Hey i need to show that every covering map of the Torus $T^2=S^1 \times S^1$ is regular. I really have no idea here how to start here. I guess i have to show that if p is a covering map than it follows, that $p*(\pi_1(Y))$ is a normal subgroup in $\pi_1(T)$ ?
I know that the latter is isomorphic to the Group $G:=\{a,b: aba^{-1}b^{-1}\}$ does it just follow because G is Abelian ? and so is $\pi_1(T)$ so every sub group is normal ? Can i assume that $p*(\pi(Y))$ is always a sub group in $\pi(T)$ ?
Yes, $\pi_1(T^2)=\mathbb{Z}\times\mathbb{Z}$ is abelian, so every subgroup is normal. Hence every covering map is regular (known as a normal cover in Hatcher).
You can assume $p_*(\pi_1(Y))$ is a subgroup of $G$ because $p_*$ is the induced map, which is a homomorphism.