万有覆叠空间

本书暂定概念：尚未进入正式概念库。下面是本书给出的定义原文。

定义 · 定义 4.12

设 $X$ 与 $\tilde{X}$ 为连通拓扑空间。若存在连续映射 $p:\tilde{X}\to X$ 满足下列条件，则称二元组 $(\tilde{X},p)$（或简记为 $\tilde{X}$）为 $X$ 的覆叠空间(covering space)：

(1) $p$ 是满射（映上）；(2) 对每个 $x\in X$，存在包含 $x$ 的连通开集 $U\subset X$，使得 $p^{-1}(U)$ 是 $\tilde{X}$ 中若干开集的不交并，而其中每个开集都被 $p$ 同胚地映到 $U$ 上。

特别地，若 $\tilde{X}$ 是单连通的，则称 $(\tilde{X},p)$ 为 $X$ 的万有覆叠空间(universal covering space)。[注记：某些群已知是拓扑空间，它们称为拓扑群(topological group)。例如 $\mathrm{SO}(n)$ 与 $\mathrm{SU}(n)$ 都是拓扑群。若 定义 4.12 中的 $X$ 与 $\tilde{X}$ 恰为拓扑群，且 $p:\tilde{X}\to X$ 是群同态，则相应的（万有）覆叠空间称为（万有）覆叠群(covering group)。]

例如，$\mathbb{R}$ 是 $S^1$ 的万有覆叠空间，见4.3 节。由于 $S^1$ 与 $\mathrm{U}(1)$ 等同，若把 $\mathbb{R}$ 看作加法群，则 $\mathbb{R}$ 是 $\mathrm{U}(1)$ 的万有覆叠群。映射 $p:\mathbb{R}\to \mathrm{U}(1)$ 可以取为 $p:x\to {\mathrm{e}}^{\mathrm{i}2\pi x}$。显然 $p$ 是满射，并且若 $U=\{{\mathrm{e}}^{\mathrm{i}2\pi x}|x\in (x_0-0.1,x_0+0.1)\}$，则

$$p^{-1}(U)=\bigcup _{n\in \mathbb{Z}}(x_0-0.1+n,x_0+0.1+n)$$

它是 $\mathbb{R}$ 的若干开集的不交并。容易证明，$p$ 关于 $\mathbb{R}$ 中的加法与 $\mathrm{U}(1)$ 中的乘法也是同态。因此，$(\mathbb{R},p)$ 是 $\mathrm{U}(1)=S^1$ 的万有覆叠群。

英文原文

Let $X$ and $\tilde{X}$ be connected topological spaces. The pair $(\tilde{X},p)$ , or simply $\tilde{X}$ , is called the covering space of $X$ if there exists a continuous map $p:\tilde{X}\to X$ such that

(1) $p$ is surjective (onto)
(2) for each $x\in X$ , there exists a connected open set $U\subset X$ containing $x$ , such that $p^{-1}(U)$ is a disjoint union of open sets in $\tilde{X}$ , each of which is mapped homeomorphically onto $U$ by $p$ .

In particular, if $\tilde{X}$ is simply connected, $(\tilde{X},p)$ is called the universal covering space of $X$ . [Remarks: Certain groups are known to be topological spaces. They are called topological groups. For example $\mathrm{SO}(n)$ and $\mathrm{SU}(n)$ are topological groups. If $X$ and $\tilde{X}$ in definition 4.12 happen to be topological groups and $p:\tilde{X}\to X$ to be a group homomorphism, the (universal) covering space is called the (universal) covering group.]

For example, $\mathbb{R}$ is the universal covering space of $S^1$ , see section 4.3. Since $S^1$ is identified with $\mathrm{U}(1)$ , $\mathbb{R}$ is a universal covering group of $\mathrm{U}(1)$ if $\mathbb{R}$ is regarded as an additive group. The map $p:\mathbb{R}\to \mathrm{U}(1)$ may be $p:x\to {\mathrm{e}}^{\mathrm{i}2\pi x}$ . Clearly $p$ is surjective and if $U=\{{\mathrm{e}}^{\mathrm{i}2\pi x}|x\in (x_0-0.1,x_0+0.1)\}$ , then

$$p^{-1}(U)=\bigcup _{n\in \mathbb{Z}}(x_0-0.1+n,x_0+0.1+n)$$

which is a disjoint union of open sets of $\mathbb{R}$ . It is easy to show that $p$ is also a homomorphism with respect to addition in $\mathbb{R}$ and multiplication in $\mathrm{U}(1)$ . Hence, $(\mathbb{R},p)$ is the universal covering group of $\mathrm{U}(1)=S^1$ .

出处：4.6 高阶同伦群的一般性质 · node a-f93e176dae385677718561c6 · 源节点 SHA-256 db6ae9484c3a14f0…