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


Solutions contributed by Alvaro Muniz.

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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exercise 0.0.2

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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exercise 0.0.3

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 0.0.4: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 0.0.5: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 0.0.6: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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