Brown 的渐近极限函子与真同调
Brown's Asymptotic Limit Functor and Proper Homology
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中文总结 AI 辅助
本文构建了 Brown 提出的缺失的真同调理论，通过真 Hurewicz 定理将其与真同伦联系，并证明 $\wp$-函子优于经典极限，且应用于开可缩 3-流形的真基本群性质。
中文摘要 AI 辅助
1974 年，E. M. Brown 引入了 $\wp$-函子来研究端点的真同伦群，并提出了一个平行的真同调理论，但该理论至今未被发展。在本文中，我们构建了这一缺失的真同调，并通过发展真 Hurewicz 定理将其与真同伦联系起来。绕过抽象的 pro-范畴机制，我们证明 $\wp$ 相较于经典极限 $\varprojlim$ 和 $\varprojlim^1$ 具有重大优势：它是正合的，能检测 pro-平凡性，满足基数二分性，并通过一个 4 项正合序列与这些经典极限相联系。作为应用，我们证明了除 $\mathbb{R}^3$ 外的任何开可缩 $3$-流形的 Brown--Grossman 真基本群是不可数且完美的。
英文摘要
In 1974, E. M. Brown introduced the $\wp$-functor to study the proper homotopy groups of an end, suggesting a parallel proper homology theory that has since remained undeveloped. In this paper, we construct this missing proper homology and relate it to proper homotopy by developing a proper Hurewicz theorem. Bypassing abstract pro-categorical machinery, we show that $\wp$ has major advantages over the classical limits $\varprojlim$ and $\varprojlim^1$: it is exact, detects pro-triviality, satisfies a cardinality dichotomy, and connects to these classical limits through a 4-term exact sequence. As an application, we prove that the Brown--Grossman proper fundamental group of any open contractible $3$-manifold other than $\mathbb{R}^3$ is uncountable and perfect.
