We know that the fundamental group of $S^1\vee \Bbb RP^2$ is $\langle a,b\mid b^2\rangle$. I am attemping to find the covering spaces of $S^1\vee \Bbb RP^2$ corresponds to:
(a) the subgroup $\langle a\rangle$ of $\langle a,b\mid b^2\rangle$
(b) the subgroup $\langle b\rangle$
(c) the subgroup $\langle a, bab\rangle$
For (a) I think the double cover $S^1\vee S^1\vee S^2$ has the desired fundamental group. But it looks like a normal covering space and $\langle a\rangle $ is not a normal subgroup. So is it something wrong here?
For (b) I have found the covering space corresponds to the normal subgroup generated by $b$, it is a helix with an $\Bbb RP^2$ at each integer point. But no idea for the subgroup generated by $b$.
For (c) I have no idea.
May I please ask how can I find them? Thanks!
EDIT: Now I think for the covering space corresponds to $\langle b\rangle$, recall the universal cover of $S^1\vee \Bbb RP^2$, it is an infinite tree with an $S^2$ attaching on each vertices. I think the desired covering space is to link an $\Bbb RP^2$ with two branches linking to the tree which looks like the universal cover of $S^1\vee\Bbb RP^2$.
To construct a covering space corresponds to $\langle a\rangle$, first considering the universal cover of $S^1\vee\Bbb RP^2$, fix a "central" sphere $S^2$, it has four branches, keep two of them unchanged, and for the other two branches, remove them and attach an $S^1$ instead.
Is that correct? I think it is quite hard to discribe...
Thanks for patience of reading this. And thanks for any help.