Abstract. The dominant topologies in the Euclidean path integral for quantum gravity differ sharply according to the sign of the cosmological constant. For , saddle points can occur only for topologies with vanishing first Betti number and finite fundamental group. For , 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 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.

PACS numbers: 0460G, 9880H, 0420G, 0240K

Print publication: Issue 9 (September 1998)
Received 28 October 1997