What can we say about the validity of these statements depending on the dimension?
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For any smooth manifold $(M, X)$ (where $M$ is the set and $X$ a maximal atlas on $M$) and any homeomorphism $h: M \to N$ there exists a differential structure (i.e. a maximal atlas) $Y$ of $M$ such that $h: (M,X) \to (M,Y)$ is a diffeomorphism ($X$ and $Y$ can be the same maximal atlas or differ).
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For any smooth manifold $M$ and any pair $X,Y$ of distinct differentiable structures of $M$ ($X \neq Y$) there exists an homeomorphism $h: M \to M$ such that $h: (M,X) \to (M,Y)$ is a diffeomorphism.
3. For any pair (M,X), (N,Y) of homeomorphic smooth manifolds with r: (M,X) to (M,Y) there exists another homeomorphism h: (M,X) to (N,Y) such that r is a diffeomorphism.
3'. For any pair $(M,X), (N,Y)$ of homeomorphic smooth manifolds with $r: M \to N$ an homeomorphism, there exists another homeomorphism $h: M \to N$ such that $h: (M,X) \to (M,Y)$ is a diffeomorphism.
I believe that (1) should be valid in any dimension, via transport of structure. Actually reading that article triggered my questions. In particular, if (1) holds, how can we have topological manifolds that have no differential structure? I would say that the only possibility for such manifolds to exist is that they cannot be homeomorphic to any smooth manifold.
I believe that (2) should be valid for dimensions up to 3, but I have not clear what happens for $n > 3$.
Statement (3') should be a generalization of (2), such that (2) assumes $M$ and $N$ to be the same manifold and $r$ the identity. But I am not sure if (3) is well posed, in particular if I can speak about another homeomorphism between two homeomorphic manifolds.
Distinguishing (2) and (3') from (1), allowing (2) and (3') to fail for $n > 3$ is an effort I did to concile the concept of transport of structure (that should guarantee statement (1)) with the existence of exotic spheres.
Any clarification or correction of the aforementioned statements is welcome.
Best Answer
(1) is indeed true. And there are indeed topological manifolds that are not homeomorphic to any smooth manifold.
(2) is false. The first counterexample was Milnor's exotic smooth structure on the 7-sphere.
(3) is not well formulated, I do not know what $r$ represents and I think there are typos. Perhaps the intention is something like this?
If so then (3') is equivalent to (2), by easy proofs, as follows. To prove (3') from (2), use $r : M \to N$ to transport the structure $Y$ back from $N$ to a smooth structure $Y'$ on $M$, then apply (2) with that transported structure on $N=M$ to get a diffeomorphism $h' : (M,X) \to (M,Y')$, and then let $h = r \circ h'$. To prove (2) from (3'), let $N=M$ and let $r : M \to M$ be the identity, and then (2) follows from (3').
And so knowing that (2) is false, (3') is also false.