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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?