Exercise 6, § 6

Let X be the quotient space of the space $\mathbf{R}\times \{0,1\}$ by the finest equivalence relation for which the points $(t,0)$ and $(t,1)$ are equivalent, for all $x\in \mathbf{R}^*_+$. Denote by $p:\mathbf{R}\times \{0,1\} \rightarrow X$ the canonical surjection.

a) Prove that the topological space X is homotopic to a point.

b) For which points $x\in X$ is the pointed space $(X, x)$ contractible?

c) Let $U =p(\mathbf{R}\times \{0\})$ and $V =p(\mathbf{R}\times \{V\})$. Let G be a topological group. Construct a bijection from the set $\mathscr{C}(\mathbf{R}^*_+; G)$ of continuous mappings from $\mathbf{R}^*_+$ into G onto the set of isomorphism classes of principal bundles with base X and group G.

d) Deduce that there exist principal bundles with base X and group $\mathbf{R}$ which are not trivializable.