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?
Properties preserved by diffeomorphisms but not by homeomorphisms
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1If manifolds with corners are allowed, then you get an example. – 2011-11-26
2 Answers
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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0how can anything be preserved by diffeos but not homeos when all diffeos are particular cases of homeos? – 2011-11-26
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2@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. – 2011-11-26
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0sorry i misread your statement. – 2011-11-26
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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0Julia sets are not manifolds (but nice example anyway) – 2011-11-26
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0I 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. – 2011-11-26
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0Koch snowflake is probably an easier example. – 2013-08-27