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I'm quite at a loss with this...I want to use Mayer-Vietoris with open covers $A=\Sigma_{2}\times (S^{1}\setminus \{p\})$ and $B=\Sigma_{2}\times (S^{1}\setminus \{q\})$ so that $A$ and $B$ both deformation retract to $\Sigma_{2}$ and $A\cap B$ deformation retracts to $\Sigma_{2}\times\{0,1\}$, but I don't understand how to think about the inclusion maps $H_{n}(A\cap B) \hookrightarrow H_{n}(A)\bigoplus H_{n}(B)$.

$\Sigma_2$ denotes the orientable surface of genus two.

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    @user36025, please add that information to the body of the question.2012-07-18

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Note that $A$ and $B$ are both homotopic to $\Sigma_2$, and so $H_n(A \cap B) \cong H_n(A) + H_n(B)$ for all $n$, if you choose your embeddings $i$ and $j$ wisely.

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    If $A$ is homotopic to $\Sigma_{2}$, then $H_{1}(A) \simeq \mathbb{Z}^{4}$ according to topospaces (http://topospaces.subwiki.org/wiki/Homology_of_compact_orientable_surfaces ). I'll try to think about this again tonight.2012-07-20