Boundary parallel

In mathematics, a boundary parallel, ∂-parallel, or peripheral closed n-manifold N embedded in an (n + 1)-manifold M is one for which there is an isotopy of N onto a boundary component of M.^[1]

An example

Consider the annulus ${\displaystyle I\times S^{1}}$. Let π denote the projection map

${\displaystyle \pi \colon I\times S^{1}\rightarrow S^{1},\quad (x,z)\mapsto z.}$

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.)

• 
  An example in which π is not bijective on S, but S is ∂-parallel anyway.
• 
  An example in which π is bijective on S.
• 
  An example in which π is neither surjective nor injective on S.

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

• Atoroidal
• Satellite knot