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. s 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 ) ≅ G k and ∂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 . r corollary is that for a graph L the minimal number rk H 1 ( Q ; Z ) for closed orientable 3-manifolds Q containing L × S 1 is twice the orientable genus of the graph.