2 problems

Expansion conjecture. The space $Y$ smoothly embeds in $S^4$ if and only if it is obtained by a possibly empty sequence of expansions from some $Y'$ of the displayed form, with

Let $Y$ be a closed oriented $3$--manifold. Gompf's embedding conjecture. $Y$ admits a smooth embedding into a symplectic $4$--manifold. Such an embedding would provide a route tow…