boundary parallel

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 $I \times S^1$ . Let π denote the projection map

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

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.)

Image:Annulus.circle.pi 1-injective.png|An example wherein π is not bijective on S, but S is ∂-parallel anyway.

Image:Annulus.circle.bijective-projection.png|An example wherein π is bijective on S.

Image:Annulus.circle.nulhomotopic.png|An example wherein π is not surjective on S.

References

Category:Geometric topology