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What is a geometric interpretation of $H_0, H_1, H^0, H^1$ with coefficients in $\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$,and $G$ for any abelian group?

Furthermore, what is a geometric interpretation of $H_n(X)$ when $X$ is a compact $n$-manifold?

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    For homology: https://math.stackexchange.com/questions/40149/intuition-of-the-meaning-of-homology-groups#:~:text=Intuitively%2C%20homology%20finds%20n%2Dholes,your%20space%20which%20is%20missing. For cohomology it is good to look at De Rham cohomology, which is defined specifically for differentiable manifolds: https://math.stackexchange.com/questions/1112419/intuitive-approach-to-de-rham-cohomology#:~:text=The%20intuition%20behind%20homology%20may,quotient%20of%20vector%20spaces%20procedure. – Kevin.S Jan 08 '21 at 14:18
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    Did you read answers to https://math.stackexchange.com/questions/40149/intuition-of-the-meaning-of-homology-groups?noredirect=1&lq=1 ? – Moishe Kohan Jan 08 '21 at 18:55

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