the quotient of a sphere by a contractible closed subset in it

Question

let $A \subsetneq S^n$ be a contractible closed subset of the $n$-dimensional sphere. my visual intuition for the two-dimensional sphere tells me there should be a homeomorphism $S^n / A \cong S^n$. is it true? how can one prove this?

Answer 1

It is true if the inclusion $A \hookrightarrow S^n$ is a cofibration (e.g. if $A$ is a subpolyhedron with respect to some triangulation of $S^n$).

But in general it is false. For $n \ge 3$ there exist arcs $A \subset S^n$ (an arc is a homeomorphic copy of $[0,1]$) having the property that $S^n \setminus A$ is not simply connected. But if $S^n/A \approx S^n$, then $S^n \setminus A \approx S^n \setminus \{*\} \approx \mathbb R^n$, a contradiction.

The existence of such wild arcs is not trivial. See

Rushing, T. Benny. Topological embeddings. Academic Press, 1973.