TOPOLOGY AND THE UNIVERSE

Topology may play an important role in cosmology in several different ways. First, Einstein's field equations tell us about the local geomet ry of the universe but not about its topology. Therefore, the universe may be multiply connected. Inflation predicts that the fluctuations t hat made clusters and groups of galaxies arose from random quantum flu ctuations in the early universe. These should be Gaussian random phase . This can be tested by quantitatively measuring the topology of large -scale structure in the universe using the genus statistic. If the ori ginal fluctuations were Gaussian random phase then the structure we se e today should have a spongelike topology. A number of studies by our group and others have shown that this is indeed the case. Future tests using the Sloan Digital Sky Survey should be possible. Microwave back ground fluctuations should also exhibit a characteristic symmetric pat tern of hot and cold spots. The COBE data are consistent with this pat tern and the MAP and PLANCK satellites should provide a definitive tes t. If the original inflationary state was metastable then it should de cay by making an infinite number of open inflationary bubble universes . This model makes a specific prediction for the power spectrum of flu ctuations in the microwave background which can be checked by the MAP and PLANCK satellites. Finally, Gott and Li have proposed how a multip ly connected cosmology with an early epoch of closed timelike curves m ight allow the universe to be its own mother.