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?](https://mathoverflow.net/questions/368325/status-of-a-conjecture-of-c-t-c-wall).



Unknotted $S^{n-2}$ in $S^n$
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 $D^2 \times S^2$.


