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In Milnor's paper "On manifolds homeomorphic to the 7-sphere" it is proven that there are manifolds homeomorphic but not diffeomorphic to the standard 7-sphere. His construction involves sphere bundles over $S^4$. Also, Kervaire and Milnor proved that there are exactly 28 h-cobordism (therefore diffeomorphism) classes of homotopy spheres in dimension 7.

Does every class contain an exotic sphere arising as the total space of an $S^3$-bundle over $S^4$?, if not, how can one determine the number of classes with such representatives?, how do they look like?

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  • $\begingroup$ When you say "$S^3$-bundle over $S^4$," I would assume you mean a sphere bundle associated with a rank 4 vector bundle over $S^4$, right? $\endgroup$
    – John Klein
    Commented May 16, 2012 at 2:51
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    $\begingroup$ John Klein: linear $S^3$-bundles are the only ones that make sense here as any smooth $S^3$-bundle is linear by Smale conjecture. $\endgroup$
    – Igor Belegradek
    Commented May 16, 2012 at 3:33
  • $\begingroup$ @Igor: yes, I'm perfectly aware of that. But for completely dumb (and pedantic) reasons, any exotic $7$-sphere always fibers over $S^4$ with structure group $\text{Top}(S^3)$. $\endgroup$
    – John Klein
    Commented May 16, 2012 at 4:13

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This is done in the paper "An invariant for certain smooth manifolds" by James Eells and Nicolaas Kuiper. They introduce and study the so called $\mu$-invariant which is strong enough to classify homotopy $7$-spheres up to oriented diffeomorphism. A theorem on page 103 says that out of 28 oriented differomorphism types of homotopy 7-spheres precisely 16 are realized by $S^3$-bundles over $S^4$. I am not sure what is the best way to visualize the exotic spheres that aren't sphere bundles but e.g. if memory serves, all homotopy $7$-spheres are Brieskorn spheres.

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    $\begingroup$ Yes, it's true all homotopy 7-spheres are Brieskorn spheres. $\endgroup$
    – John Klein
    Commented May 16, 2012 at 2:49
  • $\begingroup$ the formula is given in this mathoverflow answer: mathoverflow.net/questions/14574/… $\endgroup$
    – Will Sawin
    Commented May 16, 2012 at 3:24
  • $\begingroup$ Will Sawin: what formula? $\endgroup$
    – Igor Belegradek
    Commented May 16, 2012 at 3:33
  • $\begingroup$ @ Igor, I think Will Sawin means the equation for the Brieskorn spheres. $\endgroup$
    – Xiaolei Wu
    Commented May 16, 2012 at 4:10
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    $\begingroup$ @Igor: aha, I made an error with respect to what it means to reduce the structure group, namely, if $\Sigma$ denotes and\ exotic $7$-sphere, then a choice of homeomorphism $\Sigma \cong S^7$ together with the Hopf map $S^7 \to S^4$ gives a Top fiber bundle $\Sigma \to S^4$ with fiber $S^3$. The reduction of structure group says that this Top bundle lifts to a smooth one, but the total space of the lift (which is a smooth manifold, possibly $S^7$, if the lift is chosen suitably) is only homeomorphic to $\Sigma$. My mistake was in thinking it might be diffeomorphic. I retract my previous remark. $\endgroup$
    – John Klein
    Commented May 16, 2012 at 16:35

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