On 4-manifolds homotopy equivalent to circle bundles over 3-manifolds

Abstract

We give criteria for a closed 4-manifold to be homotopy equivalent to the total space of an S1-bundle over a closed 3-manifold. In the aspherical case the conditions are that the Euler characteristic be 0 and that the fundamental group have an infinite cyclic normal subgroup such that the quotient group has one end and finite cohomological dimension. Under further assumptions on this quotient group we characterize the total spaces of such bundles over $$\mathop {SL}\limits^ \sim $$ -or H2 × E1-manifolds and over E3-, Nil3- or Sol3-manifolds up to s-cobordism and homeomorphism respectively.