Noncompact manifolds that are inward tame

TL;DR

This paper investigates the properties of non-compact manifolds with compact boundary, focusing on inward tameness, and characterizes those admitting a near pseudo-collar structure, extending previous results and introducing new group-theoretic conditions.

Contribution

It provides a complete characterization of n-manifolds (n>5) with near pseudo-collar structures and introduces new conditions related to perfect groups and the Quillen Plus Construction.

Findings

Manifolds with inward tameness have stable homology at infinity.

Such manifolds have 'almost perfectly semistable' fundamental groups at each end.

The paper constructs examples demonstrating the necessity of new conditions.

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.