Complexes equivalent to $S^{2k-1}$-fibrations over $S^{2k}$

TL;DR

This paper establishes criteria for when CW complexes are homotopy equivalent to total spaces of $S^{2k-1}$-fibrations over $S^{2k}$, and classifies their homotopy types based on attaching maps.

Contribution

It provides necessary and sufficient conditions for such CW complexes to have the homotopy type of these fibrations, and classifies their homotopy types via stable homotopy classes of attaching maps.

Findings

Determined the order of attaching maps for these fibrations.

Established conditions for homotopy equivalence to $S^{2k-1}$-fibration total spaces.

Classified homotopy types by stable homotopy classes for most cases.

Abstract

In this paper, necessary and sufficient conditions are obtained for the attaching map $f$ of the top cell of a CW complex to have the homotopy type of the total space of $S^{2k-1}$-fibration over $S^{2k}$ for any $k\geq 2$. As an application, the order of any attaching map of the top cell of the total space of an $S^{2k-1}$-fibration over $S^{2k}$ is determined and when $k\ne 2,4$, the homotopy types of the total spaces of $S^{2k-1}$-fibrations over $S^{2k}$ are classified by the stable homotopy classes of the attaching maps.