In the following covering space, can $bab$ be a generator of a covering space of $S^1\vee S^1$? Is it more natural to let $bab$ as the generator instead of $bab^{-1}$ as pointed out by Hatcher in his book Algebraic Topology page 58?
The $b^{-1}$ means the reverse path taken in the opposite direction of $b$ right? Why can't we use the other path $b$ above of the first $b$?
Could someone please help me clarify the confusion? Thanks.
Maybe it feels more natural to include $bab$ since it helps you cover the entire graph, but there is a much simpler reason to include $bab^{-1}$ instead. The reason is that $bab$ wraps around both circles, while $bab^{-1}$ wraps around only one circle, so altogether the generators in $\langle a, b^2, bab^{-1}\rangle$ each wrap around one distinct circle.
Note that $bab^{-1}b^2 = bab$, so both presentations are technically equivalent.