\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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\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=art_surgeryExactTriangle}{snippets/art\_surgeryExactTriangle}}} } \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}} 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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{\Coh}{Coh} \DeclareMathOperator{\CritVal}{CritV} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\FS}{FS} \DeclareMathOperator{\Vect}{Vect} \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}, 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We restrict ourselves to $\dim_\RR(X)=2$ so that we may draw pictures. However, the pictures are only for intuition (and in fact the sketch of proof we give only work when $\dim(X)\geq 4$). Let $L_0$ be some test Lagrangian, which intersects both $L_1$ and $L_2$ as in \cref{fig:roundingCorner}. If the surgery neck is chosen to lie in a neighborhood disjoint from the intersections $L_0\cap (L_1\cup L_2)$, then these intersections are in bijection with the intersections $L_0\cap (L_1\# L_2)$. Therefore $\CF(L_0, L_1\# L_2)=\CF(L_0, L_1)[1]\oplus \CF(L_, L_2)$ as vector spaces. \begin{figure} \label{fig:roundingCorner} \centering \begin{tikzpicture}\begin{scope}[] \fill[gray!20] (-2,2) rectangle (2,-2); \fill[red!20] (0,0) .. controls (0,0.5) and (0,1) .. (0,1.5) .. controls (-1,1.5) and (-1.5,1) .. (-1.5,0) .. controls (1,0) and (0.5,0) .. (0,0); \draw (0, 2)--(0,-2); \draw (2,0) -- (-2,0); \draw (2,1.5) .. controls (1,1.5) and (0.5,1.5) .. (0,1.5) .. controls (-1,1.5) and (-1.5,1) .. (-1.5,0) .. controls (-1.5,-0.5) and (-1.5,-1.5) .. (-1.5,-2); \node[right] at (2,1.5) {$L_0$}; \node[right] at (2,0) {$L_1$}; \node[above] at (0,2) {$L_2$}; \node[below right] at (0,0) {$p_{12}$}; \node[above right] at (0,1.5) {$p_{02}$}; \node[below left] at (-1.5,0) {$p_{10}$}; \node[fill=black, circle, scale=.3] at (0,0) {}; \node[fill=black, circle, scale=.3] at (0,1.5) {}; \node[fill=black, circle, scale=.3] at (-1.5,0) {}; \node at (1,-1.5) {$X$}; \end{scope} \begin{scope}[shift={(6.5,0)}] \fill[gray!20] (-2,2) rectangle (2,-2); \draw (2,1.5) .. controls (1,1.5) and (0.5,1.5) .. (0,1.5) .. controls (-1,1.5) and (-1.5,1) .. (-1.5,0) .. controls (-1.5,-0.5) and (-1.5,-1.5) .. (-1.5,-2); \node[right] at (2,1.5) {$L_0$}; \node[right] at (2,0) {$L_1\#L_2$}; \draw[fill=blue!20] (0,1.5) .. controls (0,0.5) and (-0.5,0) .. (-1.5,0) .. controls (-1.5,1) and (-1,1.5) .. (0,1.5); \draw (0,2) .. controls (0,1.5) and (0,2) .. (0,1.5) .. controls (0,0.5) and (-0.5,0) .. (-1.5,0) .. controls (-2,0) and (-2,0) .. (-2,0); \draw (0,-2) .. controls (0,-2) and (0,-2) .. (0,-1.5) .. controls (0,-0.5) and (0.5,0) .. (1.5,0) .. controls (2,0) and (2,0) .. (2,0); \node[above right] at (0,1.5) {$p_{02}$}; \node[below left] at (-1.5,0) {$p_{10}$}; \node[fill=black, circle, scale=.3] at (0,1.5) {}; \node[fill=black, circle, scale=.3] at (-1.5,0) {}; \node at (1,-1.5) {$X$}; \end{scope} \end{tikzpicture}\caption{By rounding the corner, we can compare holomorphic triangles with holomorphic strips on the surgery.} \end{figure} The intuition from \cite{fukaya2007lagrangian} is that there is a bijection between certain holomorphic triangles with boundary on $L_0, L_1, L_2$ which passes through the intersection point $p_{12}$, and holomorphic strips with boundary on $L_0$ and $L_1\# L_2$. Since holomorphic triangles contribute to the $\emprod^3$ structure coefficients, and strips to the differential, it is reasonable to hope that we can state a relation between $L_1, L_2,$ and $L_1\#L_2$ as objects of the Fukaya category. First, we observe that the intersection point $p_{12}$ determines a morphism in $\hom(L_2, L_1)$. Since we've assumed that $L_1$ and $L_2$ intersect at only one point, we know that $\emprod^1(p_{12})=0$. We can therefore form the twisted complex $\cone(p_{12})$. We now provide justification for why this is isomorphic to $L_1\# L_2$. We have already observed that for our test Lagrangian $L_0$ we had an isomorphism of vector spaces between $\hom(L_0, L_1)\oplus \hom(L_0, L_2)$ and $\hom(L_0, L_1\#_\lambda L_2)$. The differential on $\hom(L_0, L_1\#_\lambda L_2)$ comes from counting holomorphic strips, which we break into two types: those which avoid a neighborhood of the surgery neck, and those which pass through the surgery neck. \begin{proposition} \label{prp:stripsInLagrangianSurgery} Let $\dim_\RR(X)\geq 4$, and let $L_1, L_2$ be exact Lagrangian submanifolds which intersect at a single point $p_{12}$. then we can choose an almost complex structure $J$ so that whenever $p_{01}, q_{01}\in L_0\cap L_1$ are intersections, and $u: [0,1]\times \RR\to X$ is a $J$ holomorphic strip, then the boundary of $u$ is disjoint from a small neighborhood of $L_1\cap L_2$. In particular, $u$ gives a $J$-holomorphic strip with boundary on $L_0, L_1\#_\lambda L_2$. \end{proposition} The more difficult portion is to understand the strips which pass through the neck. \begin{theorem}\cite{fukaya2007lagrangian} \label{thm:roundingCorner} Let $\dim(X)\geq 4$, and $L_1, L_2$ be exact Lagrangian submanifolds which intersect transversely at a single point $p_{12}$. Let $L_0$ be another exact Lagrangian submanifold which intersects $L_1, L_2$ transversely. Then for sufficiently small surgery necks, there exists a choices of almost complex structure on $X$ for which we have a bijection between \begin{itemize} \item $J$-holomorphic strips with boundary on $L_0, L_1\# L_2$ which pass through the surgery neck; \item $J$-holomorphic triangles with boundary on $L_0, L_1, L_2$. \end{itemize} \end{theorem} In fact, \cite{fukaya2007lagrangian} proves the above statement in much greater generality than we state here. The condition that $\dim(X)\geq 4$ can already be seen in in \cref{fig:roundingCorner}. Observe that if we have a pseudoholomorphic triangle whose boundary passes through the point $p_{12}$ in the wrong way, that there is no corresponding pseudoholomorphic strip with boundary on $L_0$ and $L_1\# L_2$. Ignoring the potential complications in the definition of the Fukaya category, we obtain: Let $\dim(X)\geq 4$, and $L_1, L_2$ be exact Lagrangian submanifolds which intersect transversely at a single point $p_{12}$. Then $L_1\#_{\lambda} L_2$ is isomorphic to the twisted complex $(L_1[1]\oplus L_2, \emprod^2(T^{-\lambda} p_{12}-))$ in $\Fuk(X)$.\printbibliography \end{document}