Some applications for Fadell-Dold theorm in fibration theory by using homotopy groups

TL;DR

This paper explores how homotopy groups and sequences can be used to classify fibrations and construct fiber maps, extending the application of the Fadell-Dold theorem in fibration theory.

Contribution

It demonstrates the role of homotopy sequences of fibrations in solving classification problems and constructing fiber maps in the context of the Fadell-Dold theorem.

Findings

Homotopy sequences help classify fibrations.

Constructed fiber maps are homotopy equivalences on fibers.

Extended the application of Fadell-Dold theorem using homotopy groups.

Abstract

The purpose of this paper is to give some solutions for the classification problem in fibration theory by using the homotopy sequences of fibrations (sequences of $n$-th homotopy groups $ \pi_{n}(S,s_{o}) $ of total spaces of fibrations). In particular, to show the role of homotopy sequence of $n$-th homotopy to get the required fiber map in Fadell-Dold theorem such that the restriction of this fiber map on some fiber spaces is a homotopy equivalence.