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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differential-geometry
differential-topology
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1If manifolds with corners are allowed, then you get an example. – 2011-11-26