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