Embedding 3-manifolds with boundary into closed 3-manifolds

We prove that there is an algorithm which determines whether or not a given 2-polyhedron can be embedded into some integral homology 3-sphere. This is a corollary of the following main result. Let $M$ be a compact connected orientable 3-manifold with boundary. Denote $G=\Z$, $G=\Z/p\Z$ or $G=\Q$. If $H_1(M;G)\cong G^k$ and $\bd M$ is a surface of genus $g$, then the minimal group $H_1(Q;G)$ for closed 3-manifolds $Q$ containing $M$ is isomorphic to $G^{k-g}$. Another corollary is that for a graph $L$ the minimal number $\rk H_1(Q;\Z)$ for closed orientable 3-manifolds $Q$ containing $L\times S^1$ is twice the orientable genus of the graph.