Torus Surgery Constraints and Intermediate Covers of the Cartwright--Steger Surface

We study torus surgery on the Cartwright–Steger surface and the degree-two cover construction proposed in \cite[Remark~1]{ASY}. Every torus in this surface is integrally nullhomologous, and surgery on a fixed disjoint collection of such tori cannot decrease the first Betti number. We then determine the nodal-curve subgroup and complete the fundamental-group calculation in the degree-two construction. The resulting minimal symplectic manifold has signature zero and fundamental group $\mathbb{Z}\mathrm{/}2$; for compatible symplectic gluings, its universal cover is an exotic symplectic $31{\text{CP}}^2\mathrm{\#}31{\overline{\text{CP}}}^2$. The construction on the original Cartwright–Steger surface gives a minimal symplectic manifold with signature $-1$ and fundamental group $(\mathbb{Z}\mathrm{/}2{)}^2$.