\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} \newtheorem{theorem}{Theorem}[section] \newtheorem{lem}[theorem]{Lemma} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{notation}[theorem]{Notation} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{rem}[theorem]{Remark} \newtheorem{remark}[theorem]{Remark} \crefname{rem}{Remark}{Remarks} \Crefname{rem}{Remark}{Remarks} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{example}[theorem]{Example} \newenvironment{construction}{}{} \newenvironment{exposition}{}{} \newenvironment{application}{}{} \theoremstyle{definition} \newtheorem{df}[theorem]{Definition} \newtheorem{definition}[theorem]{Definition} \titleformat*{\section}{\normalsize \bfseries \filcenter} \titleformat*{\subsection}{\normalsize \bfseries } \newtheorem{mainthm}{Theorem} \Crefname{mainthm}{Theorem}{Theorems} \newtheorem{maincor}[mainthm]{Corollary} \Crefname{maincor}{Corollary}{Corollaries} \renewcommand*{\themainthm}{\Alph{mainthm}} \makeatletter \def\namedlabel#1#2{\begingroup \def\@currentlabel{#2}% \label{#1}\endgroup } \makeatother \fancypagestyle{firstpage}{% \fancyhf{} \renewcommand\headrulewidth{0pt} \fancyfoot[R]{Original text at \texttt{ \href{http://jeffhicks.net/snippets/index.php?tag=exm_nonadmissibleHeegaardDiagram}{snippets/exm\_nonadmissibleHeegaardDiagram}}} } \newcommand{\wt}{\widetilde} \newcommand{\wh}{\widehat} \newcommand{\wb}{\overline} \newcommand{\bb}{\mathbb} \newcommand{\scr}{\mathscr} \newcommand{\RR}{\mathbb R} \newcommand{\ZZ}{\mathbb Z} \newcommand{\CC}{\mathbb C} \newcommand{\TT}{\mathbb T} \newcommand{\NN}{\mathbb N} \newcommand{\PP}{\mathbb P} \newcommand{\LL}{\mathbb L} \newcommand{\II}{\mathbb I} \newcommand{\CP}{\mathbb{CP}} \newcommand{\del}{\nabla} \newcommand{\pp}{\mathbf{m}} \newcommand{\into}{\hookrightarrow} \newcommand{\emprod}{m} \newcommand{\tensor}{\otimes} \renewcommand{\Re}{\text{Re}} \renewcommand{\Im}{\text{Im}} \newcommand{\eps}{\varepsilon} \newcommand{\CF}{CF^\bullet} \newcommand{\HF}{HF^\bullet} \newcommand{\SH}{SH^\bullet} \newcommand{\core}{\mathfrak{c}} \newcommand{\cocore}{\mathfrak{u}} \newcommand{\stp}{\mathfrak{f}} \newcommand{\li}{i} \newcommand{\ot}{\leftarrow} \newcommand{\Spinc}{\text{Spin}^c} \newcommand{\ev}{ev} \newcommand{\st}{\;:\;} \newcommand{\OP}{\mathcal O_{\mathbb P^1}} \newcommand{\OPP}{\mathcal O_{\mathbb P\times \mathbb P}} \newcommand{\gentime}{\text{\ClockLogo}} \newcommand{\q}{m} \newcommand{\HeF}{\widehat{CF}^\bullet} \newcommand{\HHeF}{\widehat{HF}^\bullet} \newcommand{\p}{\eta} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\cone}{cone} \DeclareMathOperator{\dg}{dg} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Log}{Log} \DeclareMathOperator{\Conn}{Conn} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\Flux}{Flux} \DeclareMathOperator{\Crit}{Crit} \DeclareMathOperator{\ind}{ind} \DeclareMathOperator{\codim}{codim} \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} } @article{weinstein1971symplectic, title={Symplectic manifolds and their {L}agrangian submanifolds}, author={Weinstein, Alan}, journal={Advances in Mathematics}, volume={6}, number={3}, pages={329--346}, year={1971}, publisher={Academic Press} } @article{hanlon2022aspects, title={Aspects of functoriality in homological mirror symmetry for toric varieties}, author={Hanlon, A and Hicks, J}, journal={Advances in Mathematics}, volume={401}, pages={108317}, year={2022}, publisher={Elsevier} } @article{biran2013lagrangian, title={{L}agrangian cobordism. {I}}, author={Biran, Paul and Cornea, Octav}, journal={Journal of the American Mathematical Society}, volume={26}, number={2}, pages={295--340}, year={2013} } @article{tanaka2016fukaya, title={The