  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.
