拓扑阿贝尔群的弱分裂扩张
Weak split extensions of topological Abelian groups
AI总结：
研究拓扑阿贝尔群范畴中弱分裂扩张的群结构，通过连续和结构及上同调描述，并应用于Bohr拓扑离散群构造非平凡例子。
摘要：
在拓扑阿贝尔群范畴中，我们考虑通常的扩张概念 $E=(B \to X \to A)$，以及弱分裂扩张的概念，即投影 $X \to A$ 允许连续截面 $A \to X$。给定弱分裂扩张 $E$，拓扑阿贝尔群 $X$ 同胚于 $B \times A$，尽管通常它并不代数同构于 $B \times A$。对于两个拓扑阿贝尔群 $A$ 和 $B$，我们研究 $B$ 被 $A$ 的弱分裂扩张模扩张同构构成的阿贝尔群 $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$。我们证明 $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ 可描述为定义在积空间 $B \times A$ 上的所有连续和结构（模拓扑同构）构成的群，其中 $B$ 作为拓扑子群，$A$ 作为拓扑商群。我们还给出了 $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ 的另一种描述，即商群 $Z_c(A,B)/B_c(A,B)$，其中 $Z_c(A,B)$ 由连续映射 $A \times A \to B$ 给出的上循环组成，$B_c(A,B)$ 表示相应的上边界。此外，我们将 $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ 与标准扩张群 $E_A(A,B)$ 进行比较，其中 $A$ 和 $B$ 表示底层的阿贝尔群，并通过一个六项正合序列将这些构造联系起来。尽管离散阿贝尔群的 Bohr 拓扑已被许多研究者研究，但仍有许多部分未被充分理解。这里，作为本文所发展方法的应用，我们为配备 Bohr 拓扑的离散阿贝尔群提供了非平凡 $ws$-扩张的新例子，并提出了一些相关的开放问题。
英文摘要：
In the category of topological Abelian groups, we consider the usual notion of an extension $E=(B \to X \to A)$ of $B$ by $A$, together with the notion of a weakly split extension, i.e., an extension for which the projection $X \to A$ admits a continuous section $A \to X$. Given a weakly split extension $E$, the topological Abelian group $X$ is homeomorphic to $B \times A$, although in general it is not algebraically isomorphic to $B \times A$. For two topological Abelian groups $A$ and $B$, we study the Abelian group $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ of weakly split extensions of $B$ by $A$, modulo extension isomorphisms. We show that $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ can be described as the group of all continuous sum structures defined on the product space $B \times A$ (up to topological isomorphism), with $B$ as a topological subgroup and $A$ as a topological quotient. We also provide an alternative description of $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ as a quotient $Z_c(A,B)/B_c(A,B)$, where $Z_c(A,B)$ consists of cocycles given by continuous maps $A \times A \to B$, and $B_c(A,B)$ denotes the corresponding coboundaries. Furthermore, we compare $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ with the group of standard extensions $E_A(A,B)$, where $A$ and $B$ denote the underlying Abelian groups, and relate these constructions by means of a six-term exact sequence. Although the Bohr topology of discrete Abelian groups has been investigated by many workers, there still remain many parts that are not well understood. Here, as an application of the methods developed in the paper, new examples of nontrivial $ws$-extensions for discrete Abelian groups equipped with the Bohr topology are provided and some related open questions are also proposed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
