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/pZ or G=Q. If H1(M;G)≅Gk and ∂M is a surface of genus g, then the minimal group H1(Q;G) for closed 3-manifolds Q containing M is isomorphic to Gk−g.Another corollary is that for a graph L the minimal number for closed orientable 3-manifolds Q containing L×S1 is twice the orientable genus of the graph.