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The only theory of exotic spheres that I know is of $C^\infty$ structures on them; that is, that there are plenty of spheres (in dimensions $n \geq 7$ that are homeomorphic but not diffeomorphic. To keep this question reasonable, I'll restrict to the $n=7$ case.

Are the exotic $S^7$s diffeomorphic for any $C^k$ with $k \geq 1$? Has there been any serious study about the structure of (the cobordism group of) $C^k$ exotic spheres for finite positive $k$? If so, what are some references?

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    $\begingroup$ I've added the homotopy-theory tag because I'm aware that the story of exotic spheres is closely related to some questions in homotopy theory. Apologies if the tag was chosen poorly. $\endgroup$
    – user98602
    Oct 29, 2014 at 22:50
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    $\begingroup$ Since nothing interesting happens in the $C^k$ category, you might be interested in looking into the question for PL structures. $\endgroup$
    – anomaly
    Oct 29, 2014 at 23:07

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Any $C^k$ structure on a manifold for $k > 0$ can be uniquely promoted (modulo diffeomorphism) to a $C^\infty$ structure; furthermore, the corresponding map between $C^k$ structures modulo equivalence to $C^\infty$ structures modulo equivalence is bijective. See this MathOverflow question, for example.

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  • $\begingroup$ This is incredible! Thank you. $\endgroup$
    – user98602
    Oct 29, 2014 at 23:05
  • $\begingroup$ No thanks are necessary, but you're quite welcome. $\endgroup$
    – anomaly
    Oct 29, 2014 at 23:07

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