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In set theory there is the concept of a bijection, a one-to-one correspondence between the elements of $2$ sets.

In topology the concept of a homeomorphism $f:X\to Y$ is quite easy to wrap your head around. It just establishes not just a bijection of the elements of $X$ and $Y$, but also of their open sets.

However, for a diffeomorphism $f:N\to M$ I find it a bit harder to establish a conceptual picture of what happens.

Now to me it seems like the pattern here should be that each of them establishes an 'additional one-to-one correspondence' between some properties of the domain and the codomain.

For example, the fact that $\mathbb{R}$ with the normal smooth structure and $\mathbb{R}$ with the atlas $\{(\mathbb{R}, x\mapsto x^{1/3})\}$ are not equal, but are diffeomorphic is something I can prove, but I find it harder to see what kind of additional one-to-one correspondence exists as a result of them being diffeomorphic. Obviously there is a one-to-one correspondence between their elements, and their open set (since a diffeomorphism is also a homeomorphism), but there must exists more correspondences between them, since not every homeomorphism is a diffeomorphism.

Now after some thinking I think the extra correspondence is between the smooth functions on both. Because if $F:M\to N$ is a diffeomorphism and $g:M\to K$ is a smooth function, we can define $h:N\to K$ by $$h=g \circ F^{-1}$$ which and well defined since $F$ is a diffeomorphism. This is clearly a bijective correspondence.

So I'm now thinking that this is the 'third correspondence' that the diffeomorphism gives us. However, I'm not sure about this since the exact same correspondence also exsists between continuous functions if $F$ is only a homeomorphism. Also, this correspondence is 'external' in some sense. Is there a more internal characterisation?

I hope someone can weight in on this.

  1. Is is 'correct' to think of bijection-homeomorphism-diffeomorphism as establishing ever more one-to-one correspondence between properties of the underlying spaces?
  2. If so, what additional correspondence does the diffeomorphism establish over the homeomorphism?
user2520938
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  • I don't believe thinking of homeomorphisms as "bijection and bijection on open sets" is the best way to go. Homeomorphisms are the isomorphisms of topological spaces. An homeomorphism basically tells you that two topological spaces are "the same", in all possible ways that matter from the point of view of topology. Anything that is true about the first is true about the second. Well, it's the same for diffeomorphism: diffeomorphic manifolds are basically the same from the point of view of calculus. – Najib Idrissi Aug 19 '15 at 11:24
  • @NajibIdrissi I use the fact that the homeomorphism gives a bijection between the sets and the open subsets (in such a way that the two bijection work nicely together) as a natural way to 'translate' topological statements about the one space to the other. The homeomorphism is the dictionary via which I can conclude things about the one space by proving them in the other. So yes I do realise that the real 'power' of the homeomorphism is the establishment of the topological equivalence of the spaces, but the concrete way in which it does this is via the bijections it gives. – user2520938 Aug 19 '15 at 11:34
  • It's only because you choose to describe the topology of a space through open sets. Gelfand duality tells you that compact spaces are the same thing as commutative C* algebras. An homeomorphism then translate to an isomorphism of C* algebras, and then you can write down what an isomorphism of C* algebras is. But conceptually, it's still "these two spaces/algebras are the same in all possible ways". It's important (IMO) to realize that the overall notion of "isomorphism" is the real deal, the description in terms of open sets is merely technical. – Najib Idrissi Aug 19 '15 at 12:30
  • I know that sometimes definition are very technical and that you should try to understand the idea behind the definition. For example the definition of a monomorphism is technical, but the idea behind it is that you're looking for something that is "injective". For a homeomorphism, the idea that you have a continuous function with a continuous inverse is the definition, but it also happen to be what you are conceptually looking for. – Najib Idrissi Aug 19 '15 at 12:34
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    The correspondence you describe between functions is "external" only if you refuse to think of functions on a space as being important objects in their own right. Not all the activity of interest is about the spaces; often it is the functions on the spaces that one really cares about. In this light, consider the following posts: http://math.stackexchange.com/questions/226736/a-theorem-due-to-gelfand-and-kolmogorov and http://mathoverflow.net/questions/21090/smooth-gelfand-duality. – KCd Aug 19 '15 at 12:48
  • In that case the isomorphism is a bijection of the set theoretic defition of the * map. (I dont know anything about c* algebras but still). I think its not so strange to think of isomorphism in general about well behaved bijections between all aspects of a space (which is also more or less the definiion in general) – user2520938 Aug 19 '15 at 12:48
  • @KCd Thanks. I realise that its not completely external, but to me its similar to how normal subgroups can be described as kernels of homomorphisms and as groups closed onder conjugation. Its satisfying to have both characterisations imo – user2520938 Aug 19 '15 at 12:51
  • @KCd Also, thanks for the link, very interesting stuff. – user2520938 Aug 19 '15 at 12:53

2 Answers2

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I found it. The additional correspondence is (as I could have guessed) between their charts:

If $F:N\to M$ is a diffeomorphism then $(U,\phi)$ is a chart on $N$ iff $(F(U),\phi\circ F^{-1})$ is a chart on $M$.

user2520938
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  • So we have one-to-one correspondence between maximal atlases. That was the result you were interested in? – Fallen Apart Aug 19 '15 at 11:05
  • Well when I wrote this question I did not yet know what I was interested in; I wanted SOME satisfying 'internal' one-to-one correspondence between $N$ and $M$. This seems to be the most natural one and fits in nicely with the open-set correspondence in the case of homeomorphisms, so yes looking back I think this is what I was interested in. – user2520938 Aug 19 '15 at 11:07
  • Very nice post, do you know some nice implications of this fact? the two spaces do not carry the same diffirentiable functions, right? – user123124 Jul 20 '19 at 16:10
  • @user1 They do. Composition with the diffeomorphism, as I outlined in the question, gives a bijection (in fact a ring-isomorphism) between the rings of differentiable functions on the two manifold. – user2520938 Jul 20 '19 at 17:48
  • but thats not entirely the same function? Say we have $f$ on $N$, then $F^{-1} \circ f$ is not the same function, or have confused myself? Or are you rather saying that there is a bijection between the two spaces of differentiable functions? i.e not in the same spirit as a homeomorphism which gives us that we have the same continous functions. – user123124 Jul 21 '19 at 06:56
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    @user1 A homeomorphism also doesn't give literally the same functions, it also works via composition. What I was saying is indeed that we have a bijection. Or actually a ring isomorphism. – user2520938 Jul 21 '19 at 07:45
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    @user1 ps, any function $F:M\to N$ that gives a ring isomorphism between the function spaces, is a diffeomorphism. – user2520938 Jul 21 '19 at 07:48
  • Ofc! Thanks alot – user123124 Jul 21 '19 at 08:19
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The main common idea between bijections, homeomorphisms and diffeomorphisms is that they have inverse : you have not only $f:X\to Y$ but also $f^{-1}:Y\to X$ such that $f\circ f^{-1}=id_Y$ and $f^{-1}\circ f=id_X$. If $X,Y$ are just sets and $f,f^{-1}$ are just functions, this is a bijection; if $X,Y$ are topological spaces and $f,f^{-1}$ are both continuous it is an homeomorphism; if $X,Y$ are manifolds and $f,f^{-1}$ are both differentiable it is a diffeomorphism. These are all examples of isomorphisms.

Arnaud D.
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  • I know that; these are just the definitions. However I find it insightfull to realise that the real common idea is that they induce a bijection on ALL structure of the domain and codomain, not just on the elements of the underlying sets. With my question I was looking the extra bijection that the diffeomorphism gives over the homeomorphism – user2520938 Aug 19 '15 at 11:17