You might want to check my papers on "shape of the universe". Topology can say things about this question. I, however, had no training in topology. Therefore, I probably did not figure out everything that can be figured out, and a "real topologist" ought to be able to do better. Perhaps even pretty easily. Essentially, there are (a) certain experimental physical facts, and (b) there also are certain assumptions you could make, about the universe. If you choose your favorite set from columns (a) and (b), then the question is (c) what can be deduced about the topology of the universe? Quite a lot can be said, and I did, but I believe quite a lot more could be said and this whole topic has been not been studied very much by very good people, so there's "low hanging fruit" to be had. (Indeed the very fact I managed to get as far as I did with no training in topology, kind of proves that.) Anyhow, the papers I am speaking of are #55, #63, #75 here: http://rangevoting.org/WarrenSmithPages/homepage/works.html