Boundary parallel

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

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 bijective on S.

An example wherein π is not surjective on S.