Noncompact manifolds that are inward tame

Abstract

We continue our study of ends of non-compact manifolds, with a focus on the inward tameness condition. For manifolds with compact boundary, inward tameness, has significant implications. For example, such manifolds have stable homology at infinity in all dimensions. We show that these manifolds have 'almost perfectly semistable' fundamental group at each end. That observation leads to further analysis of group theoretic conditions at infinity, and to the notion of a 'near pseudo-collar' structure. We obtain a complete characterization of n-manifolds (n>5) admitting such a structure, thereby generalizing earlier work. We also construct examples illustrating the necessity and usefulness of new conditions introduced here. Variations on the notion of a perfect group, with corresponding versions of the Quillen Plus Construction, form an underlying theme.