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Consider the wedge sum of two circles, $a V b$. Attach a 2-cell along the loop $aba^{-1}b^{-1}$.So we get a torus:

enter image description here

From this resulting torus, remove one open disk. To this surface, identify the boundary circle where our disk used to be, with the meritudal circle boundary (with this, I mean the a-side of the picture). What is the fundamental group of this surface? My current reasoning is that it should be something like $<a,b,c|aba^{-1}b^{-1}cca>$, but I'm not sure if this is correct. c is here a loop around the removed disk. Any approach for problems like this would be nice.

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What do you get using Seifert-van Kampen? –  Qiaochu Yuan Jul 26 '11 at 21:12

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up vote 3 down vote accepted

This is close. I would replace $aba^{−1}b^{−1}cca$ by $aba^{−1}b^{−1}ca^{\pm 1}c^{-1}$. The proof is to cut a straight line from the upper left corner to the new circle and label it $c$. Then the relation will be gotten by reading around the boundary of the resulting polygon. You'll either get $a$ or $a^{-1}$ depending on how you identify the boundary of the hole with $a$.

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Right, thanks! Don't you mean ( in this case that is, since the b-edges are oriented downwards) the upper left corner? –  Dedalus Jul 26 '11 at 21:47
    
Yes, sorry. I'll edit. –  Grumpy Parsnip Jul 26 '11 at 22:33

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