I apologize for the poorly worded title. Let me try to explain my question better. A scientific theory must be predictive to be useful. Since we only know what happened in the past, the global topology of spacetime cannot be an input to the theory. Given space-like slices/"chunk" of the manifold, and the metric on that chunk along with the fields + physics of how they evolve, GR appears to let us solve for not just the metric on the manifold outside this 'initial chunk' ... but "for the manifold itself". For example, let's start with an initial condition in which a wormhole exists. Run the equations and we might find that the wormhole closes off in the future. We did not know the topology ahead of time ... in some sense we are solving for the manifold itself. How can Einstein's field equations (or really any local theory) result in changes to the global structure? I have a feeling I'm really approaching this wrong, and would like to have a discussion about it. I would appreciate your insight.