From: Siu Por Lam <siuporlamlam@yahoo.co.uk>
Date: Sun, 11 Mar 2007 14:26:02 +0000 (GMT)
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A belated response to a question about Hopf bundles (Date: Fri, 29 Sep 
2006)
To: dmd1@lehigh.edu, villarini <villarini@unimo.it>

The question posed by villarini was related to the
part of the Blaschke conjecture about fibering odd
spheres by great circles. This was answered in the
positive by the combined efforts of Gluck & Warner
(3-sphere), CT Yang (for odd spheres of dimensions at
least 7) and B. Mckay (for 5-sphere). Precisely, there
is a diffeomorphism carrying an odd sphere fibered by
great circles to the Hopf fibration. 

But for an odd sphere fibered by circles (not
necessarily great circles), one can settle the cases
for 3-sphere and 5-sphere up to homeomorphism. For a
possible counter-example in higher dimensions, one
might like to look at the homotopy complex projective
spaces. 

For a 5-sphere with a free smooth circle action, the
base is simply connected and has the homology of CP2.
Since  the base is smooth, the classification (M.
Freedman) of simply connected compact closed manifolds
says the base is topologically CP2. 

The argument for a 3-sphere is similar but simpler.

A reference for the Blaschke conjecture is: THE
BLASCHKE CONJECTURE AND GREAT CIRCLE FIBRATIONS OF
SPHERES, by BENJAMIN MCKAY, Math Archive, 0112027. And
there are further refernces therein.

Siu P. Lam
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ORIGINAL MESSAGE BELOW

Subject: isomorphisms of hopf bundles
From: villarini <villarini@unimo.it>
Date: Fri, 29 Sep 2006 14:50:08 +0200

I would like to post the following question to people
in the list (hoping 
it is not toot trivial!..I am not a topologist..):

Question:
let S1-->S^2n+1-->CP^n be the hopf bundle
let S1-->S-->S/S1 be another circle bundle, and let S
be diffeomorphic 
(not equivariantly) to S^2n+1

Problem: is it true that the two bundles are always
diffeomorphic, or 
homeomorphic? (I think it is true if n=1 and probably
false if n>1...)
In case of negative answer, can anyone suggest me a
counterexample?

thank you, and best regards

massimo villarini
I