The scientific debate over the "shape" of the universe (especially the debate over its finiteness) goes back at least as far as Newton's time. Einstein's field equations describe the shape of space locally, but make no claims about its global topology. Interest in this topic has skyrocketed in the last three years. In October of 1997, about 30 geometers, cosmologists, and theoretical physicists met at Case-Western Reserve University in Cleveland, Ohio at the "Cleveland Cosmology-Topology Workshop." The proceedings of that meeting were published in Classical and Quantum Gravity (Starkman et al. 1998) and give an excellent (though technical) overview of this area. This topic has taken hold in the popular literature as well, and over the last 18 months has appeared as the cover story in Science News (Peterson 1998), Notices of the American Mathematical Society (Cornish and Weeks 1998), and Scientific American (Luminet et al. 1999). In fact, this month's issue of Sky and Telescope has a featured article on cosmology and topology (Falk 1999).

Unfortunately, most astronomy texts only mention three possible models for the universe: a 3-sphere, Euclidean 3-space, and an infinite hyperbolic space. This is probably the case because it is easy to refer to the 2-dimensional versions of these manifolds (namely, the 2-sphere, the Euclidean plane, and the hyperbolic 2-manifold H² [sometimes called the "saddle surface"] - see, for example, Snow and Brownsberger 1997). Since there are three posible geometries (elliptic, Euclidean, and hyperbolic), and the universe may be open (i.e. infinite) or closed (i.e. finite or compact), then there seems to be a total of six possible types of manifolds as candidates for the shape of our universe. However, a positive curvature manifold "curves back on itself" and cannot be open. This gives us five possibilities. Table 1 gives an example of a 3-manifold for each of these categories. In particular, notice from Table 1 that a closed universe can have zero or negative curvature! There is much more to the picture than is often portrayed in undergraduate classes! We suspect that this will change in the near future as the multiply connected 3-manifolds become more widely known. Of course, an even greater impact would be the empirical determination of the global topology of our universe. We now turn to that topic.

Table 1. Possible global topologies of the universe. In each admissible case, an example of a 3-manifold is given which satisfies the given conditions on geometry and open/closed-ness.

Go to Section 6.