Exotic spheres and the topology of symplectomorphism groups

Abstract

We show that, for certain families ${\varphi }_{\mathbf{s}}$ of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of $T^{\ast }{S}^n$ with the family of pullbacks ${\varphi }_{\mathbf{s}}^{\ast }$ 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 $A_m$-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 $T^{\ast }(S^n\times S^1)$ and of an exotic symplectic structure on $T^{\ast }(S^n\times S^1)$ standard at infinity.

17 pages, 3 figures; v2 streamlined version. Accepted for publication by Journal of Topology

Cited by in corpus (1)

• Projective twists and the Hopf correspondence