Other homotopy groups

The next homotopy group we consider is $\pi_2({\cal M})$. This is relevant when considering finite energy solutions in three dimensional space. The homotopy group

$$\pi_2(S_2)=Z,$$

as well. The equivalence class with topological number $n$ corresponds to wrapping the unit sphere $n$ times with the mapping. A simple example of this mapping is taking the unitary rotation matrix

$$U(\theta, \phi)=e^{i\phi\sigma_3/2}e^{i\theta\sigma_1/2}$$

and tranforming the ``unit vector'' $\sigma_3$, representing an adjoint representation configuration into

$$U\sigma_3U^\dagger=\sin\theta \cos\phi\sigma_1+\sin\theta\sin\phi\sigma_2+\cos\theta\sigma_3.$$

This corresponds to a map of equivalence class $n=1$. This last map can also be considered as a map from $S_2\to SU(2)$. Thus, we come to the conclusion that

$$\pi_2(SU(2))=Z.$$

The second homotopy group of other $SU(N)$ groups is the same.

Let us now consider the mappings of $S_3$, that is relevant in considering finite action configuration in $D=4$ Euclidean space. Take the example of $SU(2)$. We earlier established that $SU(2)$ is equivalent to $S_3$. Thus we have

$$\pi_3(SU(2))=Z.$$

In fact, one can also show that this relationship can be extended to

$$\pi_3(SU(N))=Z.$$

In other words, broken $SU(N)$ gauge theories are an appropriate ground for finding classical lumpes localized in the four dimensional Euclidean space.