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.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.proposition 0.0.4
Let \(M\) be a closed 3-manifold. There exists a Heegaard splitting for \(M\).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- Gradient \(\nabla f\) points outward along the boundary of \(U\);
- \(f\) has only critical points of index \(0\) and \(1\).
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- \(\Sigma_g\setminus \{\alpha_i\}_{i=1}^g, \Sigma_g\setminus \{\beta_i\}_{i=1}^g\) are connected and;
- \(\alpha_i\cap \alpha_j=\emptyset = \beta_i \cap \beta_j\) whenever \(i\neq j\).
example 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.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 \(T^2\).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- 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 0.0.12
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.References
[Sin33] | James Singer. Three-dimensional manifolds and their Heegaard diagrams. Transactions of the American Mathematical Society, 35(1):88--111, 1933. |