Physicists assume that the universe can be described as a manifold. Mathematician characterize manifolds in terms of their geometry and topology. Geometry is a local quantity that measures the intrinsic curvature of a surface. General relativity relates the mass distribution of the universe to its geometry and of course, the geometry of the universe determines the dynamics of the mass. Topology is a global quantity that characterizes the shape of space (see, e.g., Weeks 1985 for a general introduction). General relativity does not at all constrain the topology of the universe.
The relationship between topology and geometry is most familiar in flat space. Cosmologists often consider flat infinite universes: a model that mathematicians denote as , which symbolizes a space that is a the product of the three orthogonal real lines. A familiar cosmological model that has the same geometry as but a different topology is the three torus or what mathematician call Most cosmological simulations are run on a three torus: if a particle tries to leave the computational cube through one side it emerges on the opposite side.
There are several different ways of thinking about the topology of a manifold. It is simplest to begin by considering topology in flat two dimensional space. One example of a flat topology is a square with identified sides. This square is the fundamental domain of the topology. Another way of thinking about this topology is to glue the sides together to create a donut, a two dimensional surface embedded in a three dimensional space. Yet another way of thinking about this topology is to tile an infinite plane with identical copies of the same fundamental domain.
Cosmologists generally consider three possible geometries for the universe: a positively curved universe, a flat universe, and a negatively curved universe. In standard parlance, closed, critical and open universes. This later nomenclature is misleading: negatively curved universes can be either infinite in spatial extent or compact. Both models have the same dynamics and expand forever: dynamics is determined by geometry. The three-sphere is a space of constant positive curvature, and the pseudosphere is a hyperbolic space with constant negative curvature.
From a topological point of view, negatively curved (hyperbolic) universes are ``generic'': most three dimensional manifolds can be viewed as homogenous negatively curved and compact (Thurston's geometrization conjecture [Thurston 1978]). Cornish, Gibbons and Weeks (1997) have recently shown that in the Hartle-Hawking approach to quantum cosmology, the smallest volume manifolds are favored. These smallest volume manifolds are the simplest (least complex) compact models.
In negatively curved manifolds, the characteristic length is the curvature scale. Thus, throughout the rest of this talk, we will use it as our unit of length. There are an infinite number of distinct compact hyperbolic topologies. There is a fundamental group, usually denoted , associated with each of these topologies. Each of these topologies also has a specific volume (measured in curvature units). We strongly recommend the publicly available SNAPPEA program (http://www.geom.umn.edu/) for anyone interested in developing a more intuitive feel for the rich structure of compact hyperbolic topology.
One of the intriguing mathematical properties of compact hyperbolic manifolds is that geodesic flows on these manifolds are maximally chaotic and mixing. Because of this property they are extensively studied by physicists and mathematicians interested in quantum chaos(Balazs & Voros 1986; Gutzweiler 1985). We have speculated that quantum chaos plays an important role in the homogenization of the early universe (Cornish, Spergel & Starkman 1996a).