Existence of a homeomorphism on ambient space mapping submanifolds

Question

Let $M,N\subseteq \mathbb{R}^n$ be smooth sub-manifolds both of which are diffeomorphic to $\mathbb{R}^d$ for some $0\leq d<n$. Let $\gamma:=n-d$. For what values of $d$ does there necessarily exist a homeomorphism $\phi:\mathbb{R}^n\rightarrow \mathbb{R}^n$ for which $$ \phi(M)= N? $$

In other words: What is the minimal co-dimension of the ambient space for which any two topologies copies of $\mathbb{R}^n$ can be mapped into one another by a homomorphism?

Answer 1

Assuming I'm interpreting your question correctly, the answer is basically never. Consider \begin{align*} M &= (-\infty, \infty) \subseteq \mathbb{R} \\ N &= (-1,1) \subseteq \mathbb{R} \end{align*}

Then $M$ and $N$ are both diffeomorphic to $\mathbb{R}$, but there is no ambient homeomorphism $\phi$ of $\mathbb{R}$ carrying $M$ to $N$, since $M$ is closed in $\mathbb{R}$ and $N$ is not closed in $\mathbb{R}$.

For other dimensions, take \begin{align*} \tilde{M} = M \times \mathbb{R}^{d-1} \subseteq \mathbb{R} \times \mathbb{R}^{n-1} \cong \mathbb{R}^n \\ \tilde{N} = N \times \mathbb{R}^{d-1} \subseteq \mathbb{R} \times \mathbb{R}^{n-1} \cong \mathbb{R}^n \end{align*} where $\mathbb{R}^{d-1} \subseteq \mathbb{R}^{n-1}$ is a standard linear embedding.

Then $\tilde{M}$ and $\tilde{N}$ are both diffeomorphic to $\mathbb{R}^d$, but $\tilde{M}$ is closed in $\mathbb{R}^n$ and $\tilde{N}$ is not, so no ambient homeomorphism taking $M$ to $N$ exists.

So it seems the only time such a homeomorphism exists is if $d = 0$.