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Let $M_1^n$ and $M_2^n$, $n>4$ be two smooth compact manifolds that are homeomorphic but not diffeomorphic. Suppose that a finite group is $G$ acting faithfully on $M_1^n$ by diffeomorphisms. Is it true that $G$ admits as well a faithful action on $M_2^n$ by diffeomorphisms?

If no, what would be the a (relatively) simple example?

For example, can one differentiate exotic structures on $S^7$ this way?

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    $\begingroup$ there is absolutely no reason to expect for something like this to be true. Finding explicit counterexamples is quite a different story. I don't know of any examples in dimensions above 4 if you don't restrict the action in any way. For example I believe it's not known if every exotic sphere admits a circle action. $\endgroup$
    – Vitali Kapovitch
    Commented May 9, 2012 at 20:05
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    $\begingroup$ @aglearner: Reinhard Schultz had a sequence of papers titled "Differentiable group actions on homotopy spheres" in 1980s, you should check them for such examples. $\endgroup$
    – Misha
    Commented May 9, 2012 at 20:45
  • $\begingroup$ you might also look at Weinberger's articles on propagating group actions, e.g. ams.org/mathscinet-getitem?mr=910951 $\endgroup$
    – Paul
    Commented May 10, 2012 at 1:41

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I think the answer is no. But I don't know the reason. I take the fact that exotic spheres in dimension 7 do not admit orientation reversing diffeomorphisms (but the standard sphere does) as a clue to say that one shouldn't expect an affirmative answer to your question.

EDIT: I gave a wrong answer and I'm not sure if I can delete the post. I decided to change my answer by a comment and make it CW. My apologies to those who read this expecting to see a right answer...

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    $\begingroup$ The OP doesn't link the actions of $G$ on $M_1$ and $M_2$ in any way. so the action of $Z_2$ on an exotic sphere is not required to be orientation reversing. $\endgroup$
    – Vitali Kapovitch
    Commented May 9, 2012 at 20:07
  • $\begingroup$ Mauricio, where can I read about the fact that exotic 7-spheres don't admit an orientation reversing diffeo? $\endgroup$
    – aglearner
    Commented May 9, 2012 at 23:14
  • $\begingroup$ In Milnor's original paper jstor.org/stable/1969983 he constructs some manifolds homeomorphic to the 7-sphere which do not admit orientation reversing diffeos (and so are exotic spheres) $\endgroup$
    – Francesco Lin
    Commented May 9, 2012 at 23:23
  • $\begingroup$ In Milnor's original paper or these notes nd.edu/~lnicolae/MS.pdf by Liviu Nicolaescu. $\endgroup$
    – Mauricio
    Commented May 9, 2012 at 23:28
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    $\begingroup$ note that not every exotic sphere does not admit an orientation reversing diffeo. most don't but some do. oriented diffeomorphism classes of exotic spheres form a finite group under connected sum. the spheres that admit orientation reversing diffeomorphisms are exactly the elements of order 2 in that group. $\endgroup$
    – Vitali Kapovitch
    Commented May 10, 2012 at 0:27

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