Talk:Boundary parallel#Unclear lead - wrong links

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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]