Extension of homeomorphisms on a spherical space

Call a "blot" set, which is the closure of its interior, the boundary is locally connected, and when you remove boundary blot remains connected. Suppose that there is a blot on the surface of the n-dimensional sphere. Is it true that every homeomorphism of the blots on itself extends to a homeomorphism of the sphere on itself? If this is not true, can it be imposed on the blot any additional reasonable conditions (eg, smoothness) to the statement was true?

edited Dec 15 2010 at 0:52

asked Dec 15 2010 at 0:36

Thomas Paul
11●2

No. For example, let $X \subset S^3$ be a closed tubular neighborhood of the unknot. Thus $X$ is a solid torus (ie $D^2 \times S^1$) and $S^3 \setminus X$ is the interior of a solid torus. Let $f : X \rightarrow X$ be the homeomorphism obtained by cutting $X$ open along the disc $D^2 \times 1$, giving the cut open space a full twist, and regluing. Then it is easy to see that $f$ does not extend to a homeomorphism of $S^3$. – Andy Putman Dec 15 2010 at 0:52