I'm trying to come up with covering spaces of $X= S^1 \lor S^1 \lor S^1$ corresponding to two subgroups of $\pi_1(X)= F_3=\langle a,b,c \rangle$. First, the subgroup $H=\langle a^2, b^2, c^2 \rangle$ and second, the normal subgroup $N= \langle \langle a^2, b^2, c^2 \rangle \rangle$ (the smallest normal subgroup containing $H$).
I've got some candidate covers in mind, but I'm not sure if I'm correct. Below is my idea for the cover corresponding to $N$.
This cover shown is normal, since the map swapping the vertices is a covering space transformation, and so the subgroups corresponding to this cover is normal in $F_3$, and it contains $H$.
If this is the correct cover, how can I show this corresponds to the smallest normal subgroup containing $H$? What about the cover corresponding to $H$ itself? No finite degree cover has worked, and I can't seem to cook up an infinite cover that works.
Any hints would be appreciated!
