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Consider the principal bundle $S^7$ with base space $\mathbb{C}P^3$ (3-dimensional complex projective space) and fiber $S^1\cong U(1)$. Can someone write to me the bundle projection $\pi:S^7\rightarrow\mathbb{C}P^3$ explicitly?

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Regard $S^7$ as the set of points in $\mathbb C^4$ of length $1$. If $p:\mathbb C^4\to\mathbb CP^3$ is the usual quotient map, then the restriction $p|_{S^7}:S^7\to\mathbb CP^3$ is the projection you are after.

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    $\begingroup$ This, of course, works for other odd values of $7$. $\endgroup$ – Mariano Suárez-Álvarez Mar 14 '11 at 22:10
  • $\begingroup$ ok, I'll do that, thanks $\endgroup$ – Dar Far Mar 14 '11 at 22:12

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