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Can one give an immersion of exotic sphere $S^7$ in a standard sphere $S^8$ of radius $1$?

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Dear Li Yan. You might want to explain what you mean by "exhaustive", and also what you mean by the notation S^8(1). I don't know what these terms mean, and I think that other MO users will also not know. Please edit your question to add this information. –  André Henriques Jun 22 '13 at 13:21
Sorry,a serious mistake! I mean " John Milnor's exotic spheres"."S^(R)"means n-sphere of radius R. Li Yan –  Li Yan Jun 22 '13 at 13:42
Note that the radius of the sphere is not important since $S^8$ of different radii are diffeomorphic. –  Lennart Meier Jun 26 '13 at 8:18
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1 Answer

Yes. By Smale-Hirsch theory it is enough to find a bundle injection $T\Sigma \to \epsilon^8$, so it is enough to find a trivialisation of $T\Sigma \oplus \epsilon^1$. It is a theorem of Kervaire and Milnor that every exotic sphere is stably framable, so $T\Sigma \oplus \epsilon^N$ is trivial for some large N, and the connectivity of $BO(8) \to BO$ means you can destabilise this to trivialise $T\Sigma \oplus \epsilon^1$.

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Wow! Does this also allow one to say something meaningful about the complement, e.g. is it two standard open balls? –  Allen Knutson Jun 26 '13 at 11:17
No--this is an immersion, not an embedding. If you had an embedding, then the allegedly exotic sphere would bound a contractible manifold, and hence be diffeomorphic to $S^7$ by the h-cobordism theorem. One probably could say something interesting about the topology of the double point set, though. –  Danny Ruberman Jun 26 '13 at 11:28
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