# 4-dimensional

## Smooth 4-dimensional Schoenflies problem ★★★★

Author(s): Alexander

\begin{problem} Let $M$ be a $3$-dimensional smooth submanifold of $S^4$, $M$ diffeomorphic to $S^3$. By the Jordan-Brouwer separation theorem, $M$ separates $S^4$ into the union of two compact connected $4$-manifolds which share $M$ as a common boundary. The Schoenflies problem asks, are these $4$-manifolds diffeomorphic to $D^4$? ie: is $M$ unknotted? \end{problem}

Keywords: 4-dimensional; Schoenflies; sphere