I heard recently that the universe is expected to be essentially flat. If this is true, I believe this means (by the 3d Poincare conjecture) that the universe cannot be simply-connected, since the 3-sphere isn't flat (i.e. doesn't admit a flat metric).

Is any/all of this true? If so (or even if not), what sorts of unexpected things might follow from the non-simply-connectedness of our universe?

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    $\begingroup$ I don't understand your argument. The Poincaré conjecture talks about closed manifolds in three dimensions. First of all, the universe (or the IR limit of the universe, for the string theorists among us) has four dimensions, and it's not very clear either that it should be compact/closed. $\endgroup$
    – Vibert
    Commented May 2, 2013 at 7:38
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/12012/2451, physics.stackexchange.com/q/1787/2451 and links therein. $\endgroup$
    – Qmechanic
    Commented May 2, 2013 at 7:43
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    $\begingroup$ You seem to think the Poincare conjecture says that the 3-sphere is the only simply connected 3-manifold. By your logic $\mathbb{R^3}$ (which can be equipped with the flat metric) isn't simply connected. $\endgroup$
    – Nikolaj-K
    Commented May 2, 2013 at 8:05
  • $\begingroup$ You missed the fact that the Poincare conjecture deals with compact manifolds. In the non-relativistic limit, where you could model space as 3 dimensional, space surely is not compact. $\endgroup$
    – Neuneck
    Commented May 2, 2013 at 8:35
  • $\begingroup$ Whoa. First of all, sorry for not having checked if my question had already been asked. But I actually did think that the universe was supposed to be compact. (I'm very much not a physicist.) And I was only thinking about 3d slices, although of course the Poincare conjecture holds in all dimensions (though I suppose in adding the time dimension we'd probably lose compactness). Anyways, does "IR limit" stand for? This isn't the same as "non-relativistic limit", is it? @Vibert $\endgroup$ Commented May 3, 2013 at 8:32