Talk:Scott core theorem

More precision needed

I assume the theorem is for differentiable manifolds.

Should'nt the manifold (and its core) be connected?

In most interesting cases, the core N must have a boundary.

Was Scott's original manifold M allowed to have a boundary?

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 the big one, for instance? Etc...

Now I just had a look at the cited article by Rubinstein and Swarup, and it gives an interesting complement:

Theorem 2: Let M be a 3-manifold with finitely-generated fundamental group and let C be a compact submanifold of dM. Then there is a core N of M with N ∩ dM = C.

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 without boundary: int(M).

Did Scott construct a core N that does not touch the boundary of M?

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 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 :(

Last question: do we work with differentiable or topological manifolds? In the latter case, at least, corners are the same as sides... — Preceding unsigned comment added by Arnaud Chéritat (talk • contribs) 13:56, 8 January 2014 (UTC)