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How can I determine the CW Structure of a torus with $g-$holes?

I understood that it consists of $4g$ $0-$ cells and $4g$ $1-$ cells. I understood well how can I form the skeletons $X_0$ and $X_1$ but I am unable to find maps $\phi : D^2 \to X $ such that $\phi_{| S^{n-1}}\text{ }\text{ }\text{ }$ is mapped to $X_1$.

I read other answers but I am unable to define the maps $\phi$. They didn't help much.

zero2infinity
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  • Have you looked at Hatcher, p. 5? – Jeroen van der Meer Feb 05 '21 at 15:47
  • Yes, It says it is formed of$4g$ $0-$cells $4g$ $1-$cells and $1$ $2-$cell. Thats all – zero2infinity Feb 05 '21 at 15:59
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    Read up on the classification of compact surfaces. One common method of classification is to show that all surfaces are polygons with pairs of edges glued together somehow. In particular, you can represent a $g$-hole torus in such a way. Then its CW structure will be much easier to understand. Book recommendation: Massey, A Basic Course in Algebraic Topology, Chapter 1 – Alex G. Feb 05 '21 at 16:21

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