Fukaya category pairs with Lagrangian cobordisms}, author={Tanaka, Hiro Lee}, journal={arXiv preprint arXiv:1607.04976}, year={2016} } @book{seidel2008fukaya, title={Fukaya categories and Picard-Lefschetz theory}, author={Seidel, Paul}, volume={10}, year={2008}, publisher={European Mathematical Society} } @article{seidel2003long, title={A long exact sequence for symplectic {F}loer cohomology}, author={Seidel, Paul}, journal={Topology}, volume={42}, pages={1003--1063}, year={2003} } @article{da2001lectures, title={Lectures on symplectic geometry}, author={da Silva, Ana Cannas}, journal={Lecture Notes in Mathematics}, volume={1764}, year={2001}, publisher={Springer} } @article{polterovich1991surgery, title={The surgery of {L}agrange submanifolds}, author={Polterovich, Leonid}, journal={Geometric \& Functional Analysis GAFA}, volume={1}, number={2}, pages={198--210}, year={1991}, publisher={Springer} } @misc{perutz2008handleslide, doi = {10.48550/ARXIV.0801.0564}, url = {https://arxiv.org/abs/0801.0564}, author = {Perutz, Timothy}, keywords = {Symplectic Geometry (math.SG), Geometric Topology (math.GT), FOS: Mathematics, FOS: Mathematics, 53D12; 53D40; 57M27; 32U40}, title = {Hamiltonian handleslides for {H}eegaard {F}loer homology}, publisher = {arXiv}, year = {2008}, copyright = {Assumed arXiv.org perpetual, non-exclusive license to distribute this article for submissions made before January 2004} } @incollection{audin1994symplectic, title={Symplectic rigidity: {L}agrangian submanifolds}, author={Audin, Mich{\`e}le and Lalonde, Fran{\c{c}}ois and Polterovich, Leonid}, booktitle={Holomorphic curves in symplectic geometry}, pages={271--321}, year={1994}, publisher={Springer} } @phdthesis{oancea2003suite, title={La suite spectrale de {L}eray-{S}erre en homologie de {F}loer des vari{\'e}t{\'e}s symplectiques compactes {\`a} bord de type contact}, author={Oancea, Alexandru}, year={2003}, school={Universit{\'e} Paris Sud-Paris XI} } @article{abouzaid2010geometric, title={A geometric criterion for generating the {F}ukaya category}, author={Abouzaid, Mohammed}, journal={Publications Math{\'e}matiques de l'IH{\'E}S}, volume={112}, pages={191--240}, year={2010} } @article{viterbo1999functors, title={Functors and computations in {F}loer homology with applications, I}, author={Viterbo, Claude}, journal={Geometric \& Functional Analysis GAFA}, volume={9}, number={5}, pages={985--1033}, year={1999}, publisher={Springer} } @misc{stacks-project, author = {The {Stacks project authors}}, title = {The Stacks project}, howpublished = {\url{https://stacks.math.columbia.edu}}, year = {2022}, } @article{wendlbeginner, title={A beginner’s overview of symplectic homology}, author={Wendl, Chris}, journal={Preprint. www. mathematik. hu-berlin. de/wendl/pub/SH. pdf} } @article{seidel2006biased, title={A biased view of symplectic cohomology}, author={Seidel, Paul}, journal={Current developments in mathematics}, volume={2006}, number={1}, pages={211--254}, year={2006}, publisher={International Press of Boston} } @article{arnol1980lagrange, title={{L}agrange and {L}egendre cobordisms. I}, author={Arnol'd, Vladimir Igorevich}, journal={Funktsional'nyi Analiz i ego Prilozheniya}, volume={14}, number={3}, pages={1--13}, year={1980}, publisher={Russian Academy of Sciences} } @article{fukaya2007lagrangian, title={{L}agrangian intersection {F}loer theory-anomaly and obstruction, chapter 10}, author={Fukaya, K and Oh, YG and Ohta, H and Ono, K}, journal={Preprint, can be found at http://www. math. kyoto-u. ac. jp/\~{} fukaya/Chapter10071117. pdf}, year={2007} } @article{biran2014lagrangian, title={Lagrangian cobordism and Fukaya categories}, author={Biran, Paul and Cornea, Octav}, journal={Geometric and functional analysis}, volume={24}, number={6}, pages={1731--1830}, year={2014}, publisher={Springer} } @article{bourgeois2009symplectic, title={Symplectic homology, autonomous {H}amiltonians, and {M}orse-{B}ott moduli spaces}, author={Bourgeois, Fr{\'e}d{\'e}ric and Oancea, Alexandru}, journal={Duke mathematical journal}, volume={146}, number={1}, pages={71--174}, year={2009}, publisher={Duke University Press} } @incollection{auroux2014beginner, title={A beginner’s introduction to {F}ukaya categories}, author={Auroux, Denis}, booktitle={Contact and symplectic topology}, pages={85--136}, year={2014}, publisher={Springer} } @article{singer1933three, title={Three-dimensional manifolds and their {H}eegaard diagrams}, author={Singer, James}, journal={Transactions of the American Mathematical Society}, volume={35}, number={1}, pages={88--111}, year={1933}, publisher={JSTOR} } @article{ozsvath2004holomorphic, title={Holomorphic disks and three-manifold invariants: properties and applications}, author={Ozsv{\'a}th, Peter and Szab{\'o}, Zolt{\'a}n}, journal={Annals of Mathematics}, pages={1159--1245}, year={2004}, publisher={JSTOR} } @article{ozsvath2004introduction, title={An introduction to {H}eegaard {F}loer homology}, author={Ozsv{\'a}th, Peter and Szab{\'o}, Zolt{\'a}n}, journal={{F}loer homology, gauge theory, and low-dimensional topology}, volume={5}, pages={3--27}, year={2004} } @article{fet1952variational, title={Variational problems on closed manifolds}, author={Fet, Abram Il'ich}, journal={Matematicheskii Sbornik}, volume={72}, number={2}, pages={271--316}, year={1952}, publisher={Russian Academy of Sciences, Steklov Mathematical Institute of Russian~…} }\end{filecontents} \addbibresource{references.bib}\begin{document} \begin{example} \label{exm:nonadmissibleHeegaardDiagram} We look at the example of $M=S^2\times S^1$. Observe that $S^2=D^2\cup_{S^1} D^2$, so we can write $M=D^2\times S^1 \cup_{\Sigma_1} D^2\times S^1$. The Heegaard diagram $(\Sigma_1, \alpha, \beta)$ consists of a torus with two meridional cycles. \begin{figure} \label{fig:nonadmissibleHeegaardDiagram} \centering \begin{tikzpicture} \begin{scope}[] \begin{scope}[] \draw[fill=gray!20] (-2,-1.5) .. controls (-3.5,-1.5) and (-4,0) .. (-4,1) .. controls (-4,2) and (-3.5,3.5) .. (-2,3.5) (-2,1.5) .. controls (-2.5,1.5) and (-2.5,0.5) .. (-2,0.5); \end{scope} \begin{scope}[xscale=-1, shift={(-0.5,0)}]] \draw[fill=gray!20] (-2,-1.5) .. controls (-3.5,-1.5) and (-4,0) .. (-4,1) .. controls (-4,2) and (-3.5,3.5) .. (-2,3.5) (-2,1.5) .. controls (-2.5,1.5) and (-2.5,0.5) .. (-2,0.5); \end{scope} \fill[gray!20] (-2,0.5) rectangle (2.5,-1.5); \fill[gray!20] (-2,3.5) rectangle (2.5,1.5); \end{scope} \begin{scope}[] \fill[orange!20] (1,-0.5) ellipse (0.5 and 1); \fill[orange!20] (-0.5,0.5) rectangle (1,-1.5); \fill[gray!20] (-0.5,-0.5) ellipse (0.5 and 1); \end{scope} \draw (-2,-1.5) -- (2.5,-1.5) (-2,0.5) -- (2.5,0.5) (-2,1.5) -- (2.5,1.5) (-2,3.5) -- (2.5,3.5); \begin{scope}[] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[red, shift={(1.5,0)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[blue, shift={(3,0)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-0.5,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[shift={(4.5,0)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[shift={(4.5,3)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[shift={(0,3)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \node[left, red] at (0,-0.5) {$\beta$}; \node[right, blue] at (1.5,-0.5) {$\alpha$}; \node at (0.5,0) {$\mathcal D$}; \node[circle, fill=black, scale=.2] at (0.5,2.5) {}; \node[right] at (0.5,2.5) {$z$}; \end{tikzpicture}\caption{A non-admissible Heegaard diagram} \end{figure} If the diagram is chosen so that $\alpha, \beta$ are disjoint, then the Lagrangian intersection Floer cohomology $\HF(\alpha, \beta)$ vanishes. \begin{figure} \label{fig:admissibleHeegaardDiagram} \centering \begin{tikzpicture} \begin{scope}[] \begin{scope}[] \draw[fill=gray!20] (-2,-1.5) .. controls (-3.5,-1.5) and (-4,0) .. (-4,1) .. controls (-4,2) and (-3.5,3.5) .. (-2,3.5) (-2,1.5) .. controls (-2.5,1.5) and (-2.5,0.5) .. (-2,0.5); \end{scope} \begin{scope}[xscale=-1, shift={(-0.5,0)}]] \draw[fill=gray!20] (-2,-1.5) .. controls (-3.5,-1.5) and (-4,0) .. (-4,1) .. controls (-4,2) and (-3.5,3.5) .. (-2,3.5) (-2,1.5) .. controls (-2.5,1.5) and (-2.5,0.5) .. (-2,0.5); \end{scope} \fill[gray!20] (-2,0.5) rectangle (2.5,-1.5); \fill[gray!20] (-2,3.5) rectangle (2.5,1.5); \end{scope} \begin{scope}[] \fill[orange!20] (1,-0.5) ellipse (0.5 and 1); \fill[orange!20] (-0.5,0.5) rectangle (1,-1.5); \fill[gray!20] (-0.5,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[red, shift={(1.5,0)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (0.5,-1.5); \draw[fill=gray!20] (-2,0.5) .. controls (-1.5,0.5) and (-1.5,0.5) .. (-1.5,0) .. controls (-1.5,-0.5) and (0.5,0) .. (0.5,-0.5) .. controls (0.5,-1) and (-1.5,-0.5) .. (-1.5,-1) .. controls (-1.5,-1.5) and (-1.5,-1.5) .. (-2,-1.5); \clip (-0.5,0) rectangle (0.5,-1); \draw[fill=orange!20] (-1.5,0) .. controls (-1.5,-0.5) and (0.5,0) .. (0.5,-0.5) .. controls (0.5,-1) and (-1.5,-0.5) .. (-1.5,-1); \clip (-1.5,0) .. controls (-1.5,-0.5) and (0.5,0) .. (0.5,-0.5) .. controls (0.5,-1) and (-1.5,-0.5) .. (-1.5,-1); \draw[fill=gray!20] (-0.5,-0.5) ellipse (0.5 and 1); \draw (-1.5,0) .. controls (-1.5,-0.5) and (0.5,0) .. (0.5,-0.5) .. controls (0.5,-1) and (-1.5,-0.5) .. (-1.5,-1); \end{scope} \begin{scope}[blue, shift={(3,0)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-0.5,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[shift={(4.5,0)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[shift={(4.5,3)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[shift={(0,3)}] \draw[dotted] (-2,-0.5) ellipse (0.5 and 1); \clip (-2,0.5) rectangle (-1,-1.5); \draw (-2,-0.5) ellipse (0.5 and 1); \end{scope} \node[right, red] at (2,-0.5) {$\beta'$}; \node[left, blue] at (1.5,-0.5) {$\alpha$}; \node at (0.5,0) {$\mathcal D$}; \node[circle, fill=black, scale=.2] at (0.5,2.5) {}; \node[right] at (0.5,2.5) {$z$}; \draw (-2,-1.5) -- (2.5,-1.5) (-2,0.5) -- (2.5,0.5) (-2,1.5) -- (2.5,1.5) (-2,3.5) -- (2.5,3.5); \end{tikzpicture}\caption{An admissible Heegaard diagram} \end{figure} However, if the diagram is chosen so that $\alpha, \beta'$ intersect transversely, the Lagrangian intersection Floer cohomology (with $\ZZ/2\ZZ$ coefficients) is $\ZZ/2\ZZ\oplus \ZZ/2\ZZ$. Note that $\beta'$ can be chosen so that it is Hamiltonian isotopic to $\beta$. The discrepancy between these two answers comes from the non-convergence of the homotopy between the composition of continuation maps \[f\circ g:\CF(\alpha, \beta')\to \CF(\alpha, \beta) \to \CF(\alpha, \beta')\] \[\id: \CF(\alpha, \beta')\to \CF(\alpha, \beta')\] over $\ZZ/2\ZZ$ coefficients. The presence of an annulus between $\alpha, \beta$ is the culprit for the non-convergence. One can make the quantities converge by using \underline{\href{https://jeffhicks.net/snippets/index.php?tag=def:novikovRing}{ Novikov coefficients}} instead of $\ZZ/2\ZZ$ coefficients. In that setting, the differential in \cref{fig:nonadmissibleHeegaardDiagram} will be exact unless the areas of the two strips agree --- that is, the Lagrangians $\alpha, \beta$ are Hamiltonian isotopic. \end{example} \printbibliography \end{document}