Welcome to MathOverflow

Question

5

2

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?

flag	asked May 9 2012 at 17:34 aglearner 3,868●8●24

3

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. – Vitali Kapovitch May 9 2012 at 20:05

3

@aglearner: Reinhard Schultz had a sequence of papers titled "Differentiable group actions on homotopy spheres" in 1980s, you should check them for such examples. – Misha May 9 2012 at 20:45

you might also look at Weinberger's articles on propagating group actions, e.g. ams.org/mathscinet-getitem?mr=910951 – Paul May 10 2012 at 1:41

Mauricio · Answer 1 · 2012-05-09 23:01:27Z UTC

2

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...

edited May 9 2012 at 23:01

Mauricio

community wiki

2 revisions

1

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. – Vitali Kapovitch May 9 2012 at 20:07

Mauricio, where can I read about the fact that exotic 7-spheres don't admit an orientation reversing diffeo? – aglearner May 9 2012 at 23:14

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) – Francesco Lin May 9 2012 at 23:23

In Milnor's original paper or these notes nd.edu/~lnicolae/MS.pdf by Liviu Nicolaescu. – Mauricio May 9 2012 at 23:28

6

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. – Vitali Kapovitch May 10 2012 at 0:27

show 1 more comment

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

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.