Differential structure on topological manifolds of dimension <= 3

[cianfa72]

TL;DR Summary

About the unique smooth structure on topological manifold of dimension less then equal 3

Hi,
From Lee book "Introduction on Smooth Manifolds" chapter 2, every topological manifold (Hausdorff, locally Euclidean, second countable) of dimension less then or equal 3 has unique smooth structure up to diffeomorphism.

A smooth structure on a manifold is defined by a maximal atlas.

So, why diffeomorhpisms are taken in account in the above statement ? It is actually equivalent to the claim that a given topological manifold of dimension less then equal 3 has got an unique maximal atlas.

 

[martinbn]

...
So, why diffeomorhpisms are taken in account in the above statement ? ...

Because there are different ones, but they are diffeomorphic. You can have two maximal atlases which are incompatible.

 

[cianfa72]

Because there are different ones, but they are diffeomorphic. You can have two maximal atlases which are incompatible.

Can you give an example of the above statement ? Thanks.

 

[martinbn]

Can you give an example of the above statement ? Thanks.

Consider the manifold $\mathbb R$ with a global chart the identity map. Let $\mathcal A_1$ be the maximal atlas of charts compatible with the given chart.

Then consider $\mathbb R$ with one global chart given by the map $x \mapsto x^3$ and $\mathcal A_2$ be the maximal atlas of charts compatible with the given chart.

 

[cianfa72]

Ok, the map $x \mapsto x^{1/3}$ is a global diffeomorphism between manifolds $(\mathbb R, \mathcal A_1) \mapsto (\mathbb R, \mathcal A_2)$.
Yet, the transition map between $\mathcal A_1$'s and $\mathcal A_2$'s global charts $$x \mapsto x^3$$ is not a diffeomorphism on $\mathbb R^n$ (endowed with its standard smooth structure).