2014/07/31 by Georgios Dimitroglou Rizell, Jonathan David Evans · 1 citation
Mathematics · #math.GT #math.SG #msc:53D12 #msc:53D35
paper · pdf · doi:10.1112/jtopol/jtv007
17 pages, 3 figures; v2 streamlined version. Accepted for publication by Journal of Topology
arxiv created 2015/02/20 · arxiv updated 2015/06/12
We show that, for certain families ϕs of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of T^*Sn with the family of pullbacks ϕ^*s gives a noncontractible family of compactly-supported symplectomorphisms. In particular, we find examples: where the Dehn twist along a parametrised Lagrangian sphere depends up to Hamiltonian isotopy on its parametrisation; where the symplectomorphism group is not simply-connected, and where the symplectomorphism group does not have the homotopy-type of a finite CW-complex. We show that these phenomena persist for Dehn twists along the standard matching spheres of the Am-Milnor fibre. The nontriviality is detected by considering the action of symplectomorphisms on the space of parametrised Lagrangian submanifolds. We find related examples of symplectic mapping classes for T^*(Sn× S1) and of an exotic symplectic structure on T^*(Sn× S1) standard at infinity.