The Real Albanese Map and Smooth Topology of the Cartwright--Steger Quotient

Let $X$ be the Cartwright–Steger surface with its real involution $c$ and Albanese map $\alpha :X\to E$, and put $Y=X\mathrm{/}⟨c⟩$. Every smooth real Albanese fiber has two ovals and is nondividing. The branch surface $B\cong {\mathrm{\#}}_3{\mathbb{R}\mathbb{P}}^2$ satisfies $j_{\ast }{\pi }_1(B)=4\mathbb{Z}\subset {\pi }_1(Y)\cong \mathbb{Z}$ and is nullhomologous modulo two. Every smoothly embedded $+1$-sphere representing the positive generator of $H_2(Y;\mathbb{Z})$ meets $B$ in at least six points. Its lift has square two, pairs trivially with $K_X$, and has no connected embedded representative of genus zero or one. Circle surgery on an Albanese generator is homeomorphic to ${\mathbb{C}\mathbb{P}}^2$ for either normal framing. Blowing down a positive sphere, if one exists, gives a $\mathbb{Z}[\mathbb{Z}]$-homology $S^1\times S^3$ homotopy equivalent to $S^1\times S^3$.