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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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    Which four circles are you seeing? It seems to me you'd get a genus 4 surface out of this, not genus 3.2012-10-26
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    Now, that i think about it, i'm not actually sure. But, it seems more what you were saying, that it will be a surface of genus 4. So, in this case then it is homeomorphic to a connected sum of four tori, so the fundamental group would be simply $\pi_1(\mathbb{T})\star\pi_1(\mathbb{T})\star\pi_1(\mathbb{T})\star\pi_1(\mathbb{T})=\mathbb{Z}^2\star\mathbb{Z}^2\star\mathbb{Z}^2\star\mathbb{Z}^2.$ Right? With $\star$ I have denoted the free product.2012-10-26
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    A presentation for the fundamental group would then be: $\langle\alpha_1,\beta_1,\alpha_2,\beta_2,\alpha_3,\beta_3,\alpha_4,\beta_4: \alpha_1 \beta_1\alpha_1^{-1}\beta_1^{-1}...\alpha_4\beta_4\alpha_4^{-1}\beta_{4}^{-1}\rangle$2012-10-26
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    that would do it. As I say, I'm not sure of this analysis.2012-10-26
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    Yeah, I'm not so sure myself either...let's see if someone else gives some more insight???2012-10-26
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    The connected sum isn't this nice in dimension 2. If you work through the SVK argument you end up stuck with an annulus for the intersection of your two open sets. Which gives you more relations see [here](http://math.stackexchange.com/questions/2474/fundamental-group-of-the-double-torus) for instance.2012-10-26
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    I guess, my first problem is to see how exactly this adjunction space is giving us a genus 4 surface...I mean, I can see it intuitively it makes sense, but how to show it more rigorously?2012-10-26
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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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