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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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dcgeorge
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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 07:50
  • @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 08:40
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    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 08:44
  • ...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 08:48
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    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 Jan 22 '13 at 10:46
  • @twistor59, I'm mot sure if it's a purely mathematical question or not. The goal is to understand how the rules of topology might apply to the spacetime manifold. Chris White answered my previous question with the statement that "You can no more tear spacetime than you can tear the abstract notion of the x,y-plane." but I'm still wondering why. If there is such a rule, does it apply to 3-manifolds in general or just to the spacetime manifold in particular?
    The bottom line is that I still can not see a clear and definite reason why cutouts could not exist in the spacetime manifold.     – dcgeorge Jan 23 '13 at 16:19
  • @Vibert, As for the last sentence, I understand it to mean that it's the 2-sphere that does not bound a 3-ball. The 2-sphere is embedded in the 3-manifold but does not, itself, bound a 3-ball. So, there is no 3-ball (i.e., no manifold) inside the 2-sphere.

    The 2-sphere, it seems to me, would be an internal boundary of the 3-manifold and the whole arrangement a description of a cutout in the 3-manifold. The 3-manifold may or may not be bounded externally.

     – dcgeorge Jan 23 '13 at 16:24
  • @Michael Brown, sorry about the shifting contexts; I don't know what stackexchange realm would be more appropriate. – dcgeorge Jan 23 '13 at 16:27
  • Possible duplicate: http://physics.stackexchange.com/q/1787/2451 – Qmechanic Jan 23 '13 at 17:32
  • @Vibert, The comment you left yesterday is mysteriously gone today. Perhaps you deleted it?

    Anyway, thanks for showing me how to formally describe a cutout in a 3-manifold. I'm glad that we are in agreement.

    I also agree about the Poincaré conjecture. I got sidetracked reading about his proof while I was trying to learn something about topology and the part about 3-dimensional singularities caught my eye. It just looked to me like an example how cutouts could exist in a manifold, that's all. I wasn't trying to imply that it meets the requirements for the conjecture..

     – dcgeorge Jan 24 '13 at 22:03
  • @dcgeorge: I indeed deleted it, because I wasn't 100% sure that I had properly understood what you meant, and I don't like writing false statements on this website. Good luck with your further studies. – Vibert Jan 24 '13 at 23:57
  • @twistor59, since you posted your comments, I've found that topological change does seem to be forbidden in GR. Then I ran across this paper by Gary T. Horowitz who says "... the problem of topology change has been turned around. The question is not whether topology change can occur, but rather how do we stop topology from changing?" : http://arxiv.org/pdf/hep-th/9109030v1.pdf . Are you familiar with the paper? – dcgeorge Mar 08 '13 at 19:01
  • @dcgeorge No I haven't seen that paper before - it looks like an interesting read. Thanks for the tipoff! – twistor59 Mar 08 '13 at 19:33
  • I would argue that cutouts and boundaries are a subset of singularities. Singularities can really take many forms. I don't know of any dynamic process that would cause a gap to appear in a spacetime, though. – Zo the Relativist Mar 25 '14 at 14:56
  • @Jerry Schirmer. "I don't know of any dynamic process that would cause a gap to appear in a spacetime,..." How about simple cavitation? If the spacetime manifold is a fluid-like entity, couldn't the negative pressure from expansion of the universe pull it apart or boil it? – dcgeorge Sep 07 '14 at 18:19

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