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The Sum over Topologies in Three-Dimensional Euclidean Quantum Gravity

In Hawking&#39;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 $Λ$, but for dramatically different reasons: for $Λ>0$, the divergent behavior comes from the contributions of very low volume, topologically complex manifolds, while for $Λ<0$ 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.