The Geometry of Loop Spaces III: Diffeomorphisms of Contact Manifolds.

We study the diffeomorphism and isometry groups of manifolds $\overline {M_p}$, $p\in\mathbb Z$, which are circle bundles over a closed $4n$-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 $|p| \gg 0.$ For the Kodaira-Thurston manifold, we explicitly compute that this result holds for all $p$. We also give the first examples of nonvanishing Wodzicki-Pontryagin forms.