Exercise 3, § 5

Let $A$ be a topological space and let $\sigma : A\rightarrow A$ be a homeomorphism. Let $E_{\sigma}$ be the quotient of the space $\mathbf{R}\times A$ by the action of the group $\mathbf{Z}$ such that $1\cdot (t, a) = (t+ 1, \sigma (a))$. Let $p$ be the mapping of $E_{\sigma}$ into $\mathbf{R}/\mathbf{Z}$ which maps the class of an element $(t, a)$ of $\mathbf{R}\times A$ onto the class of $t$ in $\mathbf{R}/\mathbf{Z}$.

a) Prove that $E_{\sigma}$ is a locally trivial fibered space with basis $\mathbf{R}/\mathbf{Z}$ of fiber-type A.

In the remainder of this exercise, it is assumed that A is a discrete topological space.

b) Prove that every covering space of $\mathbf{R}/\mathbf{Z}$ is isomorphic to a covering space of the form $E_{\sigma}$, where A is a discrete topological space.

c) Prove that $E_{\sigma}$ is connected if and only if $\sigma$ has at most one orbit.

d) Let B be a discrete topological space and let $\tau$ be a permutation of B. Construct a bijection between the set of morphisms (resp. of isomorphisms) of covering spaces $f: E_{\sigma}\rightarrow E_{\tau}$ and the set of mappings (resp. of bijective mappings) $\varphi : A\rightarrow B$ such that $\tau (f(a)) =f(\sigma (a))$ for all $a\in A$.