# How does the cokernel of the J-homomorphism count exotic spheres?

The wikipedia article on the J-homomorphism says that "the cokernel of the J-homomorphism is of interest for counting exotic spheres". I'd like to think this makes some sort of philosophical sense; as I understand it, the homomorphism comes from a Hopf construction, which isn't really a smooth sort of thing, but on the other hand it still feels like constructions using the (stable) (special) orthogonal group should somehow keep us in the non-exotic world. Is this at all close to the right intuition?

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You're not really asking a concrete question. It's probably better to open a book like Kosinski's "Differential Manifolds" where these constructions are explained in detail. Kosinski was Dovered recently so can be found on sites like Amazon for $5, new. – Ryan Budney Oct 10 '10 at 18:12 Another good source for this stuff is Ranicki's book "Algebraic and Geometric Surgery". However, Kervaire and Milnor's original paper is actually probably even more readable than the secondary literature. – Andy Putman Oct 10 '10 at 18:53 * vote to close (apparently I can't even close my own question, it still needs 5 votes) – Aaron Mazel-Gee Oct 10 '10 at 19:36 One of the many nice features of Kosinski's book is he gets rid of the rather ugly "smoothing of corners" issue which is present in Smale's proof of the h-cobordism theorem. – Ryan Budney Oct 10 '10 at 22:53 Too layman for an answer, but a very rough intuition is this - if you glue two copies of the$n+1\$-ball using a self-diffeomorphism of its boundary, you get a topological sphere which is non-exotic if this diffeomorphism is an orhtogonal transformation. –  მამუკა ჯიბლაძე Jan 16 at 9:13