structure; the covering map S3 → RP3 is a map of groups Spin(3) → SO(3), where Spin(3) is a Lie group that is the universal cover of SO(3). 3-torus The 3-dimensional torus is the product of 3 circles. That is: The 3-torus, T3 can be described as a quotient of R3 under integral shifts in any coordinate. That is, the 3-torus is R3 modulo the action of the integer lattice Z3 (with the action being taken as vector addition). Equivalently, the 3-torus is obtained from the 3-dimensional cube by gluing the opposite faces together. A 3-torus in this sense is an example of a 3-dimensional compact manifold. It is also an example of a compact abelian Lie group. This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication. Hyperbolic 3-space Hyperbolic space is a homogeneous space that can be characterized by a constant negative curvature. It is the model of hyperbolic geometry. It is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and models of elliptic geometry (like the 3-sphere) that have a constant positive curvature. When embedded to a Euclidean space (of a higher dimension), every point of a hyperbolic space is a saddle point. Another distinctive property is the amount of space covered by the 3-ball in hyperbolic 3-space: it increases exponentially with respect to the radius of the ball, rather than polynomially. Poincaré dodecahedral space The Poincaré homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere. Being a spherical 3-manifold, it is the only homology 3-sphere (besides the 3-sphere itself) with a finite fundamental group. Its fundamental group is known as the binary icosahedral group and has order 120. This shows the Poincaré conjecture cannot be stated in homology terms alone. In 2003, lack of structure on the largest scales (above 60 degrees) in the cosmic microwave background as observed for one year by the WMAP spacecraft led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere. In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP