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

example 0.0.1

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 0.0.2.
figure 0.0.2: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 \(p, q\).