Due February 21.

A. Let \(p:E \to B\) be a regular cover. The group of covering translations \(\mathrm{Aut}(p)\) acts on \(E\). Show that the orbit space of this action is homeomorphic to \(B\), i.e. \(B \cong E/\mathrm{Aut}(p) \).

B. Given a group \(G\) and a normal subgroup \(N\), show that there exists a normal covering space \(E\to B\) with \(\pi_1(B) \cong G\) and \(\pi_1(E) \cong N\), and covering transformation group \(\mathrm{Aut}(p) \cong G/N\).

C. For a path-connected, locally path-connected, and semilocally simply-connected space \(B\), call a path-connected covering space \(E\to B\) abelian if it is normal and has abelian covering transformation group. Show that \(B\) has an abelian covering that is ‘universal’ in the sense that it covers every abelian cover of \(B\). Find the universal abelian cover of \(S^1 \vee S^1\).

D. Construct a two-sheeted covering of the Klein bottle by the torus.

E. Read the section in Hatcher about representing covering transformations by permutations. Let \(B\) be the wedge of two circles, so that \(\pi_1(B) = \langle a, b\rangle\). Construct a three-sheeted cover corresponding to the representation \(\rho: \pi_1(B) \to S_3\) given by \(\rho(a) = (2\,3)\) and \(\rho(b) = (1\,2\,3)\).