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Dominant Topologies in Euclidean Quantum Gravity

The dominant topologies in the Euclidean path integral for quantum gravity differ sharply according on the sign of the cosmological constant. For $Λ>0$, saddle points can occur only for topologies with vanishing first Betti number and finite fundamental group. For $Λ<0$, 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&#39;&#39; grows fast enough to overwhelm this suppression. The value $Λ=0$ 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 wave function.