THE SMALE CONJECTURE FOR SEIFERT FIBERED SPACES WITH HYPERBOLIC BASE ORBIFOLD

Let M be a closed orientable 3-manifold admitting an H-2 x R or (SL2) over tilde (R) geometry, or equivalently a Seifert fibered space with a hyperbolic base 2-orbifold. Our main result is that the connected component of the identity map in the diffeomorphism group Diff(M) is either contractible or homotopy equivalent to S-1, according as the center of pi(1)(M) is trivial or infinite cyclic. Apart from the remaining case of non-Haken infranilmanifolds, this completes the homeomorphism classifications of Diff(M) and of the space of Seifert fiberings SF(M) for compact orientable aspherical 3-manifolds. We also prove that when M has an H-2 x R or (SL2) over tilde (R) geometry and the base orbifold has underlying manifold the 2-sphere with three cone points, the inclusion Isom(M) -> Diff(M) is a homotopy equivalence.