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

symplectic Dehn twist

SECTIONS
Let \(Y\) be a symplectic manifold, and let \(S^{n}\subset Y\) be a Lagrangian sphere. We now describe a symplectomorphism called symplectic Dehn twist: \[\tau_{S^n}: Y\to Y,\] described in [Sei03].

1: construction of the symplectic dehn twist

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 ((Dehn twist as surgery)). 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. 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.

definition 1.0.1

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.\]
figure 1.0.2:Performing a Dehn twist on the zero section of \(T^*S^1\)

theorem 1.0.3

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 \[ \cdots \to \CF(S, L)\otimes S \to L \xrightarrow{\ev} \tau_S(L)\xrightarrow{[1]}\cdots.\] 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).

References

[Sei03]Paul Seidel. A long exact sequence for symplectic Floer cohomology. Topology, 42:1003--1063, 2003.