Local sections of Serre fibrations with 2-manifold fibers

TL;DR

This paper extends Whitney's theorem on Serre fibrations, demonstrating that fibrations with fibers homeomorphic to a fixed compact 2-manifold admit a global section, broadening the class of fibrations with guaranteed sections.

Contribution

The paper generalizes Whitney's result from interval fibers to fibers homeomorphic to any fixed compact 2-manifold, providing new conditions for the existence of global sections.

Findings

Extended Whitney's theorem to 2-manifold fibers.

Proved existence of global sections for a broader class of fibrations.

Established conditions under which fibrations admit sections.

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]. Results of this paper extend Whitney theorem to the case when all fibers are homeomorphic to a given compact two-dimensional manifold.