Quotients of S2×S2$S^2\times {S^2}$

We consider closed topological 4-manifolds M $M$ with universal cover S 2 × ${S^2\times {S^2}}$ and Euler characteristic χ ( ) = 1 $\chi (M) 1$ . All such manifolds π ≅ Z / 4 $\pi =\pi _1(M)\cong {\mathbb {Z}/{4}}$ are homotopy equivalent. In this case, we show that there four homeomorphism types, propose a candidate for smooth example is not homeomorphic to the geometric quotient. If \cong \mathbb {Z}/{2} \times {Z}/{2}$ , three types (and between 6 24 types).