Birman–Hilden Bundles. I

A topological fibered space is 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 present a series of sufficient conditionsfor a fiber bundle over the circleto be a Birman–Hilden space.