Remarks on the homotopy type of groups of symplectic diffeomorphisms

Abstract:

Let $(X,\omega )$ be a symplectic manifold without boundary, $G(X)$ the identity component of its group of compactly supported diffeomorphisms, and ${H_\omega }(X)$ the subgroup of $G(X)$ consisting of all symplectic diffeomorphisms. In this note, we give examples in which ${H_\omega }(X)$ is not homotopy equivalent to $G(X)$.