On 4-manifolds homotopy equivalent to surface bundles over surfaces (Q1187114)

When \(B\), \(E\), \(F\) are connected finite complexes and \(p: E\to B\) is a fiber bundle projection with fiber \(F\), then \(\chi(E)=\chi(B)\chi(F)\) and there is an exact sequence \(\pi_ 2(B)\to\pi_ 1(F)\to\pi_ 1(E)\to\pi_ 1(B)\to 1\) in which the image of \(\pi_ 2(B)\) is in the center of \(\pi_ 1(F)\). It is shown that a closed 4-manifold is homotopy equivalent to such an \(E\) which fibers over a closed surface other than \(RP^ 2\) or \(S^ 2\) if these conditions hold. For the case of \(S^ 2\) additional conditions are given, depending on the fiber, which characterize when \(M\) is homotopy equivalent to such a fibering. The question of simple homotopy equivalence is also addressed in the results. lt is also shown that when a closed 4-manifold is simple homotopy equivalent to the total space of an \(F\)-bundle over \(B\), where \(B\) and \(F\) are closed aspherical surfaces, then \(M\) is topologically s-cobordant to \(E\).