CW complex for infinite holed torus? (Surface of infinite genus)

[dumbQuestion]

I am just trying to figure out how to make a CW complex for this. For the n-genus orientable manifold (connect sum of n-tori) I feel like a lot of things make sense, fundamental group, CW complex, etc. But in the infinite case, things seem to fall apart. For example, I can not figure out how the fundamental group is a free group. I was hoping to figure this out by first looking at the CW complex of this surface, but I'm not sure I can picture it.


IN a finite case, I just have a single 0 cell (1 vertex), 2n 1-cells, and a single 2 cell. BUt does this hold at the infinite case? If not what's an alternate way to visualize it?

 

[fresh_42]

I would consider projective limits here, although I'm not sure this would help. To precisely determine where and why "things fall apart" would also be of great help.