ABSTRACT: A popular tool used in freshman astronomy classes is the "balloon analogy" of the universe. In this analogy, we imagine ourselves as two-dimensional inhabitants of the surface of a swelling sphere. This model of the universe has the desirable properties that it (1) has no edge, (2) has no center, and (3) satisfies Hubble's Law. Also, this model has spherical geometry and a finite amount of "space." When discussing the other possible geometries of the universe (namely, Euclidean and hyperbolic), the two-dimensional analogies used are usually the Euclidean plane and the hyperbolic parabaloid (respectively). These surfaces have the desired curvatures and geometries. However, many students get the impression from these examples that a space with zero or negative curvature must be infinite. This is not the case.

In this presentation, an informal description of 3-manifolds and their topology will be given. A catalogue of topologically distinct manifolds will be presented, including those which have zero and negative curvature, yet have finite volume. Models of the universe in terms of these manifolds will be introduced. Finally, empirical methods for determining which 3-manifold represents the topology of our universe will be described.

1. Introduction
2. Some Definitions
3. 2-Manifolds
4. 3-Manifolds
5. The Global Toplogy of the Universe
6. Empirical Methods
7. References

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