揭示充足群胚同调中离散系数的性质与覆盖结构影响
Discrete Coefficients and Open Invariant Covers in the Homology of Ample Groupoids
- 通过紧支连续函数复形研究群胚同调，分析系数映射的单射性与满射条件
- 证明实数系数下万有系数定理在康托集上不成立，且覆盖导致同调序列失效
- 发现开不变覆盖可构造非平凡连接映射，适合拓扑群论与算子代数研究者
充足群胚的同调由其神经空间上的紧支连续函数复形计算。该复形上常施加的两个假设行为相反。我们证明：从整数链复形张量系数群到带系数链复形的比较映射始终是单射，且当且仅当每个到系数群的紧支连续函数均为局部常值时才为满射。因此，充足群胚的万有系数定理本质上是离散系数形式，甚至在康托集上对实数系数已不成立。接着我们证明：单位空间被两个闭不变子集覆盖时，群胚必分裂为三个闭不变约化之不交并，导致对应的 Mayer-Vietoris 序列连接映射恒为零，无法提供信息。若放弃闭性，两个开不变子集覆盖单位空间则对任意拓扑阿贝尔系数群均可构造有效的 Mayer-Vietoris 序列。对于整数在整数的两点紧化上的作用，我们计算了该序列，发现其连接映射是无限循环群间的同构，这是任何闭不变子集覆盖所无法实现的。
原文摘要 · Abstract (English)
The homology of an ample groupoid is computed from the complex of compactly supported continuous functions on the nerve. Two hypotheses routinely imposed on this complex behave in opposite ways. We show that the comparison map from the integral chain complex tensored with the coefficient group to the chain complex with coefficients is always injective, and that it is surjective if and only if every compactly supported continuous function into the coefficient group is locally constant. The universal coefficient theorem for ample groupoids is therefore a discrete coefficient statement, and it already fails for the real numbers on the Cantor set. We then show that a cover of the unit space by two clopen invariant subsets forces the groupoid to split as a disjoint union of three clopen invariant reductions, so that the associated Mayer-Vietoris sequence has vanishing connecting maps and computes nothing. Dropping closedness repairs this, since two open invariant subsets covering the unit space give a Mayer-Vietoris sequence for every topological abelian coefficient group. For an integer action on a two point compactification of the integers we compute that sequence and find that its connecting map is an isomorphism of infinite cyclic groups, which no cover of a unit space by clopen invariant subsets can achieve.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。