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

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.