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.

Chapter 1, section 3 of Hatcher's book has great pictures for this exact example (see page 58).  Figures 12 and 13 should be examples of what you're looking for.  Of course, the surrounding content should also be helpful, especially the picture of the universal cover found on page 59.
