Assume $M$ is a three-manifold with a single component of the boundary, if we assume that $S^2$ diffeomorphic to some submanifold $S\subset \partial M$.Prove $S$ has to be the whole boundary.
I can deal with the two manifold case, if $S^1$ diffeomorphic to some submanifold $S\subset \partial M$ (with $\partial M$ has one component) ,since the boundary of $M$ is compact smooth 1-manifold,by the classification result, therefore $\partial M$ has to diffeomorphic to $S^1$ or interval.Therefore $S$ has to be the whole circle.(since interval does not diffeomorphic to the circle).
For three-manifold case, I can argue as follows, since $S^2$ is a two dimensinal manifold without a boundary,by diffeomorphism invariance of the boundary, the two dimension submanifold $S\subset \partial M$ do not have boundary,therefore $S$ is an open subset of $\partial M$.However I have no idea how to argue that $S = \partial M$? Since topologically the $\partial M$ needs not even to looks like the $S^2$?