Embedding 3-manifolds with boundary into closed 3-manifolds

We prove that there is an algorithm which determines whether or not a given 2-poly-hedron 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 H-1 (M : G) congruent to G(k) and partial derivative 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 x S-1 is twice the orientable genus of the graph. (C) 2011 Elsevier B.V. All rights reserved.