Exercise 7, § 6

Let $G$ be a topological group and let $E$ be a principal fiber space with basis $\mathbf{S}_2$ and group G. We denote by $\mathbf{S}^+_2$ and $\mathbf{S}^-_2$ the two hemispheres, intersections of $\mathbf{S}_2$ and the half-spaces defined by the conditions $z\geqslant 0$ and $z\leqslant 0$ respectively; we identify $\mathbf{S}^+_2\cap \mathbf{S}^-_2$ with $\mathbf{S}_1$.

(a) Prove that the fiber space E possesses a section $s^+$ (resp. $s^-$) above $\mathbf{S}^+_2$ (resp. $\mathbf{S}^-_2$). Prove that there exists a unique continuous mapping $f:\mathbf{S}_1\rightarrow G$ such that $s^+(x) =s^-(x)\cdot f(x)$ for all $x\in \mathbf{S}_1$.

(b) Prove that the strict homotopy class of the map from $\mathbf{S}_1$ into G given by $x\mapsto c(x)c(1)^{-1}$ does not depend on the choice of the sections $s^+$ and $s^-$; we denote it by $c(E)$.

(c) Prove that two principal fiber spaces E and $E'$, with basis $\mathbf{S}_2$ and group G, are isomorphic if and only if $c(E) =c(E')$.

(d) Prove that, for every class $c\in \pi_1(G, e)$, there exists a principal fiber space E, with basis $\mathbf{S}_2$ and group G such that $c(E) =c$.