I Differential structures over a topological manifold

cianfa72
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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) ?
 
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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.
 
pasmith said:
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) ?
 
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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).
 
martinbn said:
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.
 

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