I 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).

 

[cianfa72]

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

Sorry to resume this thread. Yes the two maximal atlas can be incompatible at $C^1$-level or beyond, nevertheless by definition they are always $C^0$-compatible.

 

[jbergman]

Sorry to resume this thread. Yes the two maximal atlas can be incompatible at $C^1$-level or beyond, nevertheless by definition they are always $C^0$-compatible.

Yes, that seems self-evident if you know the definition of a $C^k$ atlas, i.e. a maximal set of homeomorphisms from the open sets of $M$ to $\mathbb R^n$ that are $C^k$ compatible.

What is more interesting I think, is that you can have incompatible smooth structures of a manifold that are nonetheless diffeomorphic or you can have have incompatible smooth structures that are not diffemorphic like the exotic smooth structures discussed in the thread, https://www.physicsforums.com/threa...f-exotic-smoothness-in-and-only-in-4d.936465/

 

[cianfa72]

What is more interesting I think, is that you can have incompatible smooth structures of a manifold that are nonetheless diffeomorphic or you can have have incompatible smooth structures that are not diffemorphic.

Yes, the main point I recognized only recently, is that for the notion of continuity (on which homeomorphisms are grounded on) one can define a pre-existing structure on the manifold itself (i.e. a locally euclidean topology) independently from the definition of an atlas.

On the other hand, for the notion of smooth/differentiability (i.e. differential structure) one can't define such a pre-existing structure on manifolds since it is "lifted" there by the definition of a smooth atlas (as a slogan: no atlas no differential structure on manifolds ).

Therefore, of course, it might be the case that two differential/smooth structures happen to be incompatible.

 

[lavinia]

Ok. So the next step here would be to show that for one to three dimensional manifolds that they all have a differentiable structure that is unique up to diffeomorphism.

Would someone like to start with the circle?

 

[WWGD]

Ok. So the next step here would be to show that for one to three dimensional manifolds that they all have a differentiable structure that is unique up to diffeomorphism.

Would someone like to start with the circle?

Well, we only have $S^1$, $\mathbb R$ as 1- manifolds. I'm curious as to why there are infinitely many smooth structures for n>3, but only one for Minkowski space time.

 

[cianfa72]

Well, we only have $S^1$, $\mathbb R$ as 1- manifolds.

Intuitively, thinking of 1d-manifolds as specific subsets of $\mathbb R^2$ with standard topology, it makes sense. However: can we conclude that there isn't any other Hausdorff, second-countable, locally euclidean topology for a 1d topological manifold?

 

[WWGD]

Intuitively, thinking of 1d-manifolds as specific subsets of $\mathbb R^2$ with standard topology, it makes sense. However: can we conclude that there isn't any other Hausdorff, second-countable, locally euclidean topology for a 1d topological manifold?

Another interesting result is that there are 4-manifolds that don't allow any differential structure; if the signature of the Intersection form doesn't divide 16. IIRC, Spin structures are also involved.
Edit: Maybe @lavinia knows how to create a 4-manifold with a given signature?