In his book "The Shape of Space" (2nd ed), Jeffrey Weeks talks about a technique called "Cosmic Cristallography" to find the global topology of the space we live in. Starting with the assumption that we're living in a closed 3-manifold and that the observable universe is larger than the real universe, he says that if we see multiple images of the Milky Way, the 3-manifold is multiconnected (like e.g. a 3-torus) while otherwise it would be simply connected.
I don't get this. Wouldn't we see multiple images even if we lived in a 3-sphere?
Imagine you're living on a 2-sphere and your world is 2-dimensional and there's a southbound light ray starting at the North Pole, "traveling" along a geodesic (meridian) and reaching, say, Hamburg. If there's another light ray in the opposite direction it'll travel around the South Pole but eventually also reach Hamburg, won't it?
EDIT: In reply to a comment, here a some more details.
In a previous chapter, Weeks says that the focus is on "homogeneous, isotropic 3-manifolds". He then includes a list of possible candidates with their geometries which includes the 3-sphere (elliptic) and the 3-torus (Euclidean).
The relevant sentences then are: "In a multiconnected space we see multiple images of ourselves. So testing whether the real universe is multiconnected or not is easy, right? We just point our telescopes out into the night sky. If we see images of our Milky Way galaxy out there, then the universe is multiconnected. If we don't see images of the Milky Way, then either space is simply connected, or it's multi connected but on too large a scale for us to observe it."
I emailed Jeffrey Weeks and asked him the same question. He just replied and said that this is actually an error in the book and that you'd also see multiple images if you were inside a 3-sphere. (He also said, BTW, that a 3rd edition of the book is due for 2019. I can really recommend the book which explains topology and geometry in an informal and intuitive way.)