Closed universes: their future evolution and final state*

We summarize what is currently known about the future evolution and final state of closed universes: in mathematical language, those which have a compact Cauchy surface. We show that the existence of a maximal hypersurface (a time of maximum expansion) is a necessary and sufficient condition for the existence of an all-encompassing final singularity in a universe with a compact Cauchy surface. The only topologies which can admit maximal hypersurfaces are |S³ \, and \, S² S¹, | together with more complicated topologies formed from these two types of 3-manifold by connected summation and certain identifications. The relevance of these results to inflation is also discussed.