  The dominant topologies in the Euclidean path integral for quantum gravity
differ sharply according on the sign of the cosmological constant. For
$\Lambda>0$, saddle points can occur only for topologies with vanishing first
Betti 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 wave function.
