NONCOMPACT MANIFOLDS THAT ARE INWARD TAME

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 grouptheoretic conditions at infinity, and to the notion of a "near pseudocollar" structure. We obtain a complete characterization of n-manifolds (n >= 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.