# 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?