On classifying Hurewicz fibrations and fibre bundles over polyhedron bases

TL;DR

This paper investigates the role of Lf-functions in classifying Hurewicz fibrations and fiber bundles over polyhedron bases, establishing conditions for fiber homotopy equivalence and relating to Dold's theorem.

Contribution

It introduces the concept of Lf-functions for Hurewicz fibrations and demonstrates their use in classifying fibrations and fiber bundles over polyhedron bases, linking to existing theorems.

Findings

Lf-functions help determine fiber homotopy equivalence.

Results generalize Dold's theorem for fiber bundles.

Provides criteria for fiber bundle classification over polyhedra.

Abstract

Let $f:E\longrightarrow O$ be a Hurewicz fibration with a fiber space $F_{r_{o}}$ and a lifting function $L_{f}$. The \emph{$Lf-$function} $\Theta_{L_{f}}$ of $f$ is defined by the restriction map of $L_{f}$ on the space $\Omega(O,r_{o})\times F_{r_{o}}\times \{1\}$. The purpose of this paper is to give some results which show the role of $Lf-$functions in finding a fiber homotopically equivalent relation between two fibrations, over a common polyhedron base. Furthermore we will prove the equivalently between our results and Dold's theorem in fiber bundles, over a common suspension base of polyhedron spaces.