Homotopy classification of $S^{2k-1}$-bundles over $S^{2k}$

TL;DR

This paper classifies the homotopy types of total spaces of sphere bundles over spheres for specific dimensions, introducing new criteria and formulas that lead to a counterexample to a previous conjecture.

Contribution

It provides new necessary and sufficient conditions for a CW complex to be a sphere bundle and a formula relating attaching maps and characteristic maps for certain dimensions.

Findings

Classifies homotopy types of $S^{2k-1}$-bundles over $S^{2k}$ for 2 ≤ k ≤ 6.

Introduces new criteria for recognizing total spaces of sphere bundles.

Provides a counterexample to a conjecture in the case k=4.

Abstract

In this paper, we classify the homotopy types of the total spaces of $S^{2k-1}$-bundles (or fibrations) over $S^{2k}$ for $2\leq k\leq 6$. One of the two key new ingredients in the argument is the new necessary and sufficient conditions for a CW complex to be homotopy equivalent to the total space of a sphere bundle (fibration); the other is a formula relating the attaching map of the top cell of the total space and the characteristic map of a sphere bundle for $k=2,4$. When $k=4$, the classification results provide a negative answer to the conjecture in [6].