Compactifications of manifolds with boundary

TL;DR

This paper characterizes when manifolds with boundary can be compactified or completed, extending classical results and exploring Z-compactifications, with implications for understanding manifold boundaries and completions.

Contribution

It provides a complete characterization of compactifications for manifolds with noncompact boundary and relates Z-compactifiability to manifold products, advancing the theory of manifold completion.

Findings

Complete characterization of manifold compactifications with noncompact boundary.

Conditions for Z-compactifiability are equivalent to those for M x [0,1].

Application of the Manifold Completion Theorem to Z-compactifications.

Abstract

This paper is concerned with "nice" compactifications of manifolds. Siebenmann's iconic dissertation characterized open manifolds M^m (m>5) compactifiable by addition of a manifold boundary. His theorem extends easily to cases where M^m is noncompact with compact boundary; however, when Bd(M^m) is noncompact, the situation is more complicated. The goal becomes a "completion" of M^m, ie, a compact manifold C^m and a compact subset A such that C^m\A = M^m. 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 M^m. 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.