Exercise 4, § 5

Let A be the union of the point $(0,1)$ and the set of the points of $\mathbf{R}^2$ of the form $(1/n,1)$, for $n\in \mathbf{N}^*$. Let B be the set of the points of $\mathbf{R}^2$ of the form $(t/n, t)$, for $n\in \mathbf{N}^*$ and $t\in [0,1]$. Let $C_1$ be the circle of diameter $[(0,0),(0,-1)]$ and let $C_2$ be the circle of diameter $[(0,1),(0,2)]$. Finally put $X = A\cup B\cup C_1\cup C_2$. Let $D =\{\alpha , \beta \}$ be a set with two elements and let F be the space $\mathbf{Z}\times D$ endowed with the discrete topology. a) Let $E_0$ be the complement of $\{0\}\times (\mathbf{Z}\times \{\alpha \})$ in $(A\cup B)\times F$. Prove that the first projection makes $E_0$ a trivialisable $(A\cup B)$-space fibred of fibre-type F.

b) Let $u_1$ be the permutation of F such that $u_1(n, w) = (n+ 1, w)$ for all $n\in \mathbf{Z}$ and all $w\in D$. Construct a covering of $C_1$ isomorphic to the covering $E_{u_1}$ of exercise 3.

c) Let $u_2$ be the permutation of F such that $u_2(n, \alpha ) = (n, \beta )$ and $u_2(n, \beta ) =$ $(n, \alpha )$ for all $n\in \mathbf{Z}$. Construct a covering of $C_2$ isomorphic to the covering $E_{u_2}$ of exercise 3.

d) Construct a covering E of X whose restrictions to $A\cup B$, $C_1$ and $C_2$ are isomorphic to $E_0,E_1$ and $E_2$ respectively.

e) Prove that the space E is connected.

f) Prove that the group Aut(E) acts transitively on the fibre at $(0,1)$ but not on the other fibres.