Let $M$ be a smooth manifold, and $N$ be a manifold that may or may not be smooth. Let $f:M \rightarrow N$ be a homeomorphism. If I can show that both $f$ and it's inverse are smooth, i.e. $f$ is a diffeomorphism, does that mean $N$ is necessarily a smooth manifold?
I raise this question because from every source I read, whenever one has a diffeomorphism between two manifolds, they are assumed to be smooth. One stackexchange post said that it makes no sense to construct a diffeomorphism between non-smooth manifolds, just like it makes no sense to construct a ring isomorphism between groups.
But I wish to show that $N$ is smooth by constructing a diffeomorphism with another known smooth manifold $M$. Does this approach make sense or am I misunderstanding the definitions?
Best Answer
You misunderstand. One cannot talk about smoothness of a map $f:M\to N$ if $M$ and $N$ do not have smooth structures (i.e., are smooth manifolds) to begin with. If $M$ is a smooth manifold, $N$ is a topological manifold and $f$ is a homeomorphism, then $N$ has a unique smooth structure for which $f$ is a diffeomorphism. Namely, define a suitable atlas for $N$ by precomposing the charts of an atlas for $M$ with $f^{-1}$.
The best you can do, trying to salvage your idea is: if $M$ is a smooth manifold, $S\subseteq M$ is a submanifold, and $f:M\to M$ is a diffeomorphism, then you can conclude that the image $f[S]$ is also a submanifold. But the point is that the smooth structure(s) needed to say whether $f$ is a diffeomorphism or not have to come from somewhere.