标题： 端点和端上同调 显示英文标题

标题： Ends and end cohomology

摘要： 末端和末端上同调是研究非紧空间的强大不变量。 我们提供了一个关于末端拓扑理论的自包含阐述，并证明了新的扩展，包括存在一个适当映射的穷竭。 我们将约化末端上同调定义为基于射线的空间的相对末端上同调。 我们利用这些结果证明了King的一个定理的版本，该定理计算两个流形的末端和的约化末端上同同调。 我们包含了Freudenthal关于拓扑群末端数目的基本定理的完整证明，并且我们利用零维末端上同调的结果，无需使用超限归纳法，证明了Nöbeling关于某些连续函数模的自由性的定理。 显示英文摘要

摘要： Ends and end cohomology are powerful invariants for the study of noncompact spaces. We present a self-contained exposition of the topological theory of ends and prove novel extensions including the existence of an exhaustion of a proper map. We define reduced end cohomology as the relative end cohomology of a ray-based space. We use those results to prove a version of a theorem of King that computes the reduced end cohomology of an end sum of two manifolds. We include a complete proof of Freudenthal's fundamental theorem on the number of ends of a topological group, and we use our results on dimension-zero end cohomology to prove -- without using transfinite induction -- a theorem of N\"obeling on freeness of certain modules of continuous functions.