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I'm trying to better understand singular homology groups. Standard examples like spheres and tori make it seem like $H_i(X)$ is generated by a subset that is in bijection with $i$-dimensional holes in $X$. Even more, they make it seem like those groups are freely generated by such subsets, but I guess it would be naive to think THAT might hold generally.

Could someone throw some light on the way these groups relate to holes in a topological space? Also a short comment on what changes when we use a ring of coefficients different than $\mathbb{Z}$ would be appreciated.

Tom
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    The "holes" metaphor isn't useful to take too far. It totally fails to explain, for example, what torsion in homology like $H_1(\mathbb{RP}^2) \cong \mathbb{Z}_2$ could possibly mean. – Qiaochu Yuan Oct 09 '20 at 19:53

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