4.5 Higher homotopy groups

The fundamental group classifies the homotopy classes of loops in a topological space $X$ . There are many ways to assign other groups to $X$ . For example, we may classify homotopy classes of the spheres in $X$ or homotopy classes of the tori in $X$ . It turns out that the homotopy classes of the sphere $S^n$ ( $n\ge 2$ ) form a group similar to the fundamental group.

4.5.1 Definitions

Let $I^n$ $(n\ge 1)$ denote the unit $n$ -cube $I\times \cdots \times I$

$$I^n=\left\{\left(s_1,\dots ,s_n\right)\mid 0\le s_i\le 1(1\le i\le n)\right\}.\text{\hspace{1em}(4.35)}$$

The boundary $\partial I^n$ is the geometrical boundary of $I^n$ ,

$$\partial I^n=\left\{\left(s_1,\dots ,s_n\right)\in I^n\mid \text{s o m e}{s}_i=0\text{o r}1\right\}.\text{\hspace{1em}(4.36)}$$

We recall that in the fundamental group, the boundary $\partial I$ of $I=[0,1]$ is mapped to the base point $x_0$ . Similarly, we assume here that we shall be concerned with continuous maps $\alpha :I^n\to X$ , which map the boundary $\partial I^n$ to a point $x_0\in X$ . Since the boundary is mapped to a single point $x_0$ , we have effectively obtained $S^n$ from $I^n$ ; cf figure 2.8. If $I^n/\partial I^n$ denotes the cube $I^n$ whose boundary $\partial I^n$ is shrunk to a point, we have $I^n/\partial I^n\cong S^n$ . The map $\alpha$ is called an $n$ -loop at $x_0$ . A straightforward generalization of definition 4.4 is as follows.

Definition 4.10

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 $\beta$ .

Exercise 4.6

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 $[\alpha ]$ .

Let us define the group operations. The product $\alpha \ast \beta$ of $n$ -loops $\alpha$ and $\beta$ is defined by

$$\alpha \ast \beta \left(s_1,\dots ,s_n\right)=\{\begin{array}{ll}\alpha \left(2s_1,\dots ,s_n\right)& 0\le s_1\le \frac{1}{2}\\ \beta \left(2s_1-1,\dots ,s_n\right)& \frac{1}{2}\le s_1\le 1.\end{array}\text{\hspace{1em}(4.38)}$$

The product $\alpha \ast \beta$ looks like figure 4.17(a) in $X$ . It is helpful to express it as figure 4.17(b). If we define ${\alpha }^{-1}$ by

(a)

Figure 4.17. A product $\alpha \ast \beta$ of $n$ -loops $\alpha$ and $\beta$ .

(b)

$${\alpha }^{-1}\left(s_1,\dots ,s_n\right)\equiv \alpha \left(1-s_1,\dots ,s_n\right)\text{\hspace{1em}(4.39)}$$

it satisfies

$${\alpha }^{-1}\ast \alpha (s_1,\dots ,s_n)\sim \alpha \ast {\alpha }^{-1}(s_1,\dots ,s_n)\sim c_{x_0}(s_1,\dots ,s_n)\text{\hspace{1em}(4.40)}$$

where $c_{x_0}$ is a constant $n$ -loop at $x_0\in X$ , $c_{x_0}:(s_1,\dots ,s_n)\mapsto x_0$ . Verify that both $\alpha \ast \beta$ and ${\alpha }^{-1}$ are $n$ -loops at $x_0$ .

Definition 4.11

Let $X$ be a topological space. The set of homotopy classes of $n$ -loops ( $n\ge 1$ ) at $x_0\in X$ is denoted by ${\pi }_n(X,x_0)$ and called the $n$ th homotopy group at $x_0$ . ${\pi }_n(x,x_0)$ is called the higher homotopy group if $n\ge 2$ .

The product $\alpha \ast \beta$ just defined naturally induces a product of homotopy classes defined by

$$[\alpha ]\ast [\beta ]\equiv [\alpha \ast \beta ]\text{\hspace{1em}(4.41)}$$

where $\alpha$ and $\beta$ are $n$ -loops at $x_0$ . The following exercises verify that this product is well defined and satisfies the group axioms.

Exercise 4.7

Show that the product of $n$ -loops defined by (4.41) is independent of the representatives: cf lemma 4.1.

Exercise 4.8

Show that the $n$ th homotopy group is a group. To prove this, the following facts may be verified; cf theorem 4.1.

(1) $([\alpha ]\ast [\beta ])\ast [\gamma ]=[\alpha ]\ast ([\beta ]\ast [\gamma ]).$
(2) $[\alpha ]\ast [c_x]=[c_x]\ast [\alpha ]=[\alpha ]$ .
(3) $[\alpha ]\ast [{\alpha }^{-1}]=[c_x]$ , which defines the inverse $[\alpha {]}^{-1}=[{\alpha }^{-1}]$ .

We have excluded ${\pi }_0(X,x_0)$ so far. Let us classify maps from $I^0$ to $X$ . We note $I^0=\{0\}$ and $\partial I^0=\varnothing$ . Let $\alpha ,\beta :\{0\}\to X$ be such that $\alpha (0)=x$ and $\beta (0)=y$ . We define $\alpha \sim \beta$ if there exists a continuous map $F:\{0\}\times I\to X$ such that $F(0,0)=x$ and $F(0,1)=y$ . This shows that $\alpha \sim \beta$ if and only if $x$ and $y$ are connected by a curve in $X$ , namely they are in the same (arccwise) connected component. Clearly this equivalence relation is independent of $x_0$ and we simply denote the zeroth homology group by ${\pi }_0(X)$ . Note, however, that ${\pi }_0(X)$ is not a group and denotes the number of (arccwise) connected components of $X$ .

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