The question of whether information is lost in black holes is investigated
using Euclidean path integrals. The formation and evaporation of black holes is
regarded as a scattering problem with all measurements being made at infinity.
This seems to be well formulated only in asymptotically AdS spacetimes. The
path integral over metrics with trivial topology is unitary and information
preserving. On the other hand, the path integral over metrics with non-trivial
topologies leads to correlation functions that decay to zero. Thus at late
times only the unitary information preserving path integrals over trivial
topologies will contribute. Elementary quantum gravity interactions do not lose
information or quantum coherence.