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?



representatives of the group of homotopy 7-spheres
This is done in the paper ["An invariant for certain smooth manifolds"](http://www.springerlink.com/content/75m7x61uq2966987/) 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](http://en.wikipedia.org/wiki/Brieskorn_sphere). 


