On classifying Hurewicz fibrations and fibre bundles over polyhedron bases

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.

💡 Research Summary

The paper investigates the classification of Hurewicz fibrations and fiber bundles whose base spaces are polyhedra, introducing a novel invariant called the Lf‑function. For a Hurewicz fibration (f\colon E\to O) with a chosen lifting function (L_f\colon \Omega(O,r_0)\times F_{r_0}\times I\to E) (where (\Omega(O,r_0)) denotes the based loop space of the base (O) and (F_{r_0}=f^{-1}(r_0)) is the fiber over the base point), the authors define the Lf‑function as the restriction
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