Given a positive integer $n$, there is a well known free action of $\mathbb T^1$ on $\mathbb S^{2n-1}$ due to Hopf, which makes $\mathbb S^{2n-1}$ a fibre bundle with the fibre $\mathbb T^1$. Moreover, for a few triples of positive integers $k,l,m$, $\mathbb S^l$ is a fibre bundle over $\mathbb S^m$ with the fibre $\mathbb S^k$.

Could you please present other, maybe less known, examples of structures of a fibre bundle on spheres $\mathbb S^n$ for various (classes of) $n$?

  • 5
    $\begingroup$ There's a theorem of Adams that tells you that the only fiber bundles which have spheres as base space, total space and fiber, are the four Hopf bundles. $\endgroup$ Feb 10, 2017 at 10:48
  • 6
    $\begingroup$ Also Browder showed: if the base is not a point and the base and fiber are CW complexes of finite type, then the fiber is homotopy equivalent to $S^1$, $S^3$, or $S^7$. $\endgroup$ Feb 10, 2017 at 11:01


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