  Observations of microwave background fluctuations can yield information not
only about the geometry of the universe, but potentially about the topology of
the universe. If the universe is negatively curved, then the characteristic
scale for the topology of the universe is the curvature radius. Thus, if we are
seeing the effects of the geometry of the universe, we can hope to soon see
signatures of the topology of the universe. The cleanest signature of the
topology of the universe is written on the microwave sky: there should be
thousands of pairs of matched circles. These circles can be used to determine
the precise topology and volume of the universe. Since we see hundreds of
slices through the fundamental domain of the universe, we can use the microwave
observations to reconstruct the initial conditions of the entire universe on
the scale of a few Megaparsecs.
