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Say you have two surfaces of genus 2, say $X$ and $Y$ and you want to attach them via homotopy attaching maps $f$ along their waist curves. Then what will the fundamental group of the adjunction space $X\bigcup_fY$ be?

Any Hints on how to proceed?

Thanks

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    Ok, so given that the adjunction space represents the connected sum of 4 tori, then the presentation group would be: $\langle \alpha_1, \beta_1,\dots, \alpha_4, \beta_4 : \alpha_1\beta_1\alpha_1^{-1}\beta_1^{-1},\dots, \alpha_4\beta_4\alpha_4^{-1}\beta_4^{-1} \rangle$ right? If this is correct, then I just need to have a better argument as to why that adjunction space is the connected sum of 4 tori??2012-10-26

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