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

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exercise 0.0.1

Consider the torus \(T^2=S^1\times S^1\), where we parameterize the circle \(S^1\) as \(\RR/\ZZ\). For coprime integers \(a, b\in \ZZ\), and \(\theta\in S^1\), we write \[L_{(a, b),\theta}=\{(a\cdot t+\theta,b\cdot t), t\in S^1\}.\] for the Lagrangian \(S^1\) of slope \((a, b)\) passing through the point \((\theta, 0)\).
  1. Compute \(\HF(L_{(a, b), \theta_1}, L_{(c, d), \theta_2})\).
  2. Write \(L_0:=L_{(1,0), 0}, L_1:= L_{(0,1), 0}\). Let \(L_2 = L_0\# L_1\). Find values \((a, b), \theta\) so that \(L_2\) is Hamiltonian isotopic to \(L_{(a, b), \theta}\).
  3. Let \(\{x_{01}\}=L_0\cap L_1\), \(\{x_{12}\}=L_1\cap L_2\), and \(\{x_{20}\}=L_2\cap L_0\). Prove that \begin{align*} m^2(x_{12}, x_{01})=0 && m^2(x_{20}, x_{12})=0 && m^2x_{01}, (x_{20})=0 \end{align*} so that we have what appears to be an exact sequence \[L_0\xrightarrow{x_{01}} L_1 \xrightarrow{x_{12}} L_2 \xrightarrow{x_{20}} L_0[1].\]
  4. What happens in the previous computation if we replace \(L_2\) with \(L_2'\) which is Lagrangian (but not Hamiltonian) isotopic to \(L_2\)?

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