Is there a nonslice knot $K\subset S^3$ that is slice in some closed oriented $3$-manifold $Y$? Here, when we say $K$ is slice in $Y$, it means that when regarded as a local knot in $Y\times\{1\}$, $K$ bounds an embedded disk in $Y\times[0,1]$. One can ask this question in both topological and smooth categories.

When $Y$ is irreducible, by taking the universal cover one sees that being slice in $Y$ is equivalent to being slice (in $S^3$). However, for $Y$ reducible I am not sure the same conclusion still holds.

In one dimension lower there is an analogous question, namely whether there is a nontrivial knot in $S^3$ that becomes trivial in some closed $3$-manifold $Y$. The answer is no, because surgering along essential spheres reduces the question to $Y$ irreducible where one can again take the universal cover to conclude. However, the same argument doesn't seem to carry directly to the $4$-dimensional case.