Dominant topologies in Euclidean quantum gravity

The dominant topologies in the Euclidean path integral for quantum gravity differ sharply according to the sign of the cosmological constant. For Lambda > 0, saddle points can occur only for topologies with vanishing first Beni number and finite fundamental group. For Lambda < 0, on the other hand, the path integral is dominated by topologies with extremely complicated fundamental groups; while the contribution of each individual manifold is strongly suppressed, the 'density of topologies' grows fast enough to overwhelm this suppression. The value Lambda = 0 is thus a sort of boundary between phases in the sum over topologies. I discuss some implications for the cosmological constant problem and the Hartle-Hawking wavefunction.