Is it known whether, for any fibration of a sphere with connected fibres, the fibre has to be a sphere? $S^1$, $S^3$ or $S^7$?
It was shown in
Browder, William. Fiberings of spheres and $H$-spaces which are rational homology spheres. Bull. Amer. Math. Soc. $\bf 68$ 1962 202–203.
that for any fibration $F \to S^n \to B$, where $B$ is a non-trivial polyhedron and $F$ is connected, $F$ must have the homotopy type of $S^1,S^3$ or $S^7$. This builds on previous work of Spanier-Whitehead and Borel.