# Examples of sphere bundles

For certain values of $k$ it is known that $\mbox{Diff}(S^k)$ is not homotopy equivalent to $O(k+1)$. So there are sphere bundles that do not arise from vector bundles.

Since I've never (knowingly) come across such a sphere bundle I'm interested in seeing some enlightening examples of sphere bundles which do not come from vector bundles.

Thank you for any contribution.

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If I may drag stable homotopy into it: the map assigning a stable spherical bundle to a stable vector bundle can be realized as a map $BO \to BGL_1 S$. Then, $BGL_1 S \simeq \Omega^\infty \Sigma^\infty S^0$, so in principle it should be easy to name a few such bundles: pick any element of $\pi_*^S$ which is guaranteed, for reasons of order, not to be in the image of this first map (since $\pi_m BO$ contains $2$-torsion at most). However, writing down geometric presentations of such bundles may be hard, and as these are stable statements, they will at best translate unstably only in a range... –  Eric Peterson Dec 10 '12 at 18:41
I would like to mention that I have posed this question on Stack Exchange a few days ago. Since no one has posted an actual example it was suggested to me that I should ask here on MathOverflow. –  Dave Dec 10 '12 at 19:30
Hatcher has some notes (though no explicit example) - check out p. 8:math.cornell.edu/~hatcher/Papers/Diff(M)2012.pdf –  Ian Agol Dec 10 '12 at 19:50
@Agol : Thank you for the link. Any references related to this question are much appreciated. –  Dave Dec 10 '12 at 19:59