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

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This discussion focuses on constructing a CW complex for an infinite genus orientable manifold, specifically an infinite holed torus. The user expresses confusion regarding the fundamental group being a free group in this context and seeks clarity on the structure of the CW complex. They compare the finite case, which consists of a single 0-cell, 2n 1-cells, and a single 2-cell, to the infinite case, questioning whether this structure remains valid. The user also considers the use of projective limits to aid in visualization and understanding.

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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?
 
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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.
 

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