Finite presentation of mapping class groups of certain three-manifolds

AbstractLet M be a compact, orientable irreducible 3-manifold. Let F be a compressible boundary component of M and V an incompressible neighborhood of F in M; let M'=(M−V)−. Since there is an exact sequence of mapping class groups: H+(Vrel∂V−F)→H+(M, V)→H+(M',∂V−F)→1, we can prove that H(M) is finitely presented by proving that +(Vrel∂V−F) and the mapping class group of a boundary-incompressible Haken 3-manifold are finitely presented. The latter case is a result of Waldhausen.To prove that H+(Vrel∂V−F) is finitely presented, we construct a simply connected simplicial complex with vertices the isotopy classes of compressing discs in V and study the action of the mapping class group on this complex