2017/12/16 by Shijie Gu, Gu, Shijie, Craig R. Guilbault +1
Mathematics · #57 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1712.05995
openalex publication_date 2017/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is concerned with "nice" compactifications of manifolds. Siebenmann's iconic dissertation characterized open manifolds Mm (m>5) compactifiable by addition of a manifold boundary. His theorem extends easily to cases where Mm is noncompact with compact boundary; however, when Bd(Mm) is noncompact, the situation is more complicated. The goal becomes a "completion" of Mm, ie, a compact manifold Cm and a compact subset A such that Cm\A = Mm. Siebenmann did some initial work on this topic, and O'Brien extended that work to an important special case. But, until now, a complete characterization had yet to emerge. We provide such a characterization. Our second main theorem involves Z-compactifications. An open question asks whether a well-known set of conditions laid out by Chapman and Siebenmann guarantee Z-compactifiability for a manifold Mm. We cannot answer that question, but we do show that those conditions are satisfied if and only if M x [0,1] is Z-compactifiable. A key ingredient is the above Manifold Completion Theorem---an application that partly explains our current interest in that topic, and also illustrates the utility of the conditions found in that theorem.