How can I prove that Hopf fibrations are the only ones with fiber, total space and base space homeomorphic to spheres?

Question

I know that Hopf fibrations (the four ones) are the only ones that have the form $S^k \to S^m \to S^n$, but I never seen a proof. Could anyone link me a paper or text where this is proved, or prove it for me?

Answer 1

http://en.wikipedia.org/wiki/Adams%27_theorem

Adams, J.F. (1960), "On the non-existence of elements of Hopf invariant one", Ann. Math. (The Annals of Mathematics, Vol. 72, No. 1) 72 (1): 20–104.

http://www.maths.ed.ac.uk/~aar/papers/adams1.pdf‎

http://dx.doi.org/10.2307%2F1970147

Answer 2

The result can be deduced from Adams' Hopf invariant one theorem using the Gysin sequence and Leray-Hirsch. The details can be found in this expository article.