\documentclass[11 pt]{article} \usepackage{amsmath,amsthm,amsfonts,amssymb,titlesec} \usepackage{hyperref} \usepackage{tikz} \usepackage{verbatim} \usepackage{accents} \usepackage[citestyle=alphabetic,bibstyle=alphabetic,backend=bibtex]{biblatex} \usepackage{todonotes} \usepackage[american]{babel} \usepackage{fancyhdr} \hypersetup{colorlinks=false} \usetikzlibrary{calc, decorations.pathreplacing,shapes.misc} \usetikzlibrary{decorations.pathmorphing} \usepackage[left=1in,top=1in,right=1in]{geometry} \usepackage[capitalize]{cleveref} \newcommand{\mathcolorbox}[2]{\colorbox{#1}{$\displaystyle #2$}} \newcommand{\xxx}{T base with combinatorial potential data } \newcommand{\Xxx}{T base with combinatorial potential data } \newcommand{\xxxc}{combinatorial potential stratified space } \newcommand{\Xxxc}{combinatorial potential stratified space } \newcommand{\argument}{symplectic character } \newcommand{\arguments}{symplectic characters } \newcommand{\snip}[2]{#1} 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\DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\TropB}{TropB} \DeclareMathOperator{\weight}{wt} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Fuk}{Fuk} \DeclareMathOperator{\str}{star} \DeclareMathOperator{\Ob}{Ob} \DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\Bl}{Bl} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Tw}{Tw} \DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\Arg}{\mathbf{M}}\begin{filecontents}{references.bib} @article{ballard2012hochschild, title={Hochschild dimensions of tilting objects}, author={Ballard, Matthew and Favero, David}, journal={International Mathematics Research Notices}, volume={2012}, number={11}, pages={2607--2645}, year={2012}, publisher={OUP} } @article{craw2007explicit, title={Explicit methods for derived categories of sheaves}, author={Craw, Alastair}, publisher={Citeseer} } 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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 \cref{fig:heegaardDiagram3Sphere}. \begin{figure} \label{fig:heegaardDiagram3Sphere} \centering \begin{tikzpicture}[scale=2] \begin{scope}[] \clip (-1.8,-1.6) rectangle (0,-0.2); \draw (-1.2,-1) ellipse (0.2 and 0.5); \end{scope} \fill[red!20] (-1.2,-1) ellipse (0.2 and 0.5); \node[] at (-1.2,-1) {$\times$}; \draw[fill=gray!20, fill opacity=.9] (-1.2,-0.5) ellipse (2 and 1); \begin{scope}[shift={(2.2,-0.6)}] \fill[white] plot[smooth, tension=0.7] coordinates { (-3.68,0.2) (-3.4,0.1) (-3.14,0.2) } plot[smooth, tension=0.7] coordinates {(-3.68,0.2) (-3.4,0.34) (-3.14,0.2)}; \draw plot[smooth, tension=0.7] coordinates {(-3.8,0.34) (-3.68,0.2) (-3.4,0.1) (-3.14,0.2) (-3,0.34)}; \draw plot[smooth, tension=0.7] coordinates {(-3.68,0.2) (-3.4,0.34) (-3.14,0.2)}; \end{scope} \draw[draw=blue, fill=blue!20, fill opacity=.9] (-1.2,-0.4) node (v1) {} ellipse (1.2 and 0.4); \fill[fill=blue!20, fill opacity=.9] (-2.4,-0.4) .. controls (-2.4,-0.2) and (-1.6,1) .. (-1.2,1) .. controls (-0.8,1) and (0,-0.2) .. (0,-0.4) .. controls (0,-0.2) and (-0.8,-0.2) .. (-1.2,-0.2) .. controls (-1.6,-0.2) and (-2.4,-0.2) .. (-2.4,-0.4); \draw (-2.4,-0.4) .. controls (-2.4,-0.2) and (-1.6,1) .. (-1.2,1) .. controls (-0.8,1) and (0,-0.2) .. (0,-0.4); \begin{scope}[] \clip (-0.8,-1.6) rectangle (v1); \draw[red] (-1.2,-1) ellipse (0.2 and 0.5); \end{scope} \node[left] at (-1.2,-1) {$p$}; \node[above] at (-1.2,1) {$q$}; \node at (-1.2,1) {$\times$}; \node at (-2.25,-0.71) {$\alpha$}; \node at (-0.93,-1.18) {$\beta$}; \node at (-2.8,-0.6) {$\Sigma_1$}; \end{tikzpicture}\caption{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 \(p, q\).} \end{figure}\end{example} \printbibliography \end{document}