)\cap \Sigma_g\}_{p_i\in \Crit(f), \ind(p_i)=1}.\]

definition 0.0.6

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 We will write \(\underline \alpha, \underline \beta\) for \(\{\alpha_i\}_{i=1}^g, \{\beta_i\}_{i=1}^g\). These are called the attaching cycles of the Heegaard diagram. The example given in example 0.0.2 is of genus 0, we do not have any attaching cycles for that example.

example 0.0.7

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 0.0.8.
figure 0.0.8:A Heegaard diagram for \(S^3\). The attaching cycles \(\alpha, \beta\) are drawn in red in the torus. The disks in red and blue represent the downward and upward flow spaces of the critical points \(p, q\).
Every Heegaard diagram specifies a 3-manifold, however a single 3-manifold can have many different presentations with different Heegaard diagrams. We have already seen 2 different Heegaard diagrams for \(S^3\) in example 0.0.2 and example 0.0.7. If we are to study 3-manifolds via their Heegaard diagram, we need to understand when two diagrams give presentations of the same manifold. We first describe some operations which modify a Heegaard diagram, but produce the same 3-manifold.

definition 0.0.9

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 \(T^2\). One way to see that stabilization of a Heegaard diagram produces the same manifold comes from Morse theory. Consider \(\Sigma_g\) as the level set of a self-indexing Morse function \(f\). Suppose that we wanted to modify our Morse function to \(\tilde f\) by adding in a pair of critical points \(p, q\) so that \(\ind(p)=1\) and \(\ind(q)=2\). We imagine that the critical points would appear on opposite sides of \(\Sigma_g\), and be connected by a single flow line. Furthermore, \(\tilde \Sigma_{g+1}=\tilde f(1.5)\), the new level set, would be of genus \(g+1\). By applying surgery along either the attaching circles \(W^\downarrow(p)\cap \tilde \Sigma_{g+1}\) or \(W^\uparrow(q)\cap \tilde \Sigma_{g+1}\), we obtain \(\Sigma_g\). See figure 0.0.10.
figure 0.0.10:From the perspective of Morse theory, stabilization comes from the creation of a pair of cancelling critical points.

definition 0.0.11

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
figure 0.0.12:The cycles \(\alpha_{g-1}, \alpha_g\)and \(\alpha_g'\) are related by handleslide

theorem 0.0.13 [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. It follows that in order to construct 3-manifold invariants, one needs to find quantities associated to the Heegaard diagram which are invariant under stabilizations, isotopies, and handles slides. Heegaard-Floer cohomology provides such an invariant.

References

[Sin33]James Singer. Three-dimensional manifolds and their Heegaard diagrams. Transactions of the American Mathematical Society, 35(1):88--111, 1933.