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I understand the method of getting the homology but I dont know how to interpret it: Ive always interpreted 1-homology as the number of independent 1-cycles you can do on a surface. For example for the torus with homology $H_1(T)=\mathbb{Z \times{Z}}$ means you can have two independent 1-cycles that you can combine infinitely with each other. But with the klein bottle this intuition doesnt seem to work as the 1-homology is $H_1(K)=\mathbb{Z_2 \times{Z}}$, what does the $\mathbb{Z_2}$ term means? Does combining 2 times the 1st independent path is equivalent to not doing any path at all? How is that possible?

Mallacan
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    Homology is not just about counting cycles. You may take a look at this question: https://math.stackexchange.com/questions/1170056/intuition-of-non-free-homology-groups – Mark Nov 28 '22 at 20:20
  • @Mark Mhm, the things he explains makes sense (I would have to read deeper for a better understanding but makes sense) but I dont quite get how that would work for the Klein bottle – Mallacan Nov 28 '22 at 20:43
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    In the second answer, the last paragraph is about the Klein bottle. It's like it has some kind of "twisted cycle". – Mark Nov 28 '22 at 20:46
  • @Mark After a few diagrams and thinking it through I understand it, thanks for the recommendation of the post. – Mallacan Nov 28 '22 at 21:32

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