Talk:Boundary parallel

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Unclear lead - wrong links?

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]