Quotients of $S^2\times{S^2}$

Abstract

We consider closed topological 4-manifolds $M$ with universal cover ${S^2\times{S^2}}$ and Euler characteristic $χ(M) = 1$. All such manifolds with $π=π_1(M)\cong {\mathbb Z}/4$ are homotopy equivalent. In this case, we show that there are four homeomorphism types, and propose a candidate for a smooth example which is not homeomorphic to the geometric quotient. If $π\cong {\mathbb Z}/2 \times {\mathbb Z}/2$, we show that there are three homotopy types (and between 6 and 24 homeomorphism types).