Excellent 1-manifolds in compact 3-manifolds

Abstract

Necessary and sufficient conditions are given for a compact, properly embedded 1-manifold J in a compact, connected 3-manifold M to be homotopic rel∂ J to a 1-manifold K which is excellent in the sense that its exterior is P 2-irreducible and boundary irreducible and boundary irreducible and has the property that every properly embedded incompressible surface of zero Euler characteristic is boundary parallel. Moreover when this is the case there is a ribbon concordance from K to J and there are infinitely many such K with nonhomeomorphic exteriors. this theorem is then applied is then applied to extend results of the author on relative homology cobordisms and characterizations of certain 3-manifolds by their sets of knot groups to the nonorientable case; in the course of proving the latter result an extension to nonorientable 3-manifolds of Johannson's theorem on the determination of boundary irreducible, anannular Haken manifolds by their fundamental groups is developed. In addition the Hass-Thompson characterization of closed, orientable 3-manifolds of Heegaard genus at most one is extended to bounded and/or nonorientable 3-manifolds.