Does spacetime in general relativity contain holes?
The Max Planck Institute for Gravitational Physic says: "the most drastic consequence of Einstein's description of gravity ... is the possibility that space and time may exhibit 'holes' or 'edges' ...". http://www.einstein-online.info/spotlights/singularities
GR clearly allows spacetime to contain holes. Geodesic incompleteness is built into the system. The question remains, though, as to whether or not such holes actually exist.
I believe they do, and exactly in the sense of excised regions of the manifold. The excised regions are what we call black holes.
The radial component of the Schwarzschild metric tells us that the metric stretching of space goes to infinity at the event horizon. If the vacuum has intrinsic mass, this means that space itself thins out and disappears completely at the horizon. The region inside the event horizon, then, becomes a cutout in the manifold, i.e., a cavitation bubble, or hole, in spacetime.
The infinity at the horizon is not just an unfortunate artifact of the Schwarzschild coordinate system. A coordinate transformation to a constantly accelerating frame in flat, Minkowski spacetime (the "free-fall" coordinate system) only makes the infinity analytically removable. It does not actually remove it or change the effect of metric stretching.
So, assuming an intrinsic mass of the vacuum, it looks to me like an event horizon marks the edge of the spacetime manifold and the degeneration of the metric.
Such cavitation would likely involve a topological change as well as degeneration of the metric but it may not be a problem. Here is a paper entitled Topology Change in General Relativity by Gary T. Horowitz, Department of Physics, University of California, that appears to turn the question on its head:
"The question is not whether topology change can occur, but rather how do we stop topology from changing? Why doesn’t the space around us suddenly split into disconnected pieces?" http://arxiv.org/pdf/hep-th/9109030v1.pdf
The other main issue is the question of intrinsic mass. Does the vacuum have its own mass? Is the vacuum a substance or is it not? ... the old debate of abstract relationalism vs. substantivalism. As far as I know the issue is still undecided, so, cutouts in the manifold must remain a possibility.
In my humble opinion, this is the most interesting and most important question going in physics today. If such cutouts exist, it would force a paradigm shift in our thinking about the nature of reality.
The external metric associated with a cavity in the manifold would be (by Birkhoff's theorem) the same as that for a normal massive object or a black hole. So a cavity in the spacetime manifold would be gravitationally indistinguishable from either. This, in answer to your related question, is the way localized and special configurations of (classical) spacetime could be interpreted as matter.