Hi it would be of great assistance if :
Use Galois correspondence to find all the covering spaces of
S^1 x S^1 x RP^2 (real projective plane is RP^2)
Any help would be great. Thanks
Let .
Since X is semi-locally simply connected, there exists a Galois correspondence between subgroups of the fundamental group and path-connected covering spaces of X.
The fundamental group of X is .
We shall find all the subgroups of and connect each subgroup to its corresponding path-connected covering space of X.
("A subgroup of " "A corresponding path-connected covering space of X")
1. {e} . This is a universal cover of X.
2. .
3. .
4. .
5. , where n and m are positive integers greater than 1.
In this case, consider a covering map given by , where is a polar coordinate in the plane . The map , where n is a positive integer, wraps the circle around itself n times. Thus is a covering space of , corresponding a fundamental group .
6. , where n is a positive integer greater than 1. It is similar to (5).
7. itself.
This problem becomes much more difficult if . It is because free product groups are involved in the fundamental group of X, and it is not easy to find all subgroups of a free product group and draw its Cayley graph.