Let \(X\) be the compact surface obtained from the disk by deleting two disjoint open disks, as in the figure below. Fix orientations on two boundary components as in the figure. Identify the boundaries using each of the two possible orientations on the third component. Show that the two quotient spaces so obtained have non-isomorphic fundamental groups. Hint: abelianize.
Figure: a disk with two disjoint open disks removed; the two inner boundary components have the indicated orientations.