Quotients of S2×S2$S^2\times {S^2}$

Abstract We consider closed topological 4‐manifolds with universal cover and Euler characteristic . All such manifolds with are homotopy equivalent. In this case, we show that there are four homeomorphism types, and propose a candidate for a smooth example that is not homeomorphic to the geometric quotient. If , we show that there are three homotopy types (and between 6 and 24 homeomorphism types).