| Let M be a compact, orientable irreducible 3-manifold. Let F be a compressible boundary component of F and V an incompressible neighborhood of F in M. Since there is an exact sequence of mapping class groups, H(,+)(V rel (PAR-DIFF)V - F) (--->) H(,+)(M,V) (--->) H(,+)(M',(PAR-DIFF)V - F) (--->) 1, we can prove that H(M) is finitely presented by proving that H(,+)(V rel (PAR-DIFF)V - F) and the mapping class group of a boundary-incompressible Haken 3-manifold are finitely presented.;In the case of a boundary-incompressible Haken 3-manifold M' we use properties of the characteristic submanifold to prove that the subgroup of the mapping class group generated by Dehn twists about tori and annuli in M is finitely presented. Since this subgroup is known to have finite index in M, the desired result follows.;To prove that H(,+)(V rel (PAR-DIFF)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. |
