Exotic spheres and the topology of symplectomorphism groups

TL;DR

This paper demonstrates that certain high-dimensional symplectomorphism groups exhibit complex topological features, such as non-contractibility and non-simply-connectedness, using exotic spheres and Dehn twists in cotangent bundles.

Contribution

It provides new examples of symplectomorphism groups with nontrivial topology, showing dependence on parametrization and the existence of exotic structures, extending understanding of symplectic topology.

Findings

Dehn twists depend on parametrization up to Hamiltonian isotopy

Symplectomorphism groups are not simply-connected

Existence of exotic symplectic structures on cotangent bundles

Abstract

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