# representatives of the group of homotopy 7-spheres

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?

-
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

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.