Birman–Hilden Bundles. II

We study the structure of self-homeomorphism groups of fibered manifolds.A fibered topological spaceis a Birman–Hilden spacewhenever in each isotopic pair of its fiber-preserving(taking each fiber to a fiber)self-homeomorphismsthe homeomorphisms are also fiber-isotopic(isotopic through fiber-preserving homeomorphisms).We prove in particular thatthe Birman–Hilden class containsall compact connected locally trivial surface bundles over the circle,including nonorientable ones and those with nonempty boundary,as well as all closed orientable Haken 3-manifold bundles over the circle,including nonorientable ones.