Many interesting proofs in GR (regarding blackholes and singularities, etc.) involve topological methods. However, I don't understand how a theory embodied in Einstein's equations, which appear to me to be local rules of evolution, can ever change the topology since a patch of spacetime around any event is locally equivalent to minkowski / flat-spacetime. Said another way, how can a local theory _ever_ change the global topology? I'm really having trouble picturing this.