#### Vol. 288, No. 1, 2017

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Noncompact manifolds that are inward tame

### Craig R. Guilbault and Frederick C. Tinsley

Vol. 288 (2017), No. 1, 87–128
##### Abstract

We continue our study of ends of noncompact manifolds, with a focus on the inward tameness condition. For manifolds with compact boundary, inward tameness, by itself, has significant implications. For example, such manifolds have stable homology at infinity in all dimensions. Here we show that these manifolds have “almost perfectly semistable” fundamental group at each of their ends. That observation leads to further analysis of the group-theoretic conditions at infinity, and to the notion of a “near pseudocollar” structure. We obtain a complete characterization of $n$-manifolds ($n\ge 6$) admitting such a structure, thereby generalizing our previous work (Geom. Topol. 10 (2006), 541–556). We also construct examples illustrating the necessity and usefulness of the new conditions introduced here. Variations on the notion of a perfect group, with corresponding versions of the Quillen plus construction, form an underlying theme of this work.

##### Keywords
manifold, end, tame, inward tame, open collar, pseudocollar, near pseudocollar, semistable, perfect group, perfectly semistable, almost perfectly semistable, plus construction
##### Mathematical Subject Classification 2010
Primary: 57N15, 57N65
Secondary: 57Q10, 57Q12, 19D06