Talk:Boundary parallel

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Unclear lead - wrong links?

Latest comment: 1 month ago9 comments2 people in discussion

The text In mathematics, a closed n-manifold N embedded in an (n + 1)-manifold M is boundary parallel (or ∂-parallel, or peripheral) if there is an isotopy of N onto a boundary component of M. is unclear for several reasons:

• Boundary (topology) pertains to a subset of a topological space; the topological boundary of the entire space is the empty set. Should that be Manifold#Boundary and interior?
• Homotopy#Isotopy pertains to a pair of functions
• The pair of links boundary component violates WP:SOB

I considered In mathematics, a closed n-manifold N embedded in an (n + 1)-manifold with boundary M is boundary parallel (or ∂-parallel, or peripheral) if there is an isotopy of N onto a component of M's boundary., but that seems stilted and doesn't address the second issue.

How about In mathematics, an embedding ${\displaystyle f\colon N\rightarrow M}$  of a closed n-manifold N in an (n + 1)-manifold with boundary M is boundary parallel (or ∂-parallel, or peripheral) if there is an embedding ${\displaystyle g\colon N\rightarrow C}$  of N onto a component C of M's boundary and f is isotopic to g.

Is the concept of components of the boundary of a manifold with boundary important enough to warrant a section or anchor somewhere? Note that boundary component links to the wrong definition and probably should be a DAB page. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 08:23, 12 June 2025 (UTC) -- Revised 13:12, 12 June 2025 (UTC)Reply

Unless someone objects I'll go with In mathematics, an embedding ${\displaystyle f\colon N\rightarrow M}$  of a closed n-manifold N in an (n + 1)-manifold with boundary M is boundary parallel (or ∂-parallel, or peripheral) if there is an embedding ${\displaystyle g\colon N\rightarrow C}$  of N onto a component C of M's boundary and f is isotopic to g. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:19, 25 June 2025 (UTC)Reply

I located a copy of the cited source^[1] and see that Definition 3.4.7 is substantially different from the definition in the article. The cited definition

Definition 3.4.7. Let M be a connected 3-manifold. A 2-sphere ${\displaystyle S\subset M}$  is essential if it does not bound a 3-ball. A surface ${\displaystyle F\subset M}$  is boundary parallel if it is separating and a component of ${\displaystyle M\setminus F}$  is homeomorphic to ${\displaystyle F\times I}$ 

does not mention isotopy or even homotopy and is specific to 3 dimensions. Does that constitute WP:SYNTHESIS? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:40, 14 July 2025 (UTC)Reply

The same definition as Schultens' is given (again in the context of surfaces in 3--manifolds) in Shalen's article in the handbook of geometric topology (cf. p. 963).

At this point it seems that this definition should be in the article (for 3-manifolds it's most likely going to be equivalent to the current sourceless one). And if you cannot locate a source for the other one it should probably not be in the article (you could try to ask on mathoverflow).

(As i mentioned previously i'm not even sure there should be a full-fledged article on this notion, though there should be at least a redirect). jraimbau (talk) 14:55, 17 July 2025 (UTC)Reply

Is that this^[2] book? Is there a PDF? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:32, 17 July 2025 (UTC)Reply

That's the book. jraimbau (talk) 16:49, 17 July 2025 (UTC)Reply

That seems to have a third definition, one that I suggest we quote in place of the unsourced one currently in the article. Is there an online copy that supports cut-and-paste? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:01, 17 July 2025 (UTC)Reply

It is the same definition as Schultens' up to slight rewording. I'll try to work on the article this weekend unless you get to it first. jraimbau (talk) 06:18, 18 July 2025 (UTC)Reply

No, the terms bicollared and frontier^[a] are substantive differences. They would have been closer had the wiki text and Schultens mentioned the closure of a component, but that would still have left bicollared as a difference.

I would like to cite Shalen's definition, but don't know how to do a combined citation for the article^[3] and the book.^[2] -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:19, 18 July 2025 (UTC)Reply

Notes

1. Shalen uses the term frontier to avoid ambiguity, but there would still have been a difference had he used border or boundary.

References

1. Schultens, Jennifer (2014). "Definition 3.4.7". Introduction to 3-manifolds. Graduate studies in mathematics. Vol. 151. American Mathematical Society. ISBN 978-1-4704-1020-9. LCCN 2013046541.
2. Sher, R.B.; Daverman, Robert B., eds. (2002). "Chapter 19. Representations of 3-manifold groups" (PDF). Handbook of Geometric Topology (PDF). Elsevier. p. 963. ISBN 0-444-82432-4.
3. Shalen, Peter B. (2002). "1.5 The 3-dimensional case: Essential Surfaces" (PDF). Chapter 19. Representations of 3-manifold groups (PDF). Elsevier. p. 963. ISBN 0-444-82432-4.

Still not quite right

Latest comment: 1 month ago1 comment1 person in discussion

@Jean Raimbault and Michael Hardy: As of permalink/1241623, the definition in § Boundary-parallel embedded surfaces in 3-manifolds appears correct but does not match the definition^[1] in the cited source, which appears to be missing a "closure of" phrase.

Is there any reason to not have an editor-link for Robert Daverman and an author-link for Peter Shalen? Is there any reason not to use |url=https://webhomes.maths.ed.ac.uk/~v1ranick/papers/handgt.pdf and to give page and section links? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:31, 20 July 2025 (UTC)Reply

References

1. Schultens, Jennifer (2014). "Definition 3.4.7". Introduction to 3-manifolds. Graduate studies in mathematics. Vol. 151. American Mathematical Society. ISBN 978-1-4704-1020-9. LCCN 2013046541. Definition 3.4.7. Let M be a connected 3-manifold. A 2-sphere ${\displaystyle S\subset M}$  is essential if it does not bound a 3-ball. A surface ${\displaystyle F\subset M}$  is boundary parallel if it is separating and a component of ${\displaystyle M\setminus F}$  is homeomorphic to ${\displaystyle F\times I}$ 

Proper math notation and TeX usage

Latest comment: 28 days ago1 comment1 person in discussion

@Jean Raimbault: In LaTeX and MathJax and the stripped-down modified LaTeX used in Wikipedia articles there is this typographical difference:

${\displaystyle F\times ]-1,1[}$ 

${\displaystyle F\times \left]-1,1\right[}$ 

The first one puts the bracket that appears to the left of ${\textstyle -1}$  in the wrong place and formats the "minus" sign in the wrong way. It is as if the thing to be enclosed in brackets is to the left of that and ends with the ${\textstyle \times }$  sign. Also, it puts horizontal space to the left and right of the minus sign, that should be there in cases where the minus sign is used as a binary operation symbol. But here it is used as a unary operation symbol, which would normally mean less space between the minus sign and the character to its right. Apparently things like this fail to be immediately obvious to some users. This has substrantial practical consequences: some readers will trip over an expression not so clearly formatted. The second one uses \left and \right. Michael Hardy (talk) 16:16, 31 July 2025 (UTC)Reply