# Fundamental group of a certain torus-like surface

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:

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

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 $a$.