Hurewicz fibrations, almost submetries and critical points of smooth maps

TL;DR

This paper explores how Hurewicz fibrations impose topological constraints on spaces with CW-complex homotopy types and applies these results to analyze differentiable and metric properties of maps between Riemannian manifolds with curvature bounds.

Contribution

It establishes new topological restrictions derived from Hurewicz fibrations and uses these to infer properties of maps between Riemannian manifolds under curvature conditions.

Findings

Hurewicz fibrations restrict universal coverings topologically

Results connect fibrations to properties of maps between Riemannian manifolds

Curvature restrictions influence differentiable and metric properties

Abstract

We prove that the existence of a Hurewicz fibration between certain spaces with the homotopy type of a CW-complex implies some topological restrictions on their universal coverings. This result is used to deduce differentiable and metric properties of maps between compact Riemannian manifolds under curvature restrictions.