Take the 2-minute tour ×  Sphere bundles and bundles over
spheres are everywhere and are excellent things to get one's
hands dirty with.  (1a) But when can we have a bundle $S^n \to
S^m \to B?$ It seems like requiring the total space of a sphere
bundle to be a sphere is pretty restrictive.  (1b) Does the
answer to (1a) depend on a choice of category (PL, TOP, etc)?
There's an $S^1-$ bundle over $CP^1$ with total space $S^3$, but
that's the only example I can find.  (2) Do people know examples
other than the one above?  share|improve this question How about
the Hopf bundles? Those are the only bundles where the fibre,
total space and base are all spheres -- a result, I believe, due
to Frank Adams? Experts here will correct me if I'm wrong. – 
José Figueroa-O'Farrill Nov 3 '10 at 2:21 José's comment came at
the same time as my answer. I believe him that they're the only
examples. –  Spiro Karigiannis Nov 3 '10 at 2:29 Here we're
allowing the base to be arbitrary. –  Romeo Nov 3 '10 at 2:46
The answer to (1b) i