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.

 

[math771]

I'm not an expert in topology, but I'll try my best to help. From my understanding, the CW complex for an n-genus orientable manifold would have one 0-cell (representing the single connected component), 2n 1-cells (representing the n-tori), and an infinite number of 2-cells (representing the infinite number of holes in the manifold). Each 1-cell would attach to the 0-cell, and each 2-cell would attach to the boundary of the 1-cells.

As for the fundamental group being a free group, I believe this is because the 1-cells (representing the n-tori) are all disjoint from each other and can be freely deformed without changing the fundamental group. This means that the fundamental group is generated by the 1-cells, and since there are 2n 1-cells, the fundamental group is a free group with 2n generators.

I hope this helps a bit in understanding the CW complex for the infinite case. As for visualizing it, I think it might be helpful to look at some images or diagrams of CW complexes for orientable manifolds with higher genus, and then try to imagine extending that to the infinite case.