TITLE: Unknotted $S^{n-2}$ in $S^n$
QUESTION [12 upvotes]: I wonder is it still an open question that   a smooth sphere $\Sigma^{2}\subset S^4$ is unknotted in $S^4$ iff its complement is homotopy equivalent to $S^1$? If it is an open question, how is it related to other known conjectures in 4D?
I know for all the other $n$ this has been settled by Levine 1965 "Unknotting spheres in codimension 2" and Wall 1965 "Unknotting tori in codimension one and spheres in codimension two", see relatedly Status of a conjecture of C.T.C. Wall?.

REPLY [12 votes]: My understanding is this remains an open problem in the smooth category.
I believe there have been a few claims of proofs of this statement in the literature over the years, but as far as I know none of these arguments have been robust.   As I believe you are aware, in the topological category this was done by Mike Freedman.
If you jazz up the conjecture a little you could turn it into a recognition principle for $S^1 \times D^3$, and that could in turn be turned into a proof of the smooth 4-dimensional Poincare conjecture.  Specifically, say you have a homotopy 4-sphere.  Remove a small unknotted $S^2$ (i.e. in some embedded $D^4$ in your homotopy 4-sphere), then its exterior presumably would satisfy the recognition principle for $S^1 \times D^3$.  From this you could argue the homotopy $4$-sphere is the standard smooth $S^4$ by filling in $D^2 \times S^2$.