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

Zhongjian Zhu; Jianzhong Pan

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

Algebraic Topology 2026-04-17 v2

Authors: Zhongjian Zhu , Jianzhong Pan

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].

Keywords

homotopy classification sphere bundles total spaces CW complexes attaching maps characteristic maps algebraic topology

Cite

@article{arxiv.2508.14341,
  title  = {Homotopy classification of $S^{2k-1}$-bundles over $S^{2k}$},
  author = {Zhongjian Zhu and Jianzhong Pan},
  journal= {arXiv preprint arXiv:2508.14341},
  year   = {2026}
}

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

Abstract

Keywords

Cite

Related papers