# Isotopy and homeomorphism

Let $X$ and $Y$ be topological spaces. Suppose we have an isotopy between maps $f, g: X\to Y$.

The question is that is there a homeomorphism $h: Y\to Y$ such that $h\circ f =g$?

I am especially interested in the case when $Y$ is a surface and $f, g$ are embeddings.

Is this true or do we need more conditions? Is there any analogous result?

What exactly do you mean by isotopy if $f$ and $g$ aren't embeddings? – Jason DeVito Jan 12 '13 at 1:10
(Assume f and g are embeddings) In the smooth manifold setting, If $X$ is compact and $G$ is the isotopy youve assumed, there is an "ambient isotopy" of the identity map on Y. I.e. an isotopy $H: Y\times I \rightarrow Y$ such that $H(f,t)= G$ and $H_0 = 1$. Then $H_1$ is your homeomorphism. See Kosinski's "Differential Manifolds" theorem 5.2 in chapter 2. I'm not sure about the non compact case. Weirdly I was just discussing this in the "differential-topology" section. – Tim kinsella Jan 12 '13 at 1:18