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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?

Mauricio
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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 02:51
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    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 03:33
  • @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 04:13

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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.

Igor Belegradek
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    Yes, it's true all homotopy 7-spheres are Brieskorn spheres. – John Klein May 16 '12 at 02:49
  • the formula is given in this mathoverflow answer: http://mathoverflow.net/questions/14574/your-favorite-surprising-connections-in-mathematics/15500#15500 – Will Sawin May 16 '12 at 03:24
  • Will Sawin: what formula? – Igor Belegradek May 16 '12 at 03:33
  • @ Igor, I think Will Sawin means the equation for the Brieskorn spheres. – Xiaolei Wu May 16 '12 at 04:10
  • That is what I meant. Sorry for the confusion. – Will Sawin May 16 '12 at 06:41
  • @Igor: $\text{Top}(S^3) \simeq S^3 \times PL_3$ (using the Alexander trick). But $PL_3 \simeq O_3$. Also, $\text{Diff}(S^3) \simeq S^3 \times O_3$ by Hatcher. We infer that $\text{Diff}(S^3) \simeq \text{Top}(S^3)$. So, if this argument is correct (?) we see that a fiber bundle with structure group $\text{Top}(S^3)$ admits a reduction of structure group to $\text{Diff}(S^3)$. This would seem to contradict your answer, given that any exotic $7$-sphere fibers topologically over $S^4$ (cf. above). Where's my mistake? – John Klein May 16 '12 at 14:59
  • @John: I am not sure how what you say contradicts Eells-Kuiper. Smale conjecture does imply (via Cerf's work) that the inclusion $O(4)\to Homeo(S^3)$ is a weak homotopy equivalence. Thus any topological $S^3$ bundle is topologically isomorphic to a linear $S^3$-bundle. Any homotopy $7$-sphere $\Sigma$ is homeomorphic to $S^7$, so as you say, $\Sigma$ is homeomorphic to the total space of an $S^3$-bundle over $S^4$, e.g. the standard Hopf bundle. This does not imply that $\Sigma$ is diffeomorphic to total space of a smooth $S^3$-bundle over $S^4$. Where is the contradiction? – Igor Belegradek May 16 '12 at 16:04
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    @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. – John Klein May 16 '12 at 16:35
  • The link to springerlink.com is broken, but the article can be found at doi:10.1007/BF02412768 (Zbl 0119.18704). – The Amplitwist Apr 24 '23 at 18:24
  • Reposting the link mentioned in a previous comment so that it appears in the "Linked" questions list: Csar Lozano Huerta's answer to "Your favorite surprising connections in mathematics" – The Amplitwist Apr 24 '23 at 18:27