I am currently working on a problem about the fundamental group of a covering space of $S^1 \vee S^1$, and I just want to see if I understand the mechanics of the question.
Let the free group $F_2= \langle a,b \rangle$ and we have a covering map $\phi: F_2 \rightarrow \mathbb{Z}_2 \bigoplus \mathbb{Z}_3$ as follows: $$a \mapsto (1+2\mathbb{Z}, 0+3\mathbb{Z})$$ $$b \mapsto (0+2\mathbb{Z}, 1+3\mathbb{Z})$$
If $H \leq F_2$ is the kernel of $\phi$, then I am tasked with finding (drawing suffices) a covering space of $S^1 \vee S^1$ such that the cover has fundamental group isomorphic to $H$ under the induced homomorphism by $\phi$.
First, am I correct in believing that e.g. $1+2\mathbb{Z}$ is equivalent to $[1]$, or the congruence class of 1 mod 2?
My next thought was I need to find the elements of $F_2$ which will map to $(0,0)$ under $\phi$ since $H$ is the kernel.
I noticed that $a^2=([1]+[1],[0]+[0])=(0,0)$ and similarly for $b^3$, so any multiples of $a^2$ and $b^3$ or products thereof would be in the kernel. Does this mean that $H= \langle a^2,b^3,a^2b^3 \rangle?$
If so, I shouldn't have a problem drawing a covering space that satisfies the problem, I just want to check to make sure I understand what's going on because this is all brand new material and notation for me.