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

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exercise 0.0.1

Let \(X\) be a compact symplectic manifold. Recall that a 3-ended Lagrangian cobordism \(K: (L_0, L_1)\rightsquigarrow L_2\) is a closed Lagrangian submanifold \(K\subset X\times \CC\) with the property that there exists a compact subset \(U\subset \CC\) so that \[K|_{\pi_\CC^{-1}(\CC\setminus U)}=((L_0\times \RR_{>0})\cup( L_1\times (\sqrt{-1}+\RR_{>0}) )\cup( L_2\times (\RR_{<0})))|_{\pi_\CC^{-1}(\CC\setminus U)}.\] Suppose that \(X\) is an exact symplectic manifold, which in turn makes \(X\times \CC\) an exact symplectic manifold. Let \(K\) be an exact 3-ended Lagrangian cobordism.
figure 0.0.2:The projection to the \(\CC\) coordinate of a 3-ended Lagrangian cobordism
  1. Show that \(L_0, L_1\) and \(L_2\) are exact Lagrangian submanifolds in \(X\).
  2. Consider the curve \(\gamma^-\subset \CC\). Show that for any exact Lagrangian submanifold \(L\subset X\), \(\CF(L\times \gamma^-, K)=\CF(L, L_0)\) as a vector space.
  3. Give \(X\times \CC\) an almost complex structure of the form \(J_X\times J_\CC\). Suppose that we have a finite energy pseudoholomorphic strip \(u: \RR\times [0, 1]\to X\times \CC\) with \(u(t, 0)\in L\times \gamma^-\) and \(u(t, 1)\in K\), and ends limiting to intersections of \(L\times \gamma^-\cap K\). Show that \(\pi_\CC(u)\in \text{Im}(\gamma^-)\cap \RR_{<0}\) (the location of the red cross in the figure). From this, conclude that if \(J_X\) is chosen so that all pseudoholomorphic strips with boundary on \(L, L_2\) are regular, that \(\CF(L\times \gamma^-, K) = \CF(L_2, K)\) as chain complexes.
    figure 0.0.3:Profile of the curve \(\gamma^-\)
  4. Consider now the curve \(\gamma^+\subset \CC\). Using a similar argument, one can prove that there are no pseudoholomorphic strips \(u:\RR\times [0, 1]\to X\times \CC\) with \(\lim_{t\to\infty} u(s, t)=z_2\) and \(\lim_{t\to-\infty} u(s, t)=z_0\). What can you conclude about the relationship between \(\CF(L\times \gamma^+, K)\), \(\CF(L, L_0)\) and \(\CF(L, L_1)\)?
    figure 0.0.4:Profile of the curve \(\gamma^+\)
  5. Observe that \(L\times \gamma^-\) and \(L\times \gamma^+\) are Hamiltonian isotopic. Exhibit a long exact sequence whose terms are \(\HF(L, L_0), \HF(L, L_1)\) and \(\HF(L, L_2)\).

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