Smooth perfectness of Hamiltonian diffeomorphism groups

Abstract

A fundamental result of Banyaga states that the Hamiltonian diffeomorphism group of a closed symplectic manifold is perfect. We refine this result by proving that, locally in the $C^\infty$ topology, the number of commutators needed to express a Hamiltonian diffeomorphism is bounded, and the commutators can be chosen to depend smoothly on the diffeomorphism. As a corollary, we show that any homogeneous quasimorphism on the Hamiltonian diffeomorphism group is continuous in the $C^\infty$ topology. We establish an analogous smooth perfectness result for compactly supported Hamiltonian diffeomorphisms of open symplectic manifolds, where Banyaga proved that the kernel of the Calabi homomorphism is perfect. For symplectic manifolds with boundary, we define a natural group of Hamiltonian diffeomorphisms that need not restrict to the identity on the boundary. We extend the Calabi homomorphism to this setting and prove a corresponding refined perfectness result. These results are motivated by our work on symplectic packing stability for symplectic manifolds with boundary, where they play a key role in our symplectic embedding constructions.