Let $A$ be a nice closed subset of the sphere $S^n$; for example, we could ask $A\to S^n$ to be a cofibration. Assume that $A$ is contractible.
Is then $S^n - A$ also contractible?
It follows by Alexander duality that it has trivial (reduced) homology. But this is, of course, not enough, as the examples of wild arcs show.