Boundary parallel

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

An example

π is not bijective on S, but S is boundary-parallel anyway.

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

$\pi :I\times S^{1}\rightarrow S^{1},\qquad (x,z)\mapsto z.$

π is bijective on S.

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.)

π is not surjective on S.

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.)