Abstract. We continue our study of ends non-compact manifolds. The overarching aim is to provide an appropriate generalization of Siebenmann’s famous collaring theorem that applies to manifolds having non-stable fundamental group systems at infinity. In this paper a primary goal is finally achieved; namely, a complete characterization of pseudo-collarability for manifolds of dimension at least 6. 1.

There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly well-developed theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental

field which was then young but growing very rapidly. During the intervening years, many problems in differential and geometric topology which had seemed totally impossible have been solved, often using drastically new tools. The following is a

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arXiv version: fonts, pagination and layout may vary from GT published version Manifolds with non-stable fundamental groups at infinity, III