Undergrad Difference between diffeomorphism and homeomorphism

[trees and plants]

Could we define a morphism on manifolds for integration like an integromorphism?There is a morphism called integral morphism which is defined between schemes but the definition says it is affine and I do not know if it means it makes them integrable.

 

[Infrared]

I don't think integral morphisms of schemes have anything to do with integration (more like "integer"). Can you say what you mean by "integromorphism" (and maybe start a new thread if this is unrelated to the previous posts)?

 

[ChinleShale]

One can think of the idea of different differentiable structures in terms of altasses of coordinate charts. A differentiable structure is a maximal atlas in which the transition functions are continuously differentiable. Two such atlases may be incompatible meaning that the union of their two sets of charts is no longer a differentiable structure.

The same idea applies to complex structures on smooth manifolds. A complex structure is determined by an atlas whose transition functions are holomorphic. Two complex structures are different if their transition functions are incompatible. This idea goes back to the study of Riemann surfaces in the 19th century.

 

[trees and plants]

I don't think integral morphisms of schemes have anything to do with integration (more like "integer"). Can you say what you mean by "integromorphism" (and maybe start a new thread if this is unrelated to the previous posts)?

I mean by integromorphism a morphism that is integrable and makes the manifolds suitable for integration.

 

[mathwonk]

There is apparently a nice relation between this question of homeomorphoc versus diffeomeorphic manifolds that involves the complex structure that some such manifolds can have. If we consider a non singular compact complex "surface", this is also a smooth differentiable 4-manifold, and hence also a topological 4 - manifold. The structure of complex surface allows one to define a "canonical" homology class K of dimension 2, (by taking the homology class of the zero locus of a holomorphic section of the determinant of the holomorphic cotangent bundle, in case one exists). This class turns out to be essentially a diffeomorphism invariant but not a homeomorphism invariant. I.e. if there is a diffeomorphism between two compact complex surfaces, then apparently the induced map on homology must at worst send one canonical class to ± the other one. But there exist two such compact complex surfaces that are homeomorphic, but such that one has non zero canonical class, while the other one has zero canonical class. Two such surfaces hence, although homeomorphic, cannot be diffeomorphic. (I am not an expert on this but read about it on math overflow, so I could be making some mistakes.)
https://mathoverflow.net/questions/70429/is-canonical-class-a-topological-invariant

 

[mathwonk]

Thinking more about one of the original examples, a cube versus a sphere, I think some posters were thinking intuitively of differentiable structures that are inherited from the ambient space R^3, at least when trying to distinguish these two as different. I.e. while it is apparently true that there is only one possible smooth manifold structure (up to diffeomorphism) on the topological manifold given by the cube, and it is obtained from say radial projection to the sphere, nonetheless one could define a differentiable function on the (embedded) cube to be any continuous function that is the restriction everywhere locally of smooth function on an open ball in R^3. With this definition I am guessing the cube becomes not a smooth manifold, but a "manifold with corners". I believe such a structure would not be diffeomorphic to any smooth manifold structure on the cube, and in particular that this differentiable, but non manifold, structure on the cube, would differ from the smooth manifold structure of the (homeomorphic) sphere.

 

[ChinleShale]

Nonetheless one could define a differentiable function on the (embedded) cube to be any continuous function that is the restriction everywhere locally of smooth function on an open ball in R^3. With this definition I am guessing the cube becomes not a smooth manifold, but a "manifold with corners".

Using your definition of differentiable function for subsets of Euclidean space, it would seem that a differentiable structure can be defined for a manifold with corners in the same way as for a manifold or for a manifold with boundary. Rather than open sets in $R^{n}$ or its upper half space, the charts would map into one of the eight cornered spaces such as the points of non-negative Euclidean coordinates. Charts would include not only maps into the corners but also maps to the edges.

Interestingly, a manifold with corners is topologically a manifold with boundary so corners are a property of differentiable structures.

I wonder if there is a general definition for a manifold with corners since a solid cone does not seem diffeomorphic to the same corner.

 

[mathwonk]

I got this idea from Spivak's Calculus on Manifolds, pages 131,137, where he observes that Green's theorem holds also for a square, and Gauss's theorem for a cube. He then challenges the reader to do the exercise of generalizing the results of his chapter to the case of "manifolds with corners", including giving a precise definition of that term. He also discusses a solid cone on page 131.

 

[ChinleShale]

I got this idea from Spivak's Calculus on Manifolds, pages 131,137, where he observes that Green's theorem holds also for a square, and Gauss's theorem for a cube. He then challenges the reader to do the exercise of generalizing the results of his chapter to the case of "manifolds with corners", including giving a precise definition of that term. He also discusses a solid cone on page 131.

I was a little surprised that one can construct an atlas for manifold with corners but actually didn't check that overlapping charts are smooth in the sense of subsets of Euclidean space. So maybe this doesn't work.

But this idea of generalizing differentiable structures for subsets of Euclidean space seems tempting.

Suppose,for instance, that the subset is an embedded closed topological manifold that has no differentiable structure in the sense of atlases. There must be points on it that do not have neighborhoods that are diffeomorphic to open subsets of $R^{n}$. What do they look like? Are they like corners or cone tips or some other creased or crinkled set? As in the case of a manifold with corners, can one generalize the idea of differentiable structure for such a manifold?

 

[ChinleShale]

It is maybe worth noting that diffeomorphisms are distinguished by their degree of differentiability i.e. the highest order of their partial derivatives that exist and are continuous. A $C^{k}$ diffeomorphism has all continuous partial derivatives up to order $k$. If $k$ is infinity then all partials of all orders exist.

A famous example that has been illustrated with computer graphics is a $C^1$ embedding of a torus in 3 space. It is continuously differentiable i.e. it is $C^1$ but its higher order partial derivatives do not exist around any point. One consequence is that its field of unit normal vectors is not differentiable so it does not have a shape operator. For that one would need at least $C^2$.