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I am trying to compute the relative homology $H_{n}(X, \partial X)$ for all $n \geq 0$, where $X$ is a torus with a point removed.

$X$ is homotopy equivalent to wedge sum of two circles $S^{1} \vee S^{1}$, this is easy.

I know how the long exact sequence for relative homology looks like, but i am not sure how $\partial X$ looks like. Can anybody help me with this question, please? Thanks in advance!

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    $\begingroup$ What do you mean by $\partial X$? The boundary of $X$ as a manifold? In that case, $\partial X$ is empty... $\endgroup$ Jul 4, 2016 at 21:07
  • $\begingroup$ Yes, i mean the boundary. $\endgroup$
    – Lullaby
    Jul 4, 2016 at 21:07

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Since $X$ is an open subset of the torus, every point of $X$ has a neighborhood homeomorphic to $\mathbb{R}^2$. That is, $X$ is a manifold without boundary, so $\partial X=\emptyset$. So $H_*(X,\partial X)=H_*(X,\emptyset)$ is just the same as $H_*(X)$.

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  • $\begingroup$ Then everything is fine! Thank you very much! $\endgroup$
    – Lullaby
    Jul 4, 2016 at 21:12

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