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Diffeomorphisms between manifolds are particular homeomorphisms, so each property preserved by homeomorphisms is preserved by diffeomorphisms. Can you show me some examples of properties preserved by diffeomorphisms on manifolds that are not preserved by homeomorphisms?

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If manifolds with corners are allowed, then you get an example. –  Giuseppe Tortorella Nov 26 '11 at 19:47

2 Answers 2

If you're asking about properties of the underlying manifolds, then to answer the question one needs at least some examples of manifolds that are homeomorphic but not diffeomorphic.

There are some examples, but they are quite non-trivial. First example of such phenomenon, «exotic sphere(s)», was constructed by Milnor in 1956 (ref). Milnor also constructed some invariant of such spheres that is preserved by diffeomorphisms but not homeomorphism, but it's not elementary (it's defined in terms of signature and first Pontryagin number of the manifold the exotic sphere bounds).

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how can anything be preserved by diffeos but not homeos when all diffeos are particular cases of homeos? –  lurscher Nov 26 '11 at 20:24
@lurscher Exactly: every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism. So there can be some properties preserved by diffeomorphisms but not by (some non-smooth) homeomorphisms. –  Grigory M Nov 26 '11 at 20:28
sorry i misread your statement. –  lurscher Nov 26 '11 at 20:54

I think Haudorff dimension is such a property.

If you have a fractal subset, say $K \subseteq \mathbb{R}^2$, and $\phi : U \rightarrow V$ is a diffeomorphism, where $U, V$ are open, with $K \subseteq U$, then $HD(K) = HD(\phi(K))$.

However, an homeomorphism certainly doesn't preserve this. There are many fractals arising as Julia Sets of the complex maps $z \mapsto z^2 + c$ which are known to be homeomorphic to $\mathbb{S}^1$, but have Haudorff dimension greater than $1$. There is one example for $c = \frac{1}{4}$ here http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension.

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Julia sets are not manifolds (but nice example anyway) –  Grigory M Nov 26 '11 at 20:14
I agree. I'm actually pointing out a property of a subset of a manifold ($U$, in this case) which is preserved by diffeomorphisms but not homeomorphisms. –  student Nov 26 '11 at 20:17
Koch snowflake is probably an easier example. –  studiosus Aug 27 '13 at 9:04

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