  In Hawking's Euclidean path integral approach to quantum gravity, the
partition function is computed by summing contributions from all possible
topologies. The behavior such a sum can be estimated in three spacetime
dimensions in the limit of small cosmological constant. The sum over topologies
diverges for either sign of $\Lambda$, but for dramatically different reasons:
for $\Lambda>0$, the divergent behavior comes from the contributions of very
low volume, topologically complex manifolds, while for $\Lambda<0$ it is a
consequence of the existence of infinite sequences of relatively high volume
manifolds with converging geometries. Possible implications for
four-dimensional quantum gravity are discussed.
