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\addbibresource{references.bib}\begin{document}

\title{symplectic Dehn twist}
 \maketitle
 \thispagestyle{firstpage}Let $Y$ be a symplectic manifold, and let $S^{n}\subset Y$ be a Lagrangian sphere. 
We now describe a symplectomorphism called \emph{symplectic Dehn twist:}
\[\tau_{S^n}: Y\to Y,\]
described in \cite{seidel2003long}. 
\section{construction of the symplectic dehn twist}
\label{con:dehnTwist}

    Fix the standard metric $g$ on $S^n$, and let $B_r^*S^{n}$ be the radius  $r$ conormal ball of $S^n$. 
    We first describe a symplectomorphism of $B_r^*S^n$. Let $\pi: B^*_rS^n\to S^n$ be projection to the base.
    Consider the function 
    \begin{align*}
        f: B_r^*S^{n}\to& \RR\\
        (q, p) \mapsto& |p|_g^2.
    \end{align*}
    The function $f$ is a smooth map on $B_r^*S^n$, and the Hamiltonian flow of $f$ is the geodesic flow. 
    This is a smooth function on $B^*_rS^n\setminus S^n$. 
    On the symplectic manifold $B^*_rS^n\setminus S^n$ the time $\pi$ flow of $\sqrt{f}$ is the antipodal map on the $S^n$ base (\cref{exr:dehnTwistAsSurgery}). 
    We take a smooth function $\rho: \RR\to \RR$ with the property that $\rho \circ f = f$ when $f< \epsilon$, $\rho\circ f=\sqrt f$ when $f>r-\epsilon$, and $\rho$ is increasing. 

\[ \begin{tikzpicture}
    \draw[scale=0.5, domain=4:9, smooth, variable=\x, blue] plot ({\x}, {sqrt(\x)});
      
    \draw[scale=0.5, domain=0:1, smooth, variable=\x, blue] plot ({\x}, {\x*\x});
    \node at (0,0){};
    \draw (0.5,0.5) .. controls (0.75,1) and (1,0.75) .. (2,1);
    \draw[->] (0,0) -- node[below]{$r$} (5,0);
    \draw[->] (0,0) --node[left]{$\rho$} (0,2);
    \end{tikzpicture}\]    
    Let $H= \rho \circ f: B^*_rS^n\to \RR$, and let $\phi_H: B_r^*S^n\to B_r^*S^n$ be the time-one Hamiltonian isotopy of $H$. 
    Finally, let $-\id: S^n\to S^n$ the antipodal map, which extends to a symplectomorphism $-\id: B_r^*S^n\to B_r^*S^n$. 
    Define $-\phi_H:=-\id\circ \phi_H$. 
    Observe that the map $-\phi_H: S^n\to S^n$ is a symplectomorphism of $B_r^*{S^n}$, which acts by the identity in a neighborhood of $\partial B_r^*S^{n}$.
    It acts by the antipodal map on the zero section.
\begin{definition}
\label{def:dehnTwist}

    Given a Lagrangian sphere $S^n\subset Y$, pick $r$ small enough to identify a Weinstein neighborhood  $S^n\subset B_r^*S^n\subset X$. 
    We define symplectic Dehn twist as the symplectomorphism:
    \[\tau_{S^n}(x):=\left\{\begin{array}{cc} x & \text{for $x\not\in B_r^*S^n$}\\
        -\phi_H(x) & \text{for $x\in B_r^*S^n$}
    \end{array}\right.\]
\end{definition}
\begin{figure}
 \label{fig:symplecticDehnTwist}
 \centering
 

\begin{tikzpicture}

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    \end{scope}
    \begin{scope}[]
    
    \clip  (-1.45,-1.45) rectangle (-3.55,-2);
    \draw[ dashed]  (-2.5,-2) ellipse (1 and 0.5);
    
    \end{scope}
    \end{scope}\begin{scope}[shift={(0,1.5)}]
    \begin{scope}[]
    
    \clip  (-1.45,-2.55) rectangle (-3.55,-2);
    \draw[blue]  (-2.5,-2) ellipse (1 and 0.5);
    
    \end{scope}
    \begin{scope}[]
    
    \clip  (-1.45,-1.45) rectangle (-3.55,-2);
    \draw[blue, dashed]  (-2.5,-2) ellipse (1 and 0.5);
    
    \end{scope}
    \end{scope}\begin{scope}[shift={(0,0.5)}]
    
    \draw[red] (-2.5,-2) .. controls (-2.5,-1.5) and (-3.5,-1.5) .. (-3.5,-1);
    \draw[red, dashed] (-3.5,-1) .. controls (-3.5,-0.5) and (-1.5,-0.5) .. (-1.5,0);
    \draw[red] (-1.5,0) .. controls (-1.5,0.5) and (-2.5,0.5) .. (-2.5,1);
    \draw[red] (-2.5,-3) -- (-2.5,-2) (-2.5,1) -- (-2.5,1.5);
    \end{scope}
    \draw (-3.5,2.5) -- (-3.5,-2);
    \draw (-1.5,2.5) -- (-1.5,-2);dig
    \end{scope}
    
    \begin{scope}[shift={(-5.5,0)}]
    \draw  (-2.5,2.5) ellipse (1 and 0.5);
    
    \begin{scope}[shift={(0,-0.007)}]
    \begin{scope}[]
    
    \clip  (-1.45,-2.55) rectangle (-3.55,-2);
    \draw[]  (-2.5,-2) ellipse (1 and 0.5);
    
    \end{scope}
    \begin{scope}[]
    
    \clip  (-1.45,-1.45) rectangle (-3.55,-2);
    \draw[ dashed]  (-2.5,-2) ellipse (1 and 0.5);
    
    \end{scope}
    \end{scope}\begin{scope}[shift={(0,1.5)}]
    \begin{scope}[]
    
    \clip  (-1.45,-2.55) rectangle (-3.55,-2);
    \draw[blue]  (-2.5,-2) ellipse (1 and 0.5);
    
    \end{scope}
    \begin{scope}[]
    
    \clip  (-1.45,-1.45) rectangle (-3.55,-2);
    \draw[blue, dashed]  (-2.5,-2) ellipse (1 and 0.5);
    
    \end{scope}
    \end{scope}
    \draw (-3.5,2.5) -- (-3.5,-2);
    \draw (-1.5,2.5) -- (-1.5,-2);
    \draw[red] (-2.5,2) -- (-2.5,-2.5);
    \end{scope}
    \draw[->] (-6.5,0) -- (-4,0);
    \node[fill=white] at (-5.5,0) {$\tau_{S^1}$};
    \end{tikzpicture}\caption{Performing a Dehn twist on the zero section of \(T^*S^1\)}
 \end{figure}\begin{theorem}
\label{thm:seidelDehnTwist}
    Let $X$ be a symplectic manifold, and $S\subset X$ a Lagrangian sphere, and $L\subset X$ another Lagrangian submanifold. There is an exact triangle in the Fukaya category
    \[ \cdots \to \CF(S, L)\otimes S \to L \xrightarrow{\ev} \tau_S(L)\xrightarrow{[1]}\cdots.\]

\end{theorem}
    Here, $\CF(S, L)\otimes S$ is a twisted complex. Recall that (as a vector space) $\CF(S, L)=\bigoplus_{x\in S\cap L} \Lambda\langle x \rangle$. 
    The twisted complex $\CF(S, L)\otimes S$ is given by $\bigoplus_{x\in S\cap L} S\langle x \rangle$, which is to say that formal direct sum of copies of $S$ whose grading is determined by the intersection points $x$.
    The differential on a twisted complex is a collection of maps $\delta_{xy}^E\in \CF(S\langle x \rangle, S\langle y \rangle)$. The morphism we take is 
    \[\delta_{xy}^E= \langle m^1(x), y\rangle \id.\]
    
    We now describe the map $\ev: \CF(S, L)\otimes S\to L$. Recall that a morphism of twisted complexes is a collection of maps. We must pick for each $S\langle x \rangle$ a morphism in $\hom(S\langle x \rangle , L)$. Fortunately, there is a canonical choice (which is $x$ itself). 

\printbibliography
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