同伦类

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 · 4.1 基本群

我们来看图 4.1。其中一个圆盘上有一个洞，另一个则没有。这两个圆盘之间的差别体现在哪里呢？我们注意到，图 $4.1(b)$ 中的任意环路都可以连续地收缩为一点。相反，图 $4.1(a)$ 中的环路 $\alpha$ 由于其中存在一个洞，因而不能收缩为一点。图 $4.1(a)$ 中的某些环路可以收缩为一点，而另一些则不能。如果 $\alpha$ 能由 $\beta$ 经过连续形变得到，我们就说环路 $\alpha$ 与 $\beta$ 同伦(homotopic)。例如，$Y$ 中的任意环路都与一点同伦。事实上，“同伦”是一种等价关系，它的等价类称为同伦类(homotopy class)。在图 4.1中，与 $Y$ 相联系的只有一个同伦类。在 $X$ 中，每个同伦类都由 $n\in \mathbb{Z}$ 刻画，其中 $n$ 是环路绕洞的圈数；若环路顺时针绕行则 $n<0$，若逆时针绕行则 $n>0$，若环路不绕洞则 $n=0$。此外，$\mathbb{Z}$ 是一个加法群，其群运算（加法）具有几何意义：$n+m$ 对应于先绕洞 $n$ 圈、再绕 $m$ 圈。同伦类的集合被赋予一种群结构，称为基本群(fundamental group)。

英文原文

Let us look at figure 4.1. One disc has a hole in it, the other does not. What characterizes the difference between these two discs? We note that any loop in figure $4.1(b)$ can be continuously shrunk to a point. In contrast, the loop $\alpha$ in figure $4.1(a)$ cannot be shrunk to a point due to the existence of a hole in it. Some loops in figure $4.1(a)$ may be shrunk to a point while others cannot. We say a loop $\alpha$ is homotopic to $\beta$ if $\alpha$ can be obtained from $\beta$ by a continuous deformation. For example, any loop in $Y$ is homotopic to a point. It turns out that ‘homotopic to’ is an equivalence relation, the equivalence class of which is called the homotopy class. In figure 4.1, there is only one homotopy class associated with $Y$ . In $X$ , each homotopy class is characterized by $n\in \mathbb{Z}$ , $n$ being the number of times the loop encircles the hole; $n<0$ if it winds clockwise, $n>0$ if counterclockwise, $n=0$ if the loop does not wind round the hole. Moreover, $\mathbb{Z}$ is an additive group and the group operation (addition) has a geometrical meaning; $n+m$ corresponds to going round the hole first $n$ times and then $m$ times. The set of homotopy classes is endowed with a group structure called the fundamental group.

出处：4.1 基本群 · node a-6f37806444a85d0ad737e8b6 · 源节点 SHA-256 a93ccfe2139476e6…

定义 2 · 4.5 高阶同伦群

证明 $\alpha \sim \beta$ 是一个等价关系。$\alpha$ 所属的等价类称为 $\alpha$ 的同伦类(homotopy class)，记作 $[\alpha ]$。

英文原文

Show that $\alpha \sim \beta$ is an equivalence relation. The equivalence class to which $\alpha$ belongs is called the homotopy class of $\alpha$ and is denoted by $[\alpha ]$ .

出处：4.5 高阶同伦群 · node a-0254c8700d1dde70b07e7e9f · 源节点 SHA-256 ab81595da4050042…