The uniqueness of compact cores for 3-manifolds

Abstract

A compact core for a 3-manifold M is a compact sub-manifold N of M whose inclusion in M induces an isomorphism of fundamental groups. A uniqueness result for compact cores of orientable 3-manifolds is known. The authors show that compact cores are not unique in any reasonable sense for non-orientable 3-manifolds, but they prove a finiteness result about the number of possible cores. If M is a non-compact 3-manifold with finitely generated fundamental group, then Scott showed in [Scl] that there is a compact sub-manifold N of M with the natural map π1(N) —» τri(M) an isomorphism. See [R-S] for a simpler proof. We call such a sub-manifold a core or compact core for M. In [McC-Mi-Sw], McCullough, Miller and Swarup showed that if JVi and N2 are irreducible compact cores of a F2-irreducible 3-manifold M, then Nι and N2 are homeomorphic. In this paper, we seek to generalize this to the case when M and its compact cores have no irreducibilit y restrictions. Of course, we cannot any longer expect to prove that two cores of M are homeomorphic, because the Poincare conjecture is not resolved. Thus one core for M might be the connected sum of another core with a homotopy sphere. Also we can obtain new cores by removing a 3-ball from a core or by replacing a connected summand of a core which is a 2-sphere bundle over the circle by a disc bundle over the circle . However, we give an example showing that even if one works modulo the equivalence relation on cores generated by the above operations then uniqueness does not hold. We also show that there are only finitely many different cores in a given 3-manifold up to the equivalence relation of almost homeomorphism which we define in §1. We end by using this finiteness result to prove a natural finiteness result for the boundary of a 3-manifold which has finitely generated fundamental group. The result is the following.