# Covering space of figure 8 corresponding to $\mathbb{Z}$

Provided that certain conditions are satisfied, we know that there's a one to one correspondence between covers of a space and subgroups of the fundamental group of that space. Since $\mathbb{Z}$ is a subgroup of $F(2)$, the free group on two generators, which is the fundamental group of the figure, 8 space, what's the cover corresponding to $\mathbb{Z}$? I know that this can be satisfied if I can find a graph covering the figure 8 space whose Euler Characteristic is $0$, but such a cover seems impossible.

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Which copy of $\mathbb{Z}$? – Qiaochu Yuan Mar 17 '13 at 23:47
Actually, it says an infinite cyclic subgroup of $F(2)$, so not quite $\mathbb{Z}$. – user67200 Mar 17 '13 at 23:50
What says this? Which infinite cyclic subgroup? – Qiaochu Yuan Mar 17 '13 at 23:50
The problem I'm trying to solve. I'm supposed to find a cover of the figure 8 space corresponding to an infinite cyclic subgroup of $F(2)$. So I guess any works. – user67200 Mar 17 '13 at 23:51