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?
Thank you in advance.
As pointed out here, this is not true in general in the topological case (although it is true in the differential case, as observed by Tim above). The interesting link in that answer is this.