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

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example 0.0.1

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 0.0.2.
figure 0.0.2: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 \((\cos(\theta), \sin(\theta), 0, 0)\).