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

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.

💡 Research Summary

The paper addresses a classical classification problem in fibration theory: determining when two fibrations over the same base space share a fiber that is homotopy equivalent, thereby making the total fibrations equivalent in the sense of Fadell‑Dold. Traditional applications of the Fadell‑Dold theorem require an explicit fiberwise homotopy equivalence, a condition that is often difficult to verify directly. The author proposes a novel approach that replaces this geometric requirement with algebraic conditions on the homotopy groups of the fibers, using the long exact sequences of homotopy groups associated with each fibration.

The work begins by recalling the long exact sequence for a fibration (p:E\to B) with fiber (F_b): \