On the fundamental group calculation associated to reverse-engineering an exotic CP ^2 #3 \overline{CP }^2 and CP ^2 #2 \overline{CP }^2

I provide a detailed exposition of the fundamental group calculations in the ``reverse engineering" construction of an exotic ${\text{CP}}^2\mathrm{\#}3{\overline{\text{CP}}}^2$ and of an exotic ${\text{CP}}^2\mathrm{\#}2{\overline{\text{CP}}}^2$.