同伦

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 1 · 定义 4.4

设 $\alpha ,\beta :I\to X$ 是 $x_0$ 处的环路。若存在连续映射 $F:I\times I\to X$ 使得（下式成立），则称它们是同伦(homotopic)的，记作 $\alpha \sim \beta$。

$$F(s,0)=\alpha (s),F(s,1)=\beta (s)\forall s\in I\text{\hspace{1em}(4.3)}$$ $$F(0,t)=F(1,t)=x_0\forall t\in I.$$

称 $F$ 为 $\alpha$ 与 $\beta$ 之间的同伦。

英文原文

Let $\alpha ,\beta :I\to X$ be loops at $x_0$ . They are said to be homotopic, written as $\alpha \sim \beta$ , if there exists a continuous map $F:I\times I\to X$ such that

$$F(s,0)=\alpha (s),F(s,1)=\beta (s)\forall s\in I\text{\hspace{1em}(4.3)}$$ $$F(0,t)=F(1,t)=x_0\forall t\in I.$$

The connecting map $F$ is called a homotopy between $\alpha$ and $\beta$ .

出处：4.1 基本群 · node a-cacbf32f1ce498f5e7a5000f · 源节点 SHA-256 a637fc467f06ea87…

定义 2 · 定义 4.10

设 $X$ 是拓扑空间，$\alpha ,\beta :I^n\to X$ 是 $x_0\in X$ 处的 $n$-环路。称映射 $\alpha$ 与 $\beta$ 同伦(homotopic)，记作 $\alpha \sim \beta$，如果存在连续映射 $F:I^n\times I\to X$ 使得

$$F\left(s_1,\dots ,s_n,0\right)=\alpha \left(s_1,\dots ,s_n\right)\text{\hspace{1em}(4.37a)}$$ $$F\left(s_1,\dots ,s_n,1\right)=\beta \left(s_1,\dots ,s_n\right)\text{\hspace{1em}(4.37b)}$$ $$F\left(s_1,\dots ,s_n,t\right)=x_0\text{f o r}(s_1,\dots ,s_n)\in \partial I^n,t\in I.\text{\hspace{1em}(4.37c)}$$

$F$ 称为 $\alpha$ 与 $\beta$ 之间的同伦。

英文原文

Let $X$ be a topological space and $\alpha ,\beta :I^n\to X$ be $n$ -loops at $x_0\in X$ . The map $\alpha$ is homotopic to $\beta$ , denoted by $\alpha \sim \beta$ , if there exists a continuous map $F:I^n\times I\to X$ such that

$$F\left(s_1,\dots ,s_n,0\right)=\alpha \left(s_1,\dots ,s_n\right)\text{\hspace{1em}(4.37a)}$$ $$F\left(s_1,\dots ,s_n,1\right)=\beta \left(s_1,\dots ,s_n\right)\text{\hspace{1em}(4.37b)}$$ $$F\left(s_1,\dots ,s_n,t\right)=x_0\text{f o r}(s_1,\dots ,s_n)\in \partial I^n,t\in I.\text{\hspace{1em}(4.37c)}$$

$F$ is called a homotopy between $\alpha$ and $\beta$ .

出处：4.5 高阶同伦群 · node a-071d0e71f80ed4079e3605ae · 源节点 SHA-256 dc397a71a88da95a…