Let $X$ be a path connected topological space. Is it always true that any two homeomorphisms $f$ and $g$ from $X$ to itself are homotopic? If not, is there a minimal condition on $X$ which guarantees that this will be the case for all possible $f$ and $g$?
I know that this is true for a contractible space (indeed, and two maps on such a space are homotopic). However, if it is not true in general, I want to determine a necessary and sufficient condition for this property to hold.