I came across the problem of computing the homology groups of the closed orientable surface of genus $g$.
Here Homology of surface of genus $g$ I found a solution via cellular homology. This seems to me like the natural way of calculating something of this sort although I know that it is also possible to do this using the Mayer-Vietoris sequence.
I understand the main calculations of the solution referred to above, however the CW-structure that the surface of genus $g$ is endowed with is a mystery to me.
I understand it for the case $g=1$ where it is quite conceivable in a graphic manner.
Could someone please explain what is going on for $g \geq 2$?
Thanks in advance for any help.
Best Answer
I always found the "$4g$-gon with identified sides" approach mysterious until I watched this video. Basically, you can construct a genus $g$ surface by gluing a genus 1 surface to a genus $g-1$ surface.
At around 1:00 in the video, the speaker shows how to make a 2-holed torus by taking two 1-holed tori (i.e., rectangles with opposite sides identified), cutting them at the corner and pasting them together. I think you can continue this process recursively to get orientable surfaces of higher genus. So to get a genus 3 surface, you would take the octagon you had for a genus 2 surface, a torus (rectangle), cut them both at a corner and glue them together.