Today I posted a version of the talk I gave at LOOPS 13 to the archive (1308.2934). You can also see the original slides. In principle this will be a longer paper soon, but I have some new ideas after my attendance at LOOPS 13 and I’d like to add them before doing something official.

Basically, classical gravity has nothing to say about the topology of the universe, and one would hope that quantum gravity would. By topology of the universe I mean large-scale structures – general relativity actually can’t tell if the universe is a sphere or a torus, but it can tell us that it appears to be locally flat. There could be any arbitrarily many numbers of “holes” in the universe, as long as the universe itself and the holes were both large enough that things still look flat in our little piece of things.

This means that we have to specify (read: put in by hand) the topology of the universe. Maybe not such a big deal since the model seems very good, but if quantum gravity is really “fundamental” then we don’t really like having to tell it what to do. Since there are a large number of different options for the topology (something like “as many integers as you can count”) specifying one in particular seems a bit strange. It’d be really nice if quantum gravity told us which one is the correct one – and even better if it matched observations!

That’s what my current work is focusing on, which started with an idea from Denicola, Marcolli, and al-Yasry (1005.1057). Without more details, the basic idea is to modify a current approach to quantum gravity (loop quantum gravity) to include topological data. That would allow us to ask questions about the topology, like “where did it come from?” and “what topology do we predict the universe to have if this theory is correct?” At this stage we are just checking to make sure the various constructions of LQG still work, and in this paper I am showing how to actually calculate some topological information. Hopefully this will lead to some calculations which show that LQG a) does include topological information, and b) predicts the topological structure of the universe.

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Chris,

good idea to create a blog. I also thought about (my wife has a blog in wordpress: andreamaluga.wordpress.com).

But let me comment on your approach. I agree that GRT do not determine the topology. One gets some results if one demand something like the cosmic censorship (no naked singularities) to get globally hyperbolic space. But this approach is a little bit boring for two reasons: at first one never gets any restrictions on the spatial topology, i.e. one always obtains MxR for any 3-manifold M secondly one has a problem with quantum theory. According to quantum theory, the future is open (the many-fingered world) so one would never expect that a smooth geodesics from the past to the presents will always extend to the future (I know that a particular path do not exists but in the path integral approach, you have a family of paths meeting in one point, which is a naked singularity). One has to add naked singularities to express an open future.

But if one relax the whole approach then one is able to get some restritions on the topology. We formulate some in the paper arXiv:1206.4796 (On topological restrictions of the spacetime in cosmology). One demands only the smoothness of the spacetime and one has to single out closed curves (which can be time-like closed curves prducing acausalities). Then one obtain the class of the homology 3-spheres as possible spacetime. Other examples like the 3-torus are not possible otherwise one has to find some extra structue in the CMB.

Hey Torsten, nice to have you visit! Blogs are rather interesting as a form of “passive advertisement”, so I’m giving it a shot.

Yes, I am familiar with that paper – actually it’s not too surprising that one can find restrictions on the topology of the spatial section, but your construction is very well done. Also interesting is that the homology 3-spheres show up again in the inflation generated by exotic smoothness scenario. I think it demonstrates something about the important of the fundamental group which is often overlooked. At some point I want to see what kinds of topspin networks represent homology spheres, to try and match the LQG case with what you are working on.

As far as I can remember (see Rolfsen, Knots and links) the fundamental group of a branched cover is connected to the surgery representation. Therefore take for instance the torus knot 2,5 (with a fixed orientation) and put instead of the over and under crossings double points to get a top spin model. Then according to your method you will get a Poincare sphere. The reason is simply that a 3-fold branched cover of the 3-sphere branched over this trous knot 2,5 is the Poincare sphere.

Yes, you can also get one over the trefoil, but I don’t know exactly which representations of the symmetric group these correspond to – although I think they can be worked out by just “following the sheets” during the surgery.