# When is an $n$-dimensional manifold characterized by its $m$-dimensional submanifolds?

For which $m$, $n$ (if any) is the following true: if $M$ and $M'$ are smooth manifolds of dimension $n$, and $\Phi$ is a bijection from $M$ to $M'$ such that for any subset $S$ of $M$, $\Phi(S)$ is an embedded submanifold of $M'$ of dimension $m$ iff $S$ is an embedded submanifold of $M$ of dimension $m$, then $\Phi$ is a diffeomorphism?

This is clearly false when $m=0$ and when $m=n$. I thought it looked plausible when, for example, $m=1$ and $n\geq 2$; but I can't see how to prove it.

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I think a similar question (possibly in the topological category?) came up on MO. – Qiaochu Yuan Mar 14 '11 at 17:31

My counterexample is essentially a generalization of taking $\Phi: \mathbb R \to \mathbb R, x \mapsto x^3$. For $n=2, m=1$, we will take $M=M' = \mathbb R^2 = \mathbb C$. Using complex coordinates $z = x+iy$, let $\Phi(z) = z |z|^2$. This is a diffeomorphism of $\mathbb C \setminus \{ 0 \}$, and is a homeomorphism of $\mathbb C$. The inverse map is $\Phi^{-1}(z) = z |z|^{-2/3}$ (with $\Phi^{-1}(0) = 0$).