Compactifications of manifolds with boundary

Original · EN

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 = 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.