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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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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? – John Klein May 16 '12 at 2:51
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. – Igor Belegradek May 16 '12 at 3:33
@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 $\text{Top}(S^3)$. – John Klein May 16 '12 at 4:13
up vote 15 down vote accepted

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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Yes, it's true all homotopy 7-spheres are Brieskorn spheres. – John Klein May 16 '12 at 2:49
the formula is given in this mathoverflow answer:… – Will Sawin May 16 '12 at 3:24
Will Sawin: what formula? – Igor Belegradek May 16 '12 at 3:33
@ Igor, I think Will Sawin means the equation for the Brieskorn spheres. – Xiaolei Wu May 16 '12 at 4:10
That is what I meant. Sorry for the confusion. – Will Sawin May 16 '12 at 6:41

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