Local section of Serre fibrations with 3-manifold fibers

TL;DR

This paper extends classical results on Serre fibrations by proving the existence of local sections when fibers are fixed compact 3-manifolds, generalizing previous work on 1- and 2-dimensional fibers.

Contribution

It establishes the existence of local sections in Serre fibrations with fibers homeomorphic to a fixed compact 3-manifold, expanding the scope of Whitney's theorem.

Findings

Proves local sections exist for fibrations with 3-manifold fibers

Generalizes Whitney's theorem from 1- and 2-dimensional fibers

Advances understanding of Serre fibrations in higher dimensions

Abstract

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.