Birman–Hilden bundles. I

A topological fibered space is a Birman–Hilden space whenever in each isotopic pair of its fiber-preserving (taking each fiber to a fiber) self-homeomorphisms the homeomorphisms are also fiber-isotopic (isotopic through fiber-preserving homeomorphisms). We present a series of sufficient conditions for a fiber bundle over the circle to be a Birman–Hilden space.