  It was proved by H. Whitney in 1933 that a Serre fibration of compact metric
spaces admits a global section provided every fiber is homeomorphic to the unit
interval [0,1]. An extension of the Whitney's theorem to the case when all
fibers are homeomorphic to some fixed compact two-dimensional manifold was
proved by the authors \cite{BCS}. The main result of this paper proves the
existence of local sections in a Serre fibration with all fibers homeomorphic
to some fixed compact three-dimensional manifold.
