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

exercise 0.0.1

Consider the space \(T^*S^n\) which we describe as a symplectic submanifold of \(\CC^{n+1}\) by the equation \[\{(x_0, \ldots, x_n,y_0, \ldots, y_n) \st \sum_{i} (x_i+\sqrt{-1} y_i)^2=1 \}\]
  1. Consider the Hamiltonian given by \(H=1/2|\vec y|^2\). Write down the Hamiltonian vector field on \(T^*S^n\).
  2. Consider now the symplectic manifold with boundary \[B^*_1S^n=\{(x_0, \ldots, x_n,y_0, \ldots, y_n) \in T^*S^n, |\vec y|\leq 1\}.\] Show that there exists \(t_0\) so that \(\phi^{t}: B^*_1S^n\to B^*_1 S^n\), the time-\(t_0\) Hamiltonian flow of \(H\), acts by \(-1\) on the boundary of \(B^*_1S^n\).
  3. The symplectic Dehn twist is the map: \begin{align*} \tau^n: B^*_1 S^n\to& B^*_1 S^n\\ (\vec x, \vec p) \mapsto& -\phi^{t_0}(\vec x, \vec p). \end{align*} which fixes the boundary of \(B^*_1 S^n\). Consider the Lagrangian submanifold \[F_q:=\{(1, \ldots, 0), (0, p_1, \ldots, p_n)\}\subset B^*_1S^n.\] Show that there is a Hamiltonian isotopy which identifies \[\tau_n(F_q)\sim S^n\# F_q.\]
  4. Consider in \(T^2\) the Lagrangian submanifold \(L:=L_{(1, 0),0}\) as before. Identify a small neighborhood \(U\) of \(L\) with \(B^*_1(S^1)\), and define \(\tau_L: T^2\to T^2\) by \[\tau_L(x)=\left\{\begin{array}{cc} \tau^1(x) &\text{if \(x\in U\)}\\ x &\text{otherwise}\end{array}\right.\] For \(a, b\in \ZZ\), and \(\theta\in S^1\), find the Lagrangian submanifold \(L_{(a', b'), \theta'}\) which is Hamiltonian isotopic to \(\tau_L(L_{(a, b), \theta})\).

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