The Smale conjecture for Seifert fibered spaces with hyperbolic base orbifold

Abstract

Let $M$ be a closed orientable 3-manifold admitting an $\mathbb{H}^2 \times \mathbb{R}$ or $\widetilde{\mathrm{SL}_2}(\mathbb{R})$ 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 $\mathrm{Diff}(M)$ 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 $\mathrm{Diff}(M)$ and of the space of Seifert fiberings $\mathrm{SF}(M)$ for compact orientable aspherical 3-manifolds. We also prove that when $M$ has an $\mathbb{H}^2 \times \mathbb{R}$ or $\widetilde{\mathrm{SL}_2}(\mathbb{R})$ geometry and the base orbifold has underlying manifold the 2-sphere with three cone points, the inclusion $\mathrm{Isom}(M) \to \mathrm{Diff}(M)$ is a homotopy equivalence.