Diffeomorphism groups of circle bundles over symplectic manifolds

Satoshi Egi, Yoshiaki Maeda, Steven Rosenberg

Published 2020-11-03Version 1

We study the diffeomorphism and isometry groups of manifolds $\overline {M_p}$, $p\in{\mathbb Z}$, which are circle bundles over a closed a $4n$-dimensional integral symplectic manifold. Equivalently, $\overline{M_p}$ is a compact $(4n+1)$-dimensional contact manifold with closed Reeb orbits. We use Wodzicki-Chern-Simons forms to prove that $\pi_1({\rm Diff}(\overline{M_p}))$ and $\pi_1({\rm Isom}(\overline{M_p}))$ are infinite for $|p| \gg 0.$ We also give the first high dimensional examples of nonvanishing Wodzicki-Pontryagin forms.