The Topology of General Cosmological Models
  Is the universe finite or infinite, and what shape does it have? These
fundamental questions, of which relatively little is known, are typically
studied within the context of the standard model of cosmology where the
universe is assumed to be homogeneous and isotropic. While these two hypotheses
are well-motivated, they are not realistic on all scales of description. Here
we address the above questions in highly general cosmological models, with the
only assumption being that the average flow of matter is irrotational. Using
techniques from differential geometry, specifically extensions of the
Bonnet-Myers theorem, we derive a condition which implies a finite universe and
yields an estimate for its diameter. Comparing with Planck CMB data, it is
shown that this condition is satisfied by the observable universe with
approximately $85\%$ confidence. Furthermore, under a weaker condition
involving the interplay between curvature and diameter, together with the
assumption that the universe is finite, we provide a concise list of possible
topologies. Namely, the spatial sections then would be either the ring
topologies $S^1 \times S^2$, $S^1\tilde{\times}S^2$, $S^1\times\mathbb{RP}^2$,
$\mathbb{RP}^3 \# \mathbb{RP}^3$, or covered by the sphere $S^3$ or torus
$T^3$. In particular, under this condition the basic construction of connected
sums would be ruled out (save for one), along with the plethora of topologies
associated with negative curvature. These results are obtained from
consequences of the geometrization of 3-manifolds, by applying a generalization
of the almost splitting theorem together with a curvature formula of Ehlers and
Ellis.
