Boundary parallel

Template:Short description Template:One source 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.

An example

Consider the annulus $I\times S^1$. Let Template:Pi denote the projection map

$\pi :I\times S^1\to S^1,(x,z)\mapsto z.$

If a circle S is embedded into the annulus so that Template:Pi 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 Template:Pi restricted to S is not surjective, then S is not boundary parallel. (Again, the converse is not true.)

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

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