Teichmüller Spaces and Bundles with Negatively Curved Fibers

Let M be a closed smooth manifold. We will denote the group of all selfdiffeomorphisms of M , with the smooth topology, by DIFF (M). By a smooth bundle over X, with fiber M , we mean a locally trivial bundle for which the change of coordinates between two local sections over, say, Uα, Uβ ⊂ X is given by a continuous map Uα ∩ Uβ → DIFF (M). A smooth bundle map between two such bundles over X is bundle map such that, when expressed in a local chart as U × M → U × M , the induced map U → DIFF (M) is continuous. In this case we say that the bundles are smoothly equivalent. Smooth bundles over a space X, with fiber M , modulo smooth equivalence, are classified by [ X,B ( DIFF (M) )] , the set of homotopy classes of (continuous) maps from X