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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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    What is $\Sigma_2$?2012-07-18
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    @AlexBecker Probably [the orientable surface of genus $2$](http://en.wikipedia.org/wiki/File:Double_torus_illustration.png). It would be nice to have this explicit in the question, I agree.2012-07-18
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    @DylanMoreland That was my working hypothesis. I've drawn about 20 of those today though, so I want to be sure I'm not just seeing them everywhere.2012-07-18
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    Yes, $\Sigma_{2}$ is the orientable surface of genus 2.2012-07-18
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    @user36025, please add that information to the body of the question.2012-07-18

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