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

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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 0.0.1.
figure 0.0.1:From the perspective of Morse theory, stabilization comes from the creation of a pair of cancelling critical points.