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 I × S 1 {\displaystyle I\times S^{1}} . Let π denote the projection map
π : I × S 1 → S 1 , ( x , z ) ↦ z . {\displaystyle \pi \colon I\times S^{1}\rightarrow S^{1},\quad (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.)
Context and applications
Further reading
- Culler, Marc, and Peter B. Shalen. "Bounded, separating, incompressible surfaces in knot manifolds." Inventiones mathematicae 75 (1984): 537–545.
See also
References
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. 978-1-4704-1020-9 ↩