I recall my Professor having stated something along the lines of the following, but I am not quite certain about the precise statement she gave:


*Let $M$ be a compact, orientable 3 manifold with non-empty boundary. Then $M$ can be embedded in $\mathbb S^3$. More precisely, $M$ is diffeomorphic to $\mathbb S^3 \setminus N$, where $N$ is a finite collection of embedded open handlebodies.*


Is this statement true? If not, what would be an easy counterexample, and does there still exists a weaker form of the statement that is actually true ? Any help is appreciated.



On compact, orientable 3-manifolds with non-empty boundary
Even if you say the boundary has higher genus, it is still false. 


You can't embed $\mathbb{RP}^2$ in $S^3$ but you can embed it into $\mathbb{RP}^3$ which is orientable. (Take the quotient of $S^3$ by the antipodal map. The equatorial $S^2$ becomes a one-sided projective plane.) Drilling out a torus in a ball from $\mathbb{RP}^3$ doesn't make it embeddable in $S^3$ because any embedding would give an embedding of a disjoint $\mathbb{RP}^2$.


More generally, consider drilling out a handlebody within a ball within some compact manifold that is not $S^3$. If you could embed this, then you could embed the smaller manifold where you remove the whole ball, which Igor Belegradek's comment shows is not possible.


