A few weeks back, I posted a related question, Could metric expansion create holes, or cavities in the fabric of spacetime?, asking if metric stretching could create cutouts in the spacetime manifold. The responses involved a number of issues like ambient dimensions, changes in coordinate systems, intrinsic curvature, intrinsic mass of the spacetime manifold and the inviolability of the manifold. I appreciated the comments but, being somewhat familiar with the various issues, I felt that the question didn't get a very definitive answer.
So, if I may, I would like to ask what I hope to be a more focused question; a question about the topology of 3-manifolds in general. Are cutouts or cavities allowed in a 3-manifold or are these manifolds somehow sacrosanct in general and not allowed to be broken?
As I noted in the previous discussion, G. Perleman explored singularities in unbounded 3-manifolds and found that certain singularity structures could arise. Surprisingly, their shapes were three-dimensional and limited to simple variations of a sphere stretched out along a line.
Three-dimensional singularities, then, can be embedded inside a 3-manifold and the answer to my question seems to depend on whether or not these 3-dimensional singularities are the same things as cutouts in the manifold.
I also found the following, which seems to describe what I have in mind. It's a description of an incompressible sphere embedded in a 3-manifold: "... a 2-sphere in a 3-manifold that does not bound a 3-ball ..."
Does this not define a spherical, inner boundary of the manifold, i.e., a cutout in the manifold?