Abstract: I will discuss the sum over topologies in three-dimensional quantum gravity and its relationship with the statistical interpretation of the boundary theory. I will put forward an axiomatization of the properties that a statistical CFT ensemble should possess. I will then identify a series of surgery moves on the gravity side that correspond to statistical contractions of OPE data in the boundary theory. The resulting analysis demonstrates that, despite generating a large number of non-trivial manifolds, these moves do not produce off-shell (non-hyperbolic) manifolds or even all on-shell (hyperbolic) manifolds. The proofs rely on structure theorems of 3-manifolds, which nicely interact with the requirements of the statistical boundary ensemble. I will illustrate the application of this procedure with many examples. Our findings reveal a large space of possible choices of which manifolds can consistently be included in the gravitational path integral, reflecting a wide range of possible statistical ensembles consistent with crossing symmetry. Based on work with Alexandre Belin, Lorenz Eberhardt, Diego Liska, and Boris Post.