Boundary parallel

Short description: When a closed manifold embeded in M has an isotopy onto a boundary component of M

In mathematics, a connected submanifold of a compact manifold with boundary is said to be boundary parallel, ∂-parallel, or peripheral if it can be continuously deformed into a boundary component. This notion is important for 3-manifold topology.

Boundary-parallel embedded surfaces in 3-manifolds

If $F$ is an orientable closed surface smoothly embedded in the interior of an manifold with boundary $M$ then it is said to be boundary parallel if a connected component of $M\setminus F$ is homeomorphic to $F\setminus [0,1[$.

In general, if $(F,\partial F)$ is a topologically embedded compact surface in a compact 3-manifold $(M,\partial M)$ some more care is needed: one needs to assume that $F$ admits a bicollar, and then $F$ is boundary parallel if there exists a subset $P\subset M$ such that $F$ is the frontier of $P$ in $M$ and $P$ is homeomorphic to $F\times [0,1]$.

See also

• Atoroidal
• Satellite knot

References

• Shalen, Peter B. (2002), "Representations of 3-manifold groups", in Daverman, R. J., Handbook of geometric topology, Amsterdam: Elsevier, pp. 955-1044