TITLE: Covering spaces of $S^1 \vee S^1$.
QUESTION [11 upvotes]: I can show that $\pi_1(S^1 \vee S^1)$ is the free group $\mathbb{Z} * \mathbb{Z}$, i.e. I can prove van Kampen's theorem, which boils down to 1. equivalence of categories between covers of a manifold, and sets with an action of the fundamental group, and 2. gluing of covers.
Question. For each of the following covering spaces of $S^1 \vee S^1$, what is the corresponding subgroup of $\pi_1(S^1 \vee S^1) \cong \mathbb{Z} * \mathbb{Z}$, and is the cover is regular (i.e. normal) or not?

REPLY [2 votes]: If you look at this table in the book of Allen Hatcher, you find groups corresponding to all the subgroups you mention.  Chapter 1 is on the Fundamental Group 

Here is his definition of normal cover:

A covering space $p: Y \to X$ is normal if for each $x \in X$ and each pair of lifts $y_1, y_2 \in p^{-1}(x)$ there is a deck transformation taking $y_1 \to y_2$.
Proosition 1.39 A covering space is normal iff the image of $p_\ast: \pi_1(Y) \to \pi_1(X)$  is a normal subgroup in $\pi_1(Y)$.

There's still quite a bit of work to do -- either manipulating the algebraic symbols in the free group $\langle a, b, a^{-1}, b^{-1}\rangle$ or reasoning about how changing the base point of our loop behaves in the covering space.  At least I hope you can get started.