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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.

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Possible duplicates: and links therein. – Qmechanic Nov 30 '12 at 22:46
Assuming that the universe is based on rules of QM, structure of physical space time may be very different from the usual topological spaces. – user10001 Nov 30 '12 at 22:57
@Qmechanic : I have seen that question. As far as I can see, it only deals with the fundamental gruop of the universe. I am also interested in the higher homology/homotopy groups of the universe, which is why I made a separate question. I edited the question to make this clearer. – Espen Nielsen Dec 1 '12 at 17:20
up vote 2 down vote accepted

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:

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.

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Was the downvote because I didn't include black holes? Yes, these do give a second homology, but I didn't think that was the spirit of the question, because it's kind of trivial. – Ron Maimon Dec 2 '12 at 20:37
@ChrisGerig: There are no symbols, but there is math. If you consider the universe a 3-disk bounded by the cosmological horizon, then topologically it's a manifold with boundary, and it is deformation retractable to a point (ignoring other boundaries, like black holes on the interior), so it's homology is trivial. This is a (simple) theorem: a space which deformation retracts to a point has trivial homology. The case when you have black holes in the interior introduces nonretractable 2-cycles, it's the 3-disk with punctures topologically, and the dimension of H2 is the number of BHs. – Ron Maimon Dec 3 '12 at 19:53
@ChrisGerig: When you say "we are unsure of the topology", you are referring to two things: 1. there is some question of whether the observable universe has identifications inside the cosmological horizon, but it probably doesn't, since we would see it in WMAP 2. we don't know the topology of the unobservable universe and we never will. I am adressing point 1 by assuming the experimental data is conclusive (which it nearly is), and point 2 by rejecting the idea that the universe extends past the cosmological horizon. – Ron Maimon Dec 3 '12 at 19:55
But no real disrespect, I just prefer less over-complications and excessive rambling of physics words. Your explanation can be more clear cut than what it is, and a bunch of things can be dropped completely without impacting the intent. – Chris Gerig Dec 4 '12 at 5:20
@ChrisGerig: Ok, but I see now why you said what you said, and I agree with you regarding the content. I wasn't right in laying all the repulsion to Pauli, there's a part that's nuclear repulsion once the electron shells overlap, and this is a tradeoff. You could have made the argument explicit, though, rather than relying on authority. – Ron Maimon Dec 4 '12 at 7:36

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.

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It's called the Topological Censorship Theorem these days, at least in GR. In quantum gravity, similar statements may well be false. – user1504 Dec 1 '12 at 4:58

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