Topology and singularities in cosmological spacetimes obeying the null energy condition

We consider globally hyperbolic spacetimes with compact Cauchy surfaces in a setting compatible with the presence of a positive cosmological constant. More specifically, for 3+1 dimensional spacetimes which satisfy the null energy condition and contain a future expanding compact Cauchy surface, we establish a precise connection between the topology of the Cauchy surfaces and the occurrence of past singularities. In addition to (a refinement of) the Penrose singularity theorem, the proof makes use of some recent advances in the topology of 3-manifolds and of certain fundamental existence results for minimal surfaces.