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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 Hartman 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 Hartman Dec 10 '12 at 19:59

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up vote 5 down vote accepted

As far as I know the only explicitly-described such bundles are in Hatcher's paper:

  • Hatcher. Concordance spaces, higher simple-homotopy theory, and applications. Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ, Stanford Calif 1976), Part 1, pp. 321.

I think in Igusa's Higher Franz Reidemeister Torsion book there might also be these examples, although I don't have the book at home with me so I can't check. But that seems likely as Igusa has also developed these examples.

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Thank you, I will definitely take a look at Hatcher's paper. –  Dave Hartman Dec 10 '12 at 20:39
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These examples are fairly high-dimensional to get to the range where pseudo-isotopies are relatable to k-theory. So they take some time to get used to -- they're described in terms of isotopy extensions. –  Ryan Budney Dec 10 '12 at 20:44

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