Actions of finite groups on exotic smooth manifolds of dimension >4

2012-05-09T17:34:14Z

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?

Answer by Mauricio for Actions of finite groups on exotic smooth manifolds of dimension >4

2012-05-09T19:28:35Z

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...

Actions of finite groups on exotic smooth manifolds of dimension >4 - MathOverflow

Actions of finite groups on exotic smooth manifolds of dimension >4

Answer by Mauricio for Actions of finite groups on exotic smooth manifolds of dimension >4