This paper is concerned with compactifications of high-dimensional manifolds. Siebenmann’s iconic 1965 dissertation [L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five, Ph.D. thesis, Princeton Univ. (1965), MR 2615648] provided necessary and sufficient conditions for an open manifold [Formula: see text] ([Formula: see text]) to be compactifiable by addition of a manifold boundary. His theorem extends easily to cases where [Formula: see text] is noncompact with compact boundary; however, when [Formula: see text] is noncompact, the situation is more complicated. The goal becomes a “completion” of [Formula: see text], i.e. a compact manifold [Formula: see text] containing a compactum [Formula: see text] such that [Formula: see text]. Siebenmann did some initial work on this topic, and O’Brien [G. O’Brien, The missing boundary problem for smooth manifolds of dimension greater than or equal to six, Topology Appl. 16 (1983) 303–324, MR 722123] extended that work to an important special case. But, until now, a complete characterization had yet to emerge. Here, we provide such a characterization. Our second main theorem involves [Formula: see text]-compactifications. An important open question asks whether a well-known set of conditions laid out by Chapman and Siebenmann [T. A. Chapman and L. C. Siebenmann, Finding a boundary for a Hilbert cube manifold, Acta Math. 137 (1976) 171–208, MR 0425973] guarantee [Formula: see text]-compactifiability for a manifold [Formula: see text]. We cannot answer that question, but we do show that those conditions are satisfied if and only if [Formula: see text] is [Formula: see text]-compactifiable. A key ingredient in our proof is the above Manifold Completion Theorem — an application that partly explains our current interest in that topic, and also illustrates the utility of the [Formula: see text]-condition found in that theorem.

中文翻译：

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带边界流形的紧化

本文关注高维流形的紧化。Siebenmann 1965 年的标志性论文 [LC Siebenmann，为尺寸大于 5 的开放流形寻找边界的障碍，博士。论文，普林斯顿大学。(1965), MR 2615648] 通过添加流形边界为开放流形 [公式：参见文本]（[公式：参见文本]）提供了必要和充分条件。他的定理很容易扩展到 [Formula: see text] 具有紧边界的非紧的情况；然而，当[公式：见正文]是非紧的时，情况就更复杂了。目标成为[公式：参见文本]的“完成”，即紧凑流形[公式：参见文本]包含一个紧凑体[公式：参见文本]，使得[公式：参见文本]。Siebenmann 在这个话题上做了一些初步的工作，和奥布莱恩 [G. O'Brien，维度大于或等于六的光滑流形的缺失边界问题，拓扑应用。16 (1983) 303–324, MR 722123] 将这项工作扩展到一个重要的特殊情况。但是，直到现在，一个完整的特征还没有出现。在这里，我们提供了这样的表征。我们的第二个主要定理涉及[公式：见正文]-压缩。一个重要的悬而未决的问题是查普曼和西本曼是否提出了一组众所周知的条件 [TA Chapman 和 LC Siebenmann，寻找希尔伯特立方流形的边界，数学学报。137 (1976) 171–208, MR 0425973] 保证 [公式：见正文]-歧管的可压缩性 [公式：见正文]。我们无法回答这个问题，但我们确实表明，当且仅当 [公式：参见文本] 为 [公式：见文本]-可压缩。我们证明中的一个关键要素是上述流形完成定理——该应用程序部分解释了我们目前对该主题的兴趣，也说明了在该定理中发现的 [公式：见文本] 条件的实用性。