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arXiv · 2508.14341

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

Zhongjian Zhu·Jianzhong Pan
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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].

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Zhongjian Zhu, Jianzhong Pan. 2025-08-20. Homotopy classification of $S^{2k-1}$-bundles over $S^{2k}$. https://arxiv.org/abs/2508.14341

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