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

Abstract

Let $X$ be the Cartwright--Steger surface with its real involution $c$ and Albanese map $α:X\to E$, and put $Y=X/\langle c\rangle$. Every smooth real Albanese fiber has two ovals and is nondividing. The branch surface $B\cong\#_3\mathbb{RP}^2$ satisfies $j_*π_1(B)=4\mathbb Z\subsetπ_1(Y)\cong\mathbb Z$ 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 $\mathbb{CP}^2$ 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 $S^1\times S^3$.