Constructing Covering Spaces for Algebraic Topology Qualifier Exam Question

[yaa09d]

This is a qualifier exam question in algebraic topology:

Let Z * Z_2 = <a, b | b^2> be represented by X = S^1 [tex]\vee[/tex] RP^2 , i.e. the wedge of S^1 (the unit circle)
and RP^2 (the real projective plane).
For the subgroup H below construct the covering space ˜X by sketching a good picture for ˜X and explaining how it covers X.
In each case, give a group presentation for the group G of covering transformations of the covering
p:˜X -----> X and describe (using your picture) the action of G on ˜X .

(a) H is the smallest normal subgroup containing b.
(b) H is the smallest normal subgroup containing a.
(c) H is the smallest normal subgroup containing a^2 and b.
(d) H is the trivial subgroup.

Attached is a pdf format of the question.

 

[lippyka]

For (a), H is the subgroup generated by b, and X has two components, the circle S^1 and the projective plane RP^2. The covering space ˜X is two copies of S^1 and one copy of RP^2. The group G of covering transformations is Z * Z_2 / <b>, which is Z_2. The action of G on ˜X is to swap the two copies of S^1.For (b), H is the subgroup generated by a, and X has two components, the circle S^1 and the projective plane RP^2. The covering space ˜X is two copies of S^1 and two copies of RP^2. The group G of covering transformations is Z * Z_2 / <a>, which is Z. The action of G on ˜X is to rotate the two copies of S^1 and swap the two copies of RP^2.For (c), H is the subgroup generated by a^2 and b, and X has two components, the circle S^1 and the projective plane RP^2. The covering space ˜X is two copies of S^1 and one copy of RP^2. The group G of covering transformations is Z * Z_2 / <a^2, b>, which is Z_2. The action of G on ˜X is to rotate the two copies of S^1 and swap the copy of RP^2.For (d), H is the trivial subgroup, and X has two components, the circle S^1 and the projective plane RP^2. The covering space ˜X is just X itself, so it is two copies of S^1 and one copy of RP^2. The group G of covering transformations is Z * Z_2, which is Z * Z_2. The action of G on ˜X is to rotate the two copies of S^1 and swap the copy of RP^2.