Lagrangian unknottedness in Stein surfaces (Q431566)

The author shows that the space of Lagrangian spheres inside the cotangent bundle of the 2-sphere is contractible. Also, he studies the phenomenon of Lagrangian unknottedness in other Stein surfaces. There exist homotopic Lagrangian spheres which are not Hamiltonian isotopic, but he shows that in a typical case all such spheres are still equivalent under a symplectomorphism. Roughly speaking, the main results (1--7) are as follows:NEWLINENEWLINE(1) Let \(M\) be \(T^\ast S^2\) or the result of adding any number of 1-handles to \(T^1 S^2 \) (the unit cotangent bundle) and \(L \subset M\) be a Lagrangian sphere. Then there exists a Hamiltonian diffeomorphism of \(M\) mapping \(L\) to the zero-section.NEWLINENEWLINE(2) The topological space of Lagrangian spheres in \(T^\ast S^2\) is contractible.NEWLINENEWLINE(3) The subset of fixed-points free maps contained in the diffeomorphism group of \(S^2\) is contractible.NEWLINENEWLINELet \(W\) be the Stein manifold formed by adding to \(T^1 S^2\) a single 2-handle along the Legendrian curve in a single fiber of the boundary. Note that \(W\) has two Lagrangian spheres \(L_1,\;L_2\) (from the zero-sections in the copies of \(T^1S^2\)). Let \(L\) be a Lagrangian sphere in \(W\).NEWLINENEWLINE(4) Then, there exists a symplectomorphism \(\phi \) of \(W\) such that \(\phi(L) = L_1\).NEWLINENEWLINE(5) Then, there exists a composition of Dehn twists \(\tau \) such that \(\tau(L)\) is Hamiltonian isotopic to \(L_1\) or \(L_2\).NEWLINENEWLINE(6) Let \(L_1\) be a Lagrangian sphere in a symplectic 4-manifold \(M\). Then, any other Lagrangian sphere \(L\subset M\) which is sufficiently \(C^0\) close to \(L_1\) is Hamiltonian isotopic to \(L_1\).NEWLINENEWLINE(7) Let \(L_1\) and \(L_2\) be two Lagrangian spheres in a symplectic 4-manifold \(M\), intersecting transversally in a single point. Then for any other Lagrangian sphere \(L \subset M\) which is sufficiently \(C^0\) close to \(L_1 \cup L_2\) there exists a composition \(\tau\) of the Dehn twists \(\tau_{L_1}\) and \(\tau_{L_2}\) about \(L_1\) and \(L_2\) such that \(\tau (L)\) is Hamiltonian isotopic to \(L_1\) or \(L_2\).