(Co)homology of the universe In this post let $U$ be the universe considered as a manifold.
From what I gather we don't really have any firm evidence whether the universe is closed or open. The evidence seems to point towards it being open but closedness is within the error range of present data.
However, this seems to be an awfully crude distinction, basically of whether $H_1(U)$ is trivial or not. For argument's sake, let's consider the latter case. Do present physical theories place any restrictions on $H_1(U)$? For example, is $H_1(U)=\mathbb{Z}_2$ allowed, or $H_1(U)=\mathbb{Z}\times \mathbb{Z}$? What about $H_1(U)=A_4$, the alternating group of order 12?
I am especially interested in higher (co)homology/homotopy groups.
For example, the topological censorship theorem states that the universe is simply connected, so $H_1(U)=0$. However, there are spacetimes where space is a 3-sphere. More generally, we can postulate what would happen if $U$ was a homology sphere of a certain dimension.
 A: Well, if the universe is path-wise connected, and if the topological censorship conjecture is true, then the universe is simply connected (see this thread, for example).
If the universe is simply connected, then its fundamental group is trivial $\pi_{1}(U)=0$. 
If the fundamental group is trivial for a path-connected topological space, its first homology group is trivial by the Hurewicz theorem.
A: This question is not 100% meaningful in the sense of positivism. The issue is that space-time is limited by horizons, and whether it is open or closed outside the cosmological horizon is a question which is only meaningful to the extent that it influences the future development of the part we can observe. This is extremely important and yet sometimes controversial (for reasons I can't comprehend). It is impossible to gain experimental data regarding hypothetical stuff outside the cosmological horizon, so you can't say anything definite regarding it's topology, or anything at all.
Naive handles are probably not allowed in GR A handle is a wormhole and can lead to causality paradoxes. If you spacelike separate the entrance and the exit of the wormhole, you can use it to travel faster than light, and by using boosted wormoholes, you can use them to go back in time. But let's imagine they're allowed. Then your universe topology could have a handle come in from outside the cosmological horizon during the FRW phase. This would change the topology of the observable universe dynamically, but it isn't clear whether you should call that changing the topology of the universe in a classical GR model, because maybe the handle was "always there" and only became visible later. This is the type of annoyance physicists use positivism to resolve.
The topology of our universe is best understood by thinking of the universe not as an unbounded manifold, but as a manifold with a horizon boundary, a patch. In modern physics, the physics of each patch is expected to be complete and consistent without reference to external unobservable upatches, so that if you form a black hole, you can excise the interior and reconstruct it. You don't have to include the interior in your description explicitly, only as a figure of speech for the future of an infalling observer (or the past of an outgoing Hawking radiated photon).
Then the answer to your question is that the universe has trivial homology in the large dimensions, it is deformation retractable to a point (ignoring black holes. The excised interior of black holes don't give a first homology group, but they introduce a second homology group for space which is of size the number of black holes).
In the small dimensions of string theory, the spacetime is not topologically or homologically trivial. Note that a spatial circle in Minkowski geometry doesn't have causality paradoxes unless the gluing of the two ends is by a timelike curve, so the causality argument doesn't exclude all nontrivial topology, only traversible handles in asymptotically flat backgrounds. The topology of our microscopic universe is not known precisely, nor is it known if the question is 100% meaningful at the string scale, both because of discrete ambiguities like T-duality which allow two different geometric interpretations of a vacuum, and also because of non-geometric compactifications. But if you assume that our vacuum is something like a heterotic string compactification on a Calabi-Yau manifold (which is plausible), then you learn from the fact that there are 3 generations that the Euler characteristic is 6. All of this is more complicated in the analysis of popular orbifold models, for example see here for some caveats: http://arxiv.org/abs/hep-th/0403272.
The most significant limitation on the use of traditional homology to classify string vacua is the orbifolds themselves. An orbifold is not a manifold and gives different string propagation than manifold backgrounds. The orbifolds are the most distinctive and important modification of geometry required by strings.
