Boundary parallel
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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.[1]
An example
[edit]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.)
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An example wherein π is not bijective on S, but S is ∂-parallel anyway.
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An example wherein π is bijective on S.
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An example wherein π is not surjective on S.
Context and applications
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Further reading
[edit]- Culler, Marc, and Peter B. Shalen. "Bounded, separating, incompressible surfaces in knot manifolds." Inventiones mathematicae 75 (1984): 537-545.
See also
[edit]References
[edit]- ^ Definition 3.4.7 in Schultens, Jennifer (2014). Introduction to 3-manifolds. Graduate studies in mathematics. Vol. 151. AMS. ISBN 978-1-4704-1020-9.