Massey Exercise 5.6.2


(a) First, we determine all the covering spaces of . Note that . Since this is an abelian group, every subgroup belongs to its own trivial conjugacy class. That is, each subgroup corresponds to a unique covering space. The subgroups of are precisely for , with corresponding to the trivial subgroup. For , the subgroup is isomorphic to . For describing covering maps, let be the unit circle in .

Corresponding to the trivial group, we have the covering space where .

For each , the covering space associated with is where .


(b) Now we determine all the covering spaces of . Note . Of course, there are exactly two (non-conjugate) subgroups – the trivial one, and the whole group. Let be quotient of under the identification corresponding to identifying the edges in pairs. Corresponding to the trivial group we have the covering space , where denotes the first quadrant of , and the map is . Corresponding to the whole group we have the trivial covering space .


(c) We now determine all the covering spaces of the annalus . However, note that this space is homeomorphic to . The fundamental group is still , since this deformation retracts to .

Corresponding to the trivial group, we have the covering space , where the covering map is .

Corresponding to , we have the covering space where .