Boundary parallel

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

In general, if ${\displaystyle (F,\partial F)}$ is a topologically embedded compact surface in a compact 3-manifold ${\displaystyle (M,\partial M)}$ some more care is needed^[2]: one needs to assume that ${\displaystyle F}$ admits a bicollar^[3], and then ${\displaystyle F}$ is boundary parallel if there exists a subset ${\displaystyle P\subset M}$ such that ${\displaystyle F}$ is the frontier of ${\displaystyle P}$ in ${\displaