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Compactifications of manifolds with boundary. (arXiv:1712.05995v1 [math.GT])

来源于:arXiv
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 查看全文>>