Differentiable structures and diffeomorphisms

[RedX]

The definition of having multiple differentiable structures is that given two atlases, {(U_i ,\phi_i)} and {(V_j,\psi_j)} (where the open sets are the first entry and the homeomorphisms to an open subset of Rⁿ are the second entry), that the union {(U_i,V_j;\phi_i,\psi_j)} is not necessarily an atlas. Those atlases whose union are atlases form an equivalence class and define one type of differentiable structure.

But how can a manifold depend on the atlas? For example, how can two different explorers navigate the same world and draw up two inequivalent atlases?

Also, what is the relationship between having multiple differentiable structures and diffeomorphisms? The book I'm reading seems to imply that all homeomorphisms are diffeomorphisms if there is only a single differentiable structure. Is this true, and why? It seems to me there should be no relation between homeomorphisms and diffeomorphisms, since homeomorphisms only ask for continuity (of the map and inverse map between manifolds), but diffeomorphisms require differentiability.

 

[lavinia]

But how can a manifold depend on the atlas? For example, how can two different explorers navigate the same world and draw up two inequivalent atlases?

on the real line make two different atlases. the first just maps the entire line to itself by the identity map. The second maps the line to itself by x -> x^3.

The two atlases are incompatible because the inverse of x^3 is not differentiable.

Nevertheless these two structures are diffeomorphic because the map x -> x^3 from the second atlas to the first is a diffeomorphism.

Yet x -> x^3 is only a homeomorphism of the first structure to itself since its inverse is not differentiable at 0.

It seems to me that there is a difference between compatibility classes of atlases as diffeomorphism classes of differentiable structures.

 

[RedX]

on the real line make two different atlases. the first just maps the entire line to itself by the identity map. The second maps the line to itself by x -> x^3.

The two atlases are incompatible because the inverse of x^3 is not differentiable.

Nevertheless these two structures are diffeomorphic because the map x -> x^3 from the second atlas to the first is a diffeomorphism.

Two topological spaces can be diffeomorphic to each other, but here we're dealing with one topological space, the real line, and two incompatible atlases that we call structures. When saying that a manifold is diffeomorphic, I think what is usually meant is that it is locally diffeomorphic to R^n. It seems that you are turning two atlases into topological spaces, and then seeing whether the two spaces are diffeomorphic. But if both atlases describe the same manifold, then isn't a manifold diffeomorphic to itself (the identity map is smooth)?

 

[Bacle]

RedX wrote:

" Two topological spaces can be diffeomorphic to each other, but here we're dealing with one topological space, the real line, and two incompatible atlases that we call structures. When saying that a manifold is diffeomorphic..."

I have never seen this termed used this way; I usually see M is diffeomorphic to N, or so



... I think what is usually meant is that it is locally diffeomorphic to R^n. ..."

In my experience this is just saying that your topological space is a manifold.


...It seems that you are turning two atlases into topological spaces...

I think it is the other way around: a fixed topological space --R -- with
two different --and incompatible --structures.

...and then seeing whether the two spaces are diffeomorphic. But if both atlases describe the same manifold, then isn't a manifold diffeomorphic to itself (the identity map is smooth)? ...

I think it is more accurate to say that incompatible atlases describe different
manifolds, and it is true in this case when you fix the underlying topological space.

the underlying space is always diffeomorphic to itself --just use the identity .

There are 3 issues here, as a I see it:
the underlying topology
the underlying structure
the diffeomorphism between the manifolds.


Still, outside of dimension 4, manifolds that are homeomorphic are also
diffeomorphic.

 

[lavinia]

Two topological spaces can be diffeomorphic to each other, but here we're dealing with one topological space, the real line, and two incompatible atlases that we call structures. When saying that a manifold is diffeomorphic, I think what is usually meant is that it is locally diffeomorphic to R^n. It seems that you are turning two atlases into topological spaces, and then seeing whether the two spaces are diffeomorphic. But if both atlases describe the same manifold, then isn't a manifold diffeomorphic to itself (the identity map is smooth)?

In the example I gave, there is one topological space and two different atlases that are incompatible. the two differentiable structures that these atlases determine are diffeomorphic.

In one of the atlases the identity is not a diffeomorphism.

A differentiable manifold is always locally diffeomorphic to R^n.

 

[zhentil]

It's actually a deep result, but there is no difference between homeomorphism and diffeomorphism on smooth manifolds of dimension less than 4. I.e. the smooth structure is unique up to diffeomorphism. R^4, however, has infinitely many smooth structures (i.e. there are infinitely smooth manifolds homeomorphic to R^4 but not pairwise diffeomorphic).

Of course there's a relation between homeomorphism and diffeomorphism, in that one is a subset of the other.