Topology in cosmology Usually in cosmology, we make the hypothesis that the universe is isotropic.
Which conditions does this hypothesis impose on the topology of the universe? Does it fix completely the topology? Are all the exotic topologies ruled out?
 A: A relatively recent review by Jean-Pierre Luminet discusses this subject at some length.  Broadly, it can be summarized as follows:

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*The requirements of homogeneity and (local) isotropy do not necessarily mean that the spatial slices of the cosmos are diffeomorphic to $S^3$, $\mathbb{R}^3$, or $H^3$;  these are merely the simplest examples of spaces with positive, zero, or negative uniform curvature.


*In particular, these three spaces are the universal covering spaces for any homogeneous and (locally) isotropic space of constant curvature.  All other spaces with these properties can be obtained by "identifying" one of the universal covering space with itself in a particular way.  For example, to get the 3-torus from $\mathbb{R}^3$, you "identify" all points $(x,y,z)$ with the points $(x + a,y,z)$, and similarly for the $y$- and $z$-directions.



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*A corollary of this is that local measurements of the spatial curvature scale cannot, in themselves, rule out any particular topology.  The geometric curvature of the universal covering space and that of the topologically "interesting" space are the same under the identification.

Luminet also discusses the tactics that have been used to experimentally search for non-trivial cosmological topology.  It is not impossible to detect this structure from here on Earth, but it effectively requires the Universe to be "small enough" so that light from a given source can reach you from multiple directions.  If your observations come up empty, you may be able to place a lower bound on the size of the Universe, and this may be sufficient to rule out some types of topology (since the volumes of such spaces are related to their curvature scale.)  But in general you can't rule out non-trivial topology entirely, particularly in the case of flat geometries for which the volume of the Universe is effectively arbitrary.
A: I think the question you ask is quite important.
"Which conditions does this hypothesis (isotropy) impose on the topology of the universe?"
I have seen a lot of discussion about various conclusions that something like a 3-torus (or some other topology) cannot be (or has not been) proved to be impossible. It seems to me that it would be very useful for discussions of the non-impossibility to distinguish between (a) some speculative kind of universe is not known to be impossible, and (b) some kind of universe can be shown to have a positive (i.e, definitely greater than zero) probability that it might exist.
It does seem obvious that the (a) type speculations are impossible to confirm to be part of our universe since such things are not possible to have observable evidence supporting their existence in the observable universe.
I have looked for such (b) type statements, but I have never found any.
