at.algebraic topology – Extension of homeomorphisms

Let $f,g:mathbb{R}^nrightarrow mathbb{R}^m$ be smooth injective and let $nleq m$. Let $k in mathbb{N}$, and let $iota_m^{m+k}:mathbb{R}^mrightarrow mathbb{R}^{m+k}$ be the canonical inclusion. Suppose also that $f(mathbb{R}^n)cong mathbb{R}^ncong g(mathbb{R}^n)$ via some $C^{infty}$-diffeomorphism.

Fix a compact subset $Ksubseteq mathbb{R}^n$. 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)= phicirc iota_m^{m+k}circ g(x) qquad (forall x in K)?

It isn’t difficult to see that $kleq m+n$. However, what is the smallest such value of $k$ for which this holds? My intuition says 1…

Reduction to Extension Problem

I guess since $f$ and $g$ are homeomorphisms onto their image then, $h:=gcirc f^{-1}:f(K)rightarrow g(K)$ 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?