Boundary parallel
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.
An example
Consider the annulus . Let π denote the projection map
If a circle S is embedded into the annulus so that π restricted to S is a bijection, then S is boundary parallel. (The converse is not true.)
If, on the other hand, a circle S is embedded into the annulus so that π restricted to S is not surjective, then S is not boundary parallel. (Again, the converse is not true.)
![](../I/m/Annulus.circle.pi_1-injective.png)
An example wherein π is not bijective on S, but S is ∂-parallel anyway.
![](../I/m/Annulus.circle.bijective-projection.png)
An example wherein π is bijective on S.
![](../I/m/Annulus.circle.nulhomotopic.png)
An example wherein π is not surjective on S.
gollark: A ΛK-class critical failure scenario of much PotatOS infrastructure.
gollark: Succeeded in what?
gollark: No, the fact that it's a mostly uncontrolled and highly intelligent AI.
gollark: It is *very dangerous*.
gollark: We have tried to contain it and scrub existing instances of the code from the testing repositories.
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