Let $f,g:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be smooth injective and let $n\leq m$. Let $k \in \mathbb{N}$, and let $\iota\_m^{m+k}:\mathbb{R}^m\rightarrow \mathbb{R}^{m+k}$ be the canonical inclusion. Suppose also that $f(\mathbb{R}^n)\cong \mathbb{R}^n\cong g(\mathbb{R}^n)$ via some $C^{\infty}$-diffeomorphism.


Fix a compact subset $K\subseteq \mathbb{R}^n$. For what values of $k$, does there exist a homeomorphism $\phi:\mathbb{R}^{m+k}\rightarrow \mathbb{R}^{m+k}$ satisfying
$$
\iota\_m^{m+k}\circ f(x)= \phi\circ \iota\_m^{m+k}\circ g(x) \qquad (\forall x \in K)?
$$
It isn't difficult to see that $k\leq m+n$. However, what is the *smallest* such value of $k$ for which this holds? My intuition says 1...




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*Reduction to Extension Problem*
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I guess since $f$ and $g$ are homeomorphisms onto their image then, $h:=g\circ f^{-1}:f(K)\rightarrow g(K)$ is a homomorphism. So the problem reduces to finding an extension of $h$ to all of $\mathbb{R}^{m+k}$. But when does such an extension exist?



Extension of homeomorphisms
It sounds like you're looking for something like [the Klee trick](https://www.ams.org/journals/tran/1955-078-01/S0002-9947-1955-0069388-5/). If $K,K' \subset \mathbb{R}^n$ are compact and homeomorphic, it gives a construction of a self-homeomorphism $\phi$ of $\mathbb{R}^{2n}$ such that $\iota\_n^{2n}(K) = \phi(\iota\_n^{2n}(K'))$.


