Fiber-preserving diffeomorphisms and imbeddings
  Around 1960, R. Palais and J. Cerf proved a fundamental result relating spaces of diffeomorphisms and imbeddings of manifolds: If V is a submanifold of M, then the map from Diff(M) to Imb(V,M) that takes f to its restriction to V is locally trivial. We extend this and related results into the context of fibered manifolds, and fiber-preserving diffeomorphisms and imbeddings. That is, if M fibers over B, with compact fiber, and V is a vertical submanifold of M, then the restriction from the space FDiff(M) of fiber-preserving diffeomorphisms of M to the space of imbeddings of V into M that take fibers to fibers is locally trivial. Also, the map from FDiff(M) to Diff(B) that takes f to the diffeomorphism it induces on B is locally trivial. The proofs adapt Palais' original approach; the main new ingredient is a version of the exponential map, called the aligned exponential, which has better properties with respect to fiber-preserving maps. Versions allowing certain kinds of singular fibers are proven, using equivariant methods. These apply to almost all Seifert-fibered 3-manifolds. As an application, we reprove an unpublished result of F. Raymond and W. Neumann that each component of the space of Seifert fiberings of a Haken 3-manifold is weakly contractible. 
John Kalliongis and Darryl McCullough