# What are the deck transformations of this threefold cover of the figure 8?

Hatcher lists some examples of covers of a figure 8 (page 58). One of them corresponds with the group with two generators $a$ and $b$ and the relations $a^2, b^2, aba^-1, bab^-1$. I thought threefold covers had to have three deck transformations, but I can't seem to think of an automorphism of order three. Am I misunderstanding something?

• there are threefold covering spaces with trivial deck transformation group. To obtain the isomorphism $Deck \cong \pi_1(X)/\pi_1(\hat X)$ you want your covering space to be normal, i.e. $\pi_1(\hat X) \subset \pi_1(X)$ be a normal subgroup. Jul 20, 2014 at 9:55
• The figure 8 symbol, not the eighth figure. It's the third figure. Jul 20, 2014 at 9:55

Your statement is only true for normal covering spaces $\hat X\to X$, i.e. $\pi_1(\hat X) \subset \pi_1(X)$ is normal. Then we get the isomorphism $Deck(\hat X) \cong \pi_1(X) / \pi_1(\hat X)$, which is very interesting, have a look at its geometric interpretation.