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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?

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One clarification: when you say " Are cutouts or cavities allowed in a 3-manifold or are these manifolds somehow sacrosanct in general and not allowed to be broken?" - are you asking a purely mathematical question about manifold definition, or are you asking about the types of manifolds that are allowed in physical theories like GR?. Re your last sentence: if you remove a closed ball from $\mathbb{R}^3$ you'd get a manifold-without-boundary, but if you remove an open ball, you'd get a manifold-with-boundary. Both are topology changes. – twistor59 Jan 22 '13 at 7:50
@twistor59: to remove ambiguities for other readers: 'manifolds with boundary' are generally not considered as manifolds in the mathematical literature. In any case, neither of the two cases (3-ball minus open 2-ball or 3-ball minus closed 2-ball) meets the requirements for the Poincaré conjecture. – Vibert Jan 22 '13 at 8:40
I think the question is not very clear. The work of Perelman has to do with a very special class of manifolds, namely those that are closed and simply-connected. Apart from the 3-sphere itself, I cannot even think of any closed 3-manifold: you need something 'without a boundary' but it cannot have 'open edges' either. – Vibert Jan 22 '13 at 8:44
...and do you mean "a 2-sphere in {a 3-manifold that does not bound a 3-ball}"? That seems strange. Open 3-balls are a basis of open sets for the topology of any 3-manifold. In other words, if you give me some (sub)manifold, I can always find a 3-ball inside of it. – Vibert Jan 22 '13 at 8:48
Should this be migrated to the maths stackexchange? As far as I can tell, while the initial motivation was physical, the question itself is entirely mathematical. – Michael Brown Jan 22 '13 at 10:46

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