What are the allowed topologies for a FRW metric? Given a spacetime that has the maximal amount of spacelike translations and rotations, what are the possible topologies it may take? I am mostly wondering about the "time" topology since the spatial one is fairly well documented.
I'm fairly sure that any FRW spacetime will be a foliation by spacelike hypersurfaces, so given a spacelike hypersurface $\Sigma$, either $\mathbb R \times \Sigma$ or $S \times \Sigma$ (or possibly some $\Sigma$ bundle over $\mathbb R$ or $S$), but looking through the literature (Hawking Ellis, Straumann, Ringström), everyone seems to assume global hyperbolicity outright.
Just given the Killing vector fields, can one show that the manifold has to be foliated by spacelike hypersurfaces?
 A: As you know, 3-dimensional space need not be infinite even if the spatial curvature has sign $k = 0$ or $k = -1$. As George Ellis writes in "Issues in the philosophy of cosmology", in Butterfield & Earman eds. (2006):

Misconception 5: The space sections are necessarily infinite if $k = 0$
  or −1. This is only true if they have their ‘natural’ simply connected
  topology. If their topology is more complex (e.g. a 3-torus) they can
  be spatially finite [Ellis, 1971a; Lachieze et al., 1995]. There are
  many ways this can happen; indeed if $k = −1$ there is an infinite
  number of possibilities.

Ellis, Maartens & MacCallum's excellent textbook Relativistic Cosmology (2012) contains a section "Topology", $\S9.1.5$. For $k = 0$ they mention generalisations of the Mobius band. For $k > 0$ they mention identifying antipodal points. Any $k > 0$ case is closed. They cite Wolf (1972), Ellis (1971), and Thurston (1997), and clearly there are many more references.
The most obvious reference is Stephani et al's exact solutions book. The chapters at the start of Part II discuss spacetimes with lots of isometries. Finally, I am aware my response has more resources than answers --- perhaps another can summarise the content cited here.
