Diffeomorphism vs homeomorphism

[center o bass]

From a topological point of view a homeomphism is the best notion of equality between topological spaces. I.e. homeomorphisms preserve properties such as Euler characteristic, connectedness, compactness etc.

I've understood it such that diffeomorphisms are the best notion of equality between manifolds (basically a diffeomorphism is just a smooth homeomorphism), but what exactly are the properties that are preserved after a diffeomorphic map between manifolds (say M and N)?

If I would guess it's a diffeomorphism from M to N preserves all the properties that a homeomorphism preserves PLUS the differentiability of M (say it's C^k). But are there any more important properties to keep in mind?

 

[WWGD]

I'm not sure, but I think the difference is that homeomorphisms preserve the topology; global properties: compactness, connectedness, etc even orientability , I think (since a homeomorphism induces an isomorphism in homology, so that the fundamental class is sent to a non-zero class).; the properties that can be expressed in terms of open sets, but not necessarily the local properties --i.e., the ambient geometry. I think the standard example of a homeomorphism that is not a diffeomorphism is that of $f(x)= x^3 : \mathbb R \rightarrow \mathbb R$ (the inverse map $g(x)=x^{1/3}$ is not differentiable at 0) , and there is the interesting fact that there are homeomorphisms that are nowhere-differentiable. I think the fact that the two are homeorphic but not diffeomorphic implies that the standard copy of $(\mathbb R , id )$ is a submanifold of $\mathbb R^2$ , but the second copy is not; I guess you cannot find slice charts for the $x^3$ copy at x=0.

 

[jgens]

If I would guess it's a diffeomorphism from M to N preserves all the properties that a homeomorphism preserves PLUS the differentiability of M (say it's C^k). But are there any more important properties to keep in mind?

Additional properties preserved by such maps are invariants related to the tangent bundle and things like sets of immersions and embeddings.

 

[lavinia]

Additional properties preserved by such maps are invariants related to the tangent bundle and things like sets of immersions and embeddings.

Can you name an invariant?

 

[jgens]

Can you name an invariant?

The tangent bundle itself. Characteristic classes defined over tangent bundles. Things like that.

 

[WWGD]

Additional properties preserved by such maps are invariants related to the tangent bundle and things like sets of immersions and embeddings.

But aren't emebddings themselves preserved by the homeomorphism? Don't you need some smoothness condition so that homeomorphisms do not preserve the property? If you can describe the embeddings in terms of open sets and continuous chart maps, wouldn't the embedding be preserved under homeomorphisms?

 

[jgens]

But aren't emebddings themselves preserved by the homeomorphism? Don't you need some smoothness condition so that homeomorphisms do not preserve the property? If you can describe the embeddings in terms of open sets and continuous chart maps, wouldn't the embedding be preserved under homeomorphisms?

Not topological embeddings. Smooth embeddings (i.e. a smooth injective immersion which is a homeomorphism onto its image).

 

[lavinia]

The tangent bundle itself. Characteristic classes defined over tangent bundles. Things like that.

There are topological and combinatorial analogues of the tangent bundle e.g. microbundles.

Characteristic classes are defined in the topological and combinatorial categories.

So these do not count as invariants that are preserved under diffeomorphisms but not under homeomorphisms.

 

[lavinia]

One way to restate this question might be to ask whether their exist invariants that distinguish differentiable structures on topological manifolds. While I know nothing about this, I recall that there are Milnor invariants for differential structures on spheres of dimension 4m-1.

I also think that there is another invariant that must be zero if a manifold is to admit any differential structure at all. So it would be trivially preserved under diffeomorphism.

 

[jgens]

There are topological and combinatorial analogues of the tangent bundle e.g. microbundles.

Characteristic classes are defined in the topological and combinatorial categories.

So these do not count as invariants that are preserved under diffeomorphisms but not under homeomorphisms.

Even in the microbundle context my claim is still correct. In the paper where Milnor originally introduces microbundles he shows that the tangent bundle and Pontryagin class of a manifold are not topological invariants. If you want a link to that paper, then here you go: http://www.sciencedirect.com/science/article/pii/0040938364900059

While I know nothing about this, I recall that there are Milnor invariants for differential structures on spheres of dimension 4m-1.

These are another good example of differential (but not topological) invariants. The basic ingredients in their definition are just Pontryagin numbers and the index of bilinear forms.

 

[lavinia]

Even in the microbundle context my claim is still correct. In the paper where Milnor originally introduces microbundles he shows that the tangent bundle and Pontryagin class of a manifold are not topological invariants. If you want a link to that paper, then here you go: http://www.sciencedirect.com/science/article/pii/0040938364900059

The differentiability of a curve is not a topological invariant.

 

[lavinia]

Center of Bass

One thing worth thinking about that distinguishes differentiable manifolds is the De Rham co-chain complex. This is the complex obtained from integrating differential forms over smooth simplices. De Rham's theorem says that the cohomology of this complex is the same as Singular Cohomology with real coefficients. However, the differential forms that represent de Rham cocycles and other forms related to them often carry geometric information that is not discernible purely from Singular theory. For instance there are closed differential forms on compact Riemannian manifolds which are constructed from the curvature form of the connection that give topological invariants of the manifold such as the Euler characteristic. Other differential forms give obstructions to conformal immersions of manifold into Euclidean space.

This interplay between topology and geometry is a wide area of research. Currently, there is much research in the area of Differential Extensions of cohomology theories - not just Singular cohomology - in which differential forms in the de Rham complex and their reduction mod Z or Q give added information.