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?

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?