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Let us assume that it is allowed. Then the core is a (maximal dimensional) sub-manifold-with-boundary of a manifold-with-boundary. But what does that mean? There are several possiblilities. After a quick thinking, I can already see 3 or 4 of them. Do we allow for corners contained in the boundary of
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In the
Theorem 2 above, there is no specification on the dimension of C. This means that a very loose definition of what is a sub-manifold-with-boundary of a manifold-with-boundary. Indeed, the boundary of N, that has dimension 2, is allowed to intersect the boundary of M along a 2, 1 or 0
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So indeed M is allowed to have a boundary. Moreover in under Scott's theorem hypotheses, one can have a core N that does NOT intersect the boundary of M. This gives us a way to circumvent the problem in Scott's theorem: one does only has to define a sub-manif with bdy of a manif
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dimensional sub-manifold. Maybe corners are even allowed, if C is allowed to be a manifold with boundary... Or mabye not (in the C world, two non-transverse surfaces can intersect along a closed disk, for instance). Now I'm even more confused :(
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Last question: do we work with differentiable or topological manifolds? In the latter case, at least, corners are the same as sides...
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Now I just had a look at the cited article by
Rubinstein and Swarup, and it gives an interesting complement:
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Let M be a 3-manifold with finitely-generated fundamental group and let C be a compact submanifold of dM.
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Did Scott construct a core N that does not touch the boundary of M?
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I assume the theorem is for differentiable manifolds.
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Should'nt the manifold (and its core) be connected?
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