Exercise 8, § 6

Let X be a paracompact topological space and let S(X) be its suspension (I, p. 149, Exercise 1); denote by $p: X\times [0,1]\rightarrow S(X)$ the canonical surjection. Let $a$ be a point of X.

a) Let G be a topological group and let E be a principal fibre space with group G and base S(X). Prove that the restriction of E to the subspace $p(X\times [0,1/2])$ admits a continuous section $s_0$, and that the restriction of E to the subspace $p(X\times [1/2,1])$ admits a continuous section $s_1$.

b) Prove that there exists a unique mapping $f: X\rightarrow$ G such that $s_1(x) =s_0(x)\cdot f(x)$ for all $x\in X$.

c) Prove that the strict homotopy class in $[(X, a); (G, e)]$ of the pointed mapping $x\mapsto f(x)f(a)^{-1}$ does not depend on the choice of the sections $s_0$ and $s_1$; it is denoted by $c(E)$.

d) Prove that two principal fibre spaces E and $E'$, with base S(X) and group G, are isomorphic if and only if $c(E) =c(E')$.

e) Prove that, for every class $c\in [(X, a); (G, e)$, there exists a principal fibre space E, with base S(X) and group G, such that $c(E) =c$.