# Is any connected fibre of a fibration of a sphere also a sphere?

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 $S^7$?

• No, obviously not, because every projection map $S^n \times F \to S^n$ is a fibration. – Johannes Hahn Dec 7 '14 at 12:38
• @Johannes Hahn: I think the question wants the sphere as total space: a fibration of a sphere, not over a sphere. – Oscar Randal-Williams Dec 7 '14 at 12:46

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.
• Minor quibble: $B$ cannot be a point. – John Klein Dec 7 '14 at 15:54
• I think one should also add $S^0$ to the list. – Alex Degtyarev Dec 7 '14 at 20:21
• Right, $F$ must be connected - I added this. – Andreas Thom Dec 7 '14 at 22:24