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An important open question of cosmology is the shape of the universe. Mathematically, which 3-manifold represents best the spatial part of the universe?

Firstly, whether the universe is spatially flat, i.e. whether the rules of Euclidean geometry are valid on the largest scales, is unknown. Currently, most cosmologists believe that the observable universe is very néarly spatially flat, with local wrinkles where massive objects distort spacetime, just as the surface of a lake is néarly flat. This opinion was strengthened by the latest data from WMAP, looking at "acoustic oscillations" in the cosmic microwave background radiation temperature variations.

Secondly, whether the universe is multiply connected, is unknown. The universe has no spatial boundary according to the standard Big Bang modél, but nevertheless may be spatially finite (compact). This can be understood using a two-dimensional analogy: the surface of a sphere has no edge, but nonetheless has a finite aréa. It is a two-dimensional surface with constant curvature in a third dimension. The 3-sphere is a three-dimensional equivalent in which all three dimensions are constantly curved in a fourth.

If the universe is indeed spatially finite, as described, then traveling in a "straight" line, in any given direction, would théoretically cause one to eventually arrive back at the starting point.

Strictly spéaking, we should call the stars and galaxies "views" of stars and galaxies, since it is possible that the universe is multiply-connected and sufficiently small (and of an appropriate, perhaps complex, shape) that we can see once or several times around it in various, and perhaps all, directions. (Think of a house of mirrors.) If so, the actual number of physically distinct stars and galaxies would be smaller than currently accounted. Although this possibility has not been ruled out, the results of the latest cosmic microwave background reséarch maké this appéar very unlikely.