Exercise 2, § 5

a) Let E be a simply-connected topological space and $u: E\rightarrow G$ a continuous map such that $u(E)\subset G_r$. Show that $u$ is homotopic (General Topology, Chap. XI) to the constant map with value $e$ (consider the covering $\varphi_r: (G/T)\times \mathfrak{t}_r\rightarrow G_r$; lift $u$ to $\widetilde{u}: E\rightarrow (G/T)\times \mathfrak{t}$, then use the fact that $\mathfrak{t}$ is contractible).

$^*b)$ Prove that the group $\pi_2(G)$ is zero.

(Let $u:\mathbf{S}_2\rightarrow G$ be a map of class $C^{\infty}$; by using Prop. 1 and the transversality theorem, show that $u$ is homotopic to a map with image contained in $G_r$, then apply $a$).)\*