Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 141 / 2 / a / Solution

Orient the alpha curves and choose generators $x,y$ dual to ${\alpha }_1,{\alpha }_2$, respectively. Following ${\beta }_1$ once around the edge identifications and through the attached annulus gives, up to cyclic permutation and simultaneous inversion conventions,The dotted blue arcs pass underneath the attached annulus, so their apparent crossings with the red arc on the annulus are not intersections. The fundamental group presentation is thereforeThe exponent sums of $x,y$ in $r$ are $2,0$. Henceand abelianization sends $x$ to the order-two generator and $y$ to the infinite cyclic generator.