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 $$, 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.