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We know that if $M$ is a $d-$dimensional manifold, then $H_n(M, M\setminus \{x\})=\mathbb{Z}$ for $n=d$ and equals to $0$ else.

I can understand formally the result and its proof, but I can't understand what it is actually saying. What is the intuitive meaning of this result?

Thank you in advance

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If you understand the proof you probably understand the intuition. Only that part of $M$ in a neighbourhood of $x$ is important, since we have relative to $M-\{x\}$. And since $M$ is a manifold it looks locally like $\mathbb{R}^d$ so it says that regardless of how $M$ look globally, $$H_n(M,M-\{x\}) =H_n(D,D-\{x\})$$ where $D$ is a $d$-disk. This is just a paraphrase of the proof.

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