4.7 Examples of higher homotopy groups

In general, there are no algorithms to compute higher homotopy groups ${\pi }_n(X)$ . An ad hoc method is required for each topological space for $n\ge 2$ . Here, we study several examples in which higher homotopy groups may be obtained by intuitive arguments. We also collect useful results in table 4.1.

Table 4.1. Useful homotopy groups.

Example 4.13

If we note that ${\pi }_n(X,x_0)$ is the set of the homotopy classes of $n$ -loops $S^n$ in $X$ , we immediately find that

$${\pi }_n\left(S^n,x_0\right)\cong \mathbb{Z}n\ge 1.\text{\hspace{1em}(4.53)}$$

If $\alpha$ maps $S^n$ onto a point $x_0\in S^n$ , $[\alpha ]$ is the unit element $0\in \mathbb{Z}$ . Since both $I^n/\partial I^n$ and $S^n$ are orientable, we may assign orientations to them. If $\alpha$ maps $I^n/\partial I^n$ homeomorphically to $S^n$ in the same sense of orientation, then $[\alpha ]$ is assigned an element $1\in \mathbb{Z}$ . If a homeomorphism $\alpha$ maps $I^n/\partial I^n$ onto $S^n$ in an orientation of opposite sense, $[\alpha ]$ corresponds to an element $-1$ . For example, let $n=2$ . Since $I^2/\partial I^2\cong S^2$ , the point in $I^2$ can be expressed by the polar coordinate $(\theta ,\varphi )$ , see figure 4.19. Similarly, $X=S^2$ can be expressed by the polar coordinate $({\theta }^{\prime },{\varphi }^{\prime })$ . Let $\alpha :(\theta ,\varphi )\to ({\theta }^{\prime },{\varphi }^{\prime })$ be a 2-loop in $X$ . If ${\theta }^{\prime }=\theta$ and ${\varphi }^{\prime }=\varphi$ , the point $({\theta }^{\prime },{\varphi }^{\prime })$ sweeps $S^2$ once while the point $(\theta ,\varphi )$ scans $I^2$ once in the same orientation. This 2-loop belongs to the class $+1\in {\pi }_2(S^2,x_0)$ . If $\alpha :(\theta ,\varphi )\to ({\theta }^{\prime },{\varphi }^{\prime })$ is given by ${\theta }^{\prime }=\theta$ and ${\varphi }^{\prime }=2\varphi$ , the point $({\theta }^{\prime },{\varphi }^{\prime })$ sweeps $S^2$ twice while $(\theta ,\varphi )$ scans $I^2$ once. This 2-loop belongs to the class $2\in {\pi }_2(S^2,x_0)$ . In general, the map $(\theta ,\varphi )\mapsto (\theta ,k\varphi ),k\in \mathbb{Z}$ , corresponds to the class $k$ of ${\pi }_2(S^2,x_0)$ . A similar argument verifies (4.53) for general $n>2$ .

Figure 4.19. A point in $I^2$ may be expressed by polar coordinates $(\theta ,\varphi )$ .

Example 4.14

Noting that $S^n$ is a universal covering space of $\mathbb{R}{P}^n$ for $n>2$ , we find

$${\pi }_n\left(\mathbb{R}{P}^n\right)\cong {\pi }_n\left(S^n\right)\cong \mathbb{Z}n\ge 2.\text{\hspace{1em}(4.54)}$$

[Of course this happens to be true for $n=1$ , since $\mathbb{R}{P}^1=S^1$ .] For example, we have ${\pi }_2(\mathbb{R}{P}^2)\cong {\pi }_2(S^2)\cong \mathbb{Z}$ . Since $\mathrm{SU}(2)=S^3$ is the universal covering group of $\mathrm{SO}(3)=\mathbb{R}{P}^3$ , it follows from theorem 4.10 that (see also (4.48))

$${\pi }_3(\mathrm{SO}(3))\cong {\pi }_3(\mathrm{SU}(2))\cong {\pi }_3\left(S^3\right)\cong \mathbb{Z}.\text{\hspace{1em}(4.55)}$$

Shankar’s monopoles in superfluid ${}^3\mathrm{He}-\mathrm{A}$ correspond to non-trivial elements of these homotopy classes, see section 4.10. ${\pi }_3(\mathrm{SU}(2))$ is also employed in the classification of instantons in example 9.8.

In summary, we have table 4.1. In this table, other useful homotopy groups are also listed. We comment on several interesting facts.

(a) Since $\mathrm{Spin}(4)=\mathrm{SU}(2)\times \mathrm{SU}(2)$ is the universal covering group of SO(4), we have ${\pi }_n(\mathrm{SO}(4))={\pi }_n(\mathrm{SU}(2))\oplus {\pi }_n(\mathrm{SU}(2))$ for $n>2$
(b) There exists a map $J$ called the $J$ -homomorphism $J:{\pi }_k(\mathrm{SO}(n))\to {\pi }_{k+n}(S^n)$ , see Whitehead (1978). In particular, if $k=1$ , the homomorphism is known to be an isomorphism and we have ${\pi }_1(\mathrm{SO}(n))={\pi }_{n+1}(S^n)$ . For example, we find

$${\pi }_1(\mathrm{SO}(2))\cong {\pi }_3(S^2)\cong \mathbb{Z}$$

$${\pi }_1(\mathrm{SO}(3))\cong {\pi }_4(S^3)\cong {\pi }_4(\mathrm{SU}(2))\cong {\pi }_4(\mathrm{SO}(3))\cong {\mathbb{Z}}_2.$$

(c) The Bott periodicity theorem states that

$${\pi }_k(\mathrm{U}(n))\cong {\pi }_k(\mathrm{SU}(n))\cong \{\begin{array}{ll}\{e\}& \text{i f k i s e v e n}\\ \mathbb{Z}& \text{i f k i s o d d}\end{array}\text{\hspace{1em}(4.56)}$$

for $n\ge (k+1)/2$ . Similarly,

$${\pi }_k(\mathrm{O}(n))\cong {\pi }_k(\mathrm{SO}(n))\cong \{\begin{array}{ll}\{e\}& \text{i f}k\equiv 2,4,5,6(\mathrm{mod}8)\\ {\mathbb{Z}}_2& \text{i f}k\equiv 0,1(\mathrm{mod}8)\\ \mathbb{Z}& \text{i f}k\equiv 3,7(\mathrm{mod}8)\end{array}\text{\hspace{1em}(4.50)}$$

for $n\ge k+2$ . Similar periodicity holds for symplectic groups which we shall not give here.

Many more will be found in appendix A, table 6 of Ito (1987).

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF

来源版本：second-edition-local

来源 PDF SHA-256：fcd24c3371d092a8c699ccf5cf0765462924b6c48412804231076e8084b1940d

OCR 来源 SHA-256：0b6324f01172a3548cd34382e82fd8bfae574602b6b2cfaa357ba96bc2c35478

OCR 产物 SHA-256：8a1134c9f2c9eaa024885ad6563d43abc4e62a156b5a7ec1b3ff016dde59b26d