Spanier空间与非同伦路径T2空间的覆盖理论

小闭路空间与非同伦T2空间的覆盖理论

Small loop spaces and covering theory of non-homotopically Hausdorff spaces

在这篇文章中，我们致力于空间不是同伦豪斯多夫和研究他们的覆盖空间。引入小覆盖的概念，证明了$X$的每一个小覆盖都是范畴意义下的泛覆盖.引入半局部小闭路空间的概念，它是非同伦T2空间存在泛覆盖的充要条件，等价地说是小覆盖空间存在的充要条件.此外，我们还证明了对于半局部小闭路空间，$X$是小回路空间当且仅当$X$的每个覆盖都是平凡的当且仅当$π_1^{top}(X)$是不离散的拓扑群。

粗Spanier组和第一个形状组

Thick Spanier groups and the first shape group

本文提出了一种新的途径，通过它来探索由一般空间$X$确定的$π_1$形群同态的核$\kerΨ_X$，并建立了对于每一个局部连通的仿紧T2空间$X$，$\kerΨ_X$正是$X$的Spanier群.

关于具有商拓扑的基本群

On fundamental groups with the quotient topology

拟拓扑基本群$π_{1}^{qtop}(X,x_0)$是具有从基环空间继承的自然商拓扑的基本群，并且当$X$不允许传统的泛覆盖时，它通常是非离散的。这种拓扑化的基本群是同伦型不变量，它具有区分弱同伦等价空间和形等价空间的能力。本文阐明了群$π_{1}^{qtop}(X,x_0)$的拓扑性质与其基础空间$π_{1}^{qtop}(X,x_0)$的“$π_{1}$ -形内射性”和“同伦路径-Hausdorff”等性质之间的各种关系.一个空间$X$是$π_1$ -形内射的，如果基本群正则嵌入第一个形群中，使得$π_1(X,x_0)$的元素可以表示为逆极限中的序列.证明了局部路连通度量空间$X$是$π_1$ -形内射的充要条件是$π_{1}^{qtop}(X,x_0)$在所有开不变子群（即正规子群）的交是平凡子群的意义下是不变分离的.在$X$不是$π_1$ -形内射的情况下，同伦路径Hausdorff性质对于区分环的同伦类是有用的，并且保证了某些广义覆盖映射的存在性.证明了局部路径连通空间$X$是同伦路径Hausdorff的充要条件是$π_{1}^{qtop}(X,x_0)$满足$T_1$分离公理.

Topologized standard construction and locally quasinormal subgroups

This paper is the extended version of some results in [13, 14]. Let H be a subgroup of fundamental group. The first paper of the paper is devoted to studying weaker conditions under which homotopically Hausdorff relative to H becomes homotopically path Hausdorff relative to H. By using of these conditions, we explore the connection between whisker and quotient topologies on fundamental group. After that, we address the coincidence of two determined topologies on the standard construction XeH when H is a locally quasinormal subgroup. Finally Example 3.14 illustrates that these kinds of subgroup…

On the failure of the first Čech homotopy group to register geometrically relevant fundamental group elements

We construct a space $\mathbb{P}$ for which the canonical homomorphism $π_1(\mathbb{P},p) \rightarrow \checkπ_1(\mathbb{P},p)$ from the fundamental group to the first Čech homotopy group is not injective, although it has all of the following properties: (1) $\mathbb{P}\setminus\{p\}$ is a 2-manifold with connected non-compact boundary; (2) $\mathbb{P}$ is connected and locally path connected; (3) $\mathbb{P}$ is strongly homotopically Hausdorff; (4) $\mathbb{P}$ is homotopically path Hausdorff; (5) $\mathbb{P}$ is 1-UV$_0$; (6) $\mathbb{P}$ admits a simply connected generalized covering space …

Small Loop Transfer Spaces with Respect to Subgroups of Fundamental Groups

Let $H$ be a subgroup of $π_{1}(X,x_{0})$. In this paper, we extend the concept of $X$ being SLT space to $H$-SLT space at $x_0$. First, we show that the fibers of the endpoint projection $p_{H}:\tilde{X}_{H}\rightarrow X$ are topological group when $X$ is $H$-SLT space at $x_0$ and $H$ is a normal subgroup. Also, we show that under these conditions the concepts of homotopically path Hausdorff relative to $H$ and homotopically Hausdorff relative to $H$ coincide. Moreover, among other things, we show that the endpoint projection map $p_{H}$ has the unique path lifting property if and only if $H…