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As I've stated in a prior question of mine, I am a mathematician with very little knowledge of Physics, and I ask here things I'm curious about/things that will help me learn.

This falls into the category of things I'm curious about. Have people considered whether spacetime is simply connected? Similarly, one can ask if it contractible, what its Betti numbers are, its Euler characteristic and so forth. What would be the physical significance of it being non-simply-connected?

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I do not dare tackle this, but would guess that the four dimensional space is not, since it is divided into space like and time like regions and the twain shall never meet. Then there are all those folded spaces in string theories, Calabi Yao manifolds with lots of holes. We are certainly aiming at non-simply-connected spaces if we include them in space. – anna v Jul 8 '11 at 5:01
Possibly related: – Qmechanic Jul 8 '11 at 5:21
Anna V, the first part of your comment indicated that your talking more about causally connected, which isn't really pertinent when discussing global topological structures. – Benjamin Horowitz Jul 8 '11 at 5:34
Two slightly related facts since you mentioned "Euler characteristic and so forth" (but with little bearing on the question in the title): 1. Sometimes the space-time manifold is assumed to be spin. (For example, this fact is used in Witten's proof of the positive mass theorem.) This requires the vanishing of the second Stiefel-Whitney class, which does tell you something about the topology. – Willie Wong Jul 8 '11 at 11:00
2. There is a little theorem that states the following: given a connected (1+3)-dimensional Lorentzian manifold, its universal cover cannot be compact. (Sketch of the proof: the Lorentzian metric distinguishes time directions. So there exists a nonvanishing section of the tangent sphere bundle. Hence Euler characteristic must be 0. But using Poincare duality, the Euler characteristic of a simply connected compact 4-manifold is at least 2.) – Willie Wong Jul 8 '11 at 11:01

2 Answers 2

up vote 7 down vote accepted

I suppose there are many aspects to look at this from, anna v mentioned how Calabi-Yao manifolds in string theory (might?) have lots of holes, I'll approach the question from a purely General Relativity perspective as far as global topology.

Solutions in the Einstein Equations themselves do not reveal anything about global topology except in very specific cases (most notably in 2 (spacial dimensions) + 1 (time dimension) where the theory becomes completely topological). A metric by itself doesn't necessarily place limits on the topology of a manifold.

Beyond this, there is one theorem of general relativity, called the Topological Censorship Hypothesis that essentially states that any topological deviation from simply connected will quickly collapse, resulting in a simply connected surface. This work assumes an asymptotically flat space-time, which is generally the accepted model (as shown by supernova redshift research and things of that nature).

Another aspect of this question is the universe is usually considered homogenous and isotropic in all directions, topological defects would mean this wouldn't be true. Although that really isn't a convincing answer per say...

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Calabi-Yau manifolds usually have "higher-dimensional" holes (i.e. the ones that are detected by higher homotopy groups but not by the fundamental group). In fact, for any Calabi-Yau manifold there is a finite cover that is a product of a torus and a simply-connected Calabi-Yau manifold. – Marek Jul 8 '11 at 11:52
What about eternal string singularities? BTW, it seems, the cited paper only prevent to “probe topology” and even in such a case may have some lacuna in the proof. – Alex 'qubeat' Jul 8 '11 at 18:02
I am not familiar with eternal string singularities, are there any resources you know about them? – Benjamin Horowitz Jul 8 '11 at 18:31
It is something like that – Alex 'qubeat' Jul 8 '11 at 18:36
LoL… the paper requires a conformal completion of the spacetime with both future and past parts homeomorphic to S² × ℝ. It certainly is not so for the past of a universe formed from cosmological singularity (the starting point of Big Bang). – Incnis Mrsi Aug 22 '14 at 14:25

Benjamin Horowitz's answer covered a lot of the key points, but it's worth adding that the question of the topology of the universe has been investigated by astrophysical observations. If the universe is multiply connected, and if the length scale is shorter than the horizon scale, then we should be able to see evidence of it.

To take a simple example, imagine that the universe is geometrically flat but has the geometry of a 3-torus. Specifically, take a cubical volume, and identify opposite faces, so that if you "leave" the cube through one face you reenter through the opposite face. If the length of an edge of the cube is sufficiently small, then you could see multiple copies of any given object. Of course, if the length is much larger than the horizon, then there's no way to tell the difference between this model and one in which space is infinite.

The best way to test these models is the "circles in the sky" technique, in which you look for correlated circles in different directions in maps of the microwave background radiation. The result is negative: we don't live in a multiply-connected universe with a sufficiently short length scale to be observable.

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Recently preprint appears there also the WMAP data is used, but with claim about possibility of multiply-connected Universe with spatial topology $\mathbb T^2 \times \mathbb R$. It seems not published in a journal yet, but the idea already reproduced in Wikipedia. – Alex 'qubeat' Jul 9 '11 at 16:08
Thanks for the reference. I'd missed that. As I understand it from a quick glance, the idea here is to look at the case where the fundamental cell size is larger than the horizon (so that the circles-in-the-sky technique doesn't work) but not too much larger (or else there'd be no observable effects at all). This is in principle a sensible thing to do, but in this sort of analysis the details matter a lot, and I haven't looked carefully enough to have a sensible opinion about the details. – Ted Bunn Jul 11 '11 at 14:02

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