Differential structures over a topological manifold

[cianfa72]

TL;DR

About the definition of different differential structures over a given topological manifold

Given a topological manifold, this may or may not admit a $C^1$ atlas (i.e. starting from its maximal atlas it is or it isn't possible to rip charts from it to get an atlas of $C^1$ compatible charts).

A theorem due to Whitney states that from such a topological manifold $C^1$-atlas (if any) is always feasible to rip charts to get a $C^{\infty}$ atlas.

My question is: starting from such a $C^1$ atlas could there be the case that one can "extract" two different $C^{\infty}$ atlases whose charts are not compatible each other (even when the differential structures they define turn out to be diffeomorphic) ?

 

[pasmith]

A "$C^k$ differentiable structure" is the same thing as a maximal atlas of $C^k$ compatible atlases.

It should be fairly easy to show that if the identity is a $C^k$ diffeomorphism between two $C^k$ atlases then those atlases are $C^k$ compatible, and therefore part of the same maximal atlas.

 

[cianfa72]

It should be fairly easy to show that if the identity is a $C^k$ diffeomorphism between two $C^k$ atlases then those atlases are $C^k$ compatible, and therefore part of the same maximal atlas.

Yes of course. However I'm not sure how it answers my OP question though.

Namely: could there be two non compatible $C^{\infty}$ atlases "extracted" from the same $C^1$ maximal atlas (by ripping charts from it as Whitney's theorem implies) ?

 

[martinbn]

Take for example $M=\mathbb R$ and a function that is invertible and $C^1$, and so is its inverse, but they are not $C^2$. I think $f(x)=x^{\frac53}$ will do. Then the two charts $(\mathbb R, Id)$ and $(\mathbb R, f)$ are $C^1$ compatible, but not $C^2$. So they belong to the same maximal $C^1$ atlas, and give you two different smooth structures (of course they are diffeomorphic).

 

[cianfa72]

Take for example $M=\mathbb R$ and a function that is invertible and $C^1$, and so is its inverse, but they are not $C^2$. I think $f(x)=x^{\frac53}$ will do. Then the two charts $(\mathbb R, Id)$ and $(\mathbb R, f)$ are $C^1$ compatible, but not $C^2$. So they belong to the same maximal $C^1$ atlas, and give you two different smooth structures (of course they are diffeomorphic).

Ok, thank you.

Therefore yours is an example of two not $C^k, k \geq 2$-compatible atlases each of $C^{\infty}$ class that can be "extracted" from the same $C^1$ maximal atlas.