The following article is contributed by Peter Kramer.
Einstein’s fundamental theory of Newton’s gravitation relates the interaction of masses to the curvature of space. Modern cosmology from the big bang to black holes results from Einstein’s field equations for this relation. These differential equations by themselves do not yet settle the large-scale structure and connection of the cosmos. Theoretical physicists in recent years tried to infer information on the large-scale cosmology from Cosmic microwave background radiation (CMBR), observed by satellite observations. In the frame of large-scale cosmology, the usual objects of astronomy from solar systems to galaxy clusters are smoothed out, and conditions imprinted in the early stage of the universe dominate.
In mathematical language one speaks of cosmic topology. Topology is often considered to be esoteric. Here we present topology from the familiar experience with the twisted Möbius strip. This strip on one hand can be seen as a rectangular crystallographic lattice cell whose copies tile the plane, see Fig. 2. The Möbius strip is represented as a rectangular cell, located between the two vertical arrows, of a planar crystal. A horizontal dashed line through the center indicates a glide-reflection line. A glide reflection is a translation along the dashed line by the horizontal length of the cell, followed by a reflection in this line. The crystallographic symbol for this planar crystal is cm. In three-dimensional space the planar Möbius crystal (top panel of Fig. 1) is twisted (middle panel of Fig. 1). The twist is a translation along the dashed line, combined with a rotation by 180 degrees around that line. A final bending (bottom panel of Fig. 1) of the dashed line and a smooth gluing of the arrowed edges yields the familiar Möbius strip.
Given this Möbius template in two dimension, we pass to manifolds of dimension three. We present in Fig. 3 a new cubic manifold named N3. Three cubes are twisted from an initial one. A twist here is a translation along one of the three perpendicular directions, combined with a right-hand rotation by 90 degrees around this direction. To follow the rotations, note the color on the faces. The three neighbor cubes can be glued to the initial one. If the cubes are replaced by their spherical counterparts on the three-sphere, the three new cubes can pairwise be glued with one another, with face gluings indicated by heavy lines. The complete tiling of the three-sphere comprises 8 cubes and is called the 8-cell. The gluings shown here generate the so-called fundamental group of a single spherical cube on the three-sphere with symbol N3. This spherical cube is a candidate for the cosmic topology inferred from the cosmic microwave background radiation. A second cubic twist with a different gluing and fundamental group is shown in Fig. 4. Here, the three twists combine translations along the three directions with different rotations.
The key idea in cosmic topology is to pass from a topological manifold to its eigen- or normal modes. For the Möbius strip, these eigenmodes are seen best in the planar crystal representation of Fig. 2. The eigenmodes can be taken as sine or cosine waves of wave length which repeat their values from edge to edge of the cell. It is clear that the horizontal wavelength of these modes has as upper bound the length L of the rectangle. The full Euclidean plane allows for infinite wavelength, and so the eigenmodes of the Möbius strip obey a selection rule that characterizes the topology. Moreover the eigenmodes of the Möbius strip must respect its twisted connection.
Similarly, the eigenmodes of the spherical cubes in Fig. 3 must repeat themselves when going from cube to neighbor cube. It is intuitively clear that the cubic eigenmodes must have a wavelength smaller than the edge length of the cubes. The wave length of the eigenmodes of the full three-sphere are bounded by the equator length of the three-sphere. Seen on a single cube, the different twists and gluings of the manifolds N2 and N3 shown in Figs. 3 and 4 form different boundary value problems for the cubic eigenmodes.
Besides of these spherical cubic manifolds, there are several other competing polyhedral topologies with multiple connection or homotopy. Among them are the famous Platonic polyhedra. Each of them gives rise to a Platonic tesselation of the three-sphere. Everitt has analyzed all their possible gluings in his article Three-manifolds from platonic solids in Topology and its applications, vol 138 (2004), pp. 253-263. In my contribution Platonic topology and CMB fluctuations: Homotopy, anisotropy, and multipole selection rules, Class. Quant. Grav., vol. 27 (2010), 095013 (freely available on the arxiv) I display them and present a full analysis of their corresponding eigenmodes and selection rules.
Since terrestrial observations measure the incoming radiation in terms of its spherical multipoles as functions of their incident direction, the eigenmodes must be transformed to a multipole expansion as done in my work. New and finer data on the CMB radiation are expected from the Planck spacecraft launched in 2009. These data, in conjunction with the theoretical models, will promote our understanding of cosmic space and possible twists in its topology.