Intuition behind coverings of $S^1 \vee S^1 $

Question

I was recently studying Algebraic Topology reading Hatcher, and came across a table of diagram that talks about covering spaces of $S^1 \vee S^1$ on page 58. I don't really understand how these are acting as covering spaces. I've attached an image of the first two, if anyone has any tips on how to understand the covering done using this space, it would be very appreciated, as my intuition for this is currently very poor. Hatcher Page 58 Diagram

Answer 1

Maybe obvious, but keep these things in mind:

Covering spaces have their own structures, fundamental groups, etc. The generators shown in pointed brackets are generators of loops in the covering space, not loops the figure eight.

These shapes are the covering spaces. They are not “instructions” on how to draw covering spaces over the figure eight.

Covering spaces are not spaces/shapes alone. By definition a covering space is a space plus a map that tells you how to map the covering space into the space downstairs. In Hatcher’s figures, the arrows tell us how that map works.

A list of generators is understood to include the inverses of the generators listed.

Below is an image showing how covering space #(1) on p58 works. Follow the arrows to see what maps to what.

Here are a few conclusions. The path $\ a \ $ is a loop in the cover because it goes from the base point in the cover back to that base point. So are $\ b^2 \ $ and $\ bab^{-1}\ $. Also, $\ bab \ $ is a loop; it comes from the product of generators $\ bab^{-1} b^2 \ $.

But note that $\ b \ $ is not a loop in the cover. Nor is $\ ab \ $. And as expected, these cannot be created from the generators.