representatives of the group of homotopy 7-spheres - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T03:42:40Z http://mathoverflow.net/feeds/question/97073 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97073/representatives-of-the-group-of-homotopy-7-spheres representatives of the group of homotopy 7-spheres Mauricio 2012-05-16T00:42:10Z 2012-05-16T01:47:01Z <p>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. </p> <p>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?</p> http://mathoverflow.net/questions/97073/representatives-of-the-group-of-homotopy-7-spheres/97075#97075 Answer by Igor Belegradek for representatives of the group of homotopy 7-spheres Igor Belegradek 2012-05-16T01:47:01Z 2012-05-16T01:47:01Z <p>This is done in the paper <a href="http://www.springerlink.com/content/75m7x61uq2966987/" rel="nofollow">"An invariant for certain smooth manifolds"</a> 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 <a href="http://en.wikipedia.org/wiki/Brieskorn_sphere" rel="nofollow">Brieskorn spheres</a>. </p>