Boundary parallel

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In mathematics, a closed n-manifold embedded in an (n + 1)-manifold is boundary parallel (or ∂-parallel, or peripheral) if it can be isotoped onto a boundary component.

[edit] An example

Consider the annulus I\times S^1. Let π denote the projection map

\pi:I\times S^1\rightarrow S^1,\qquad(x,z)\mapsto z.

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.)

An example wherein π is not bijective on S, but S is ∂-parallel anyway.
An example wherein π is not bijective on S, but S is ∂-parallel anyway.
An example wherein π is bijective on S.
An example wherein π is bijective on S.
An example wherein π is not surjective on S.
An example wherein π is not surjective on S.


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