Topology changes by quantum tunneling in four dimensional gravity

We investigate topology-changing processes in 4-dimensional quantum gravity with a negative cosmological constant. By playing the “gluing-polytope game” in hyperbolic geometry, we explicitly construct an instanton-like solution without singularity. Because of cusps, this solution is non-compact but has a finite volume. Then we evaluate a topology change amplitude in the WKB approximation in terms of the volume of this solution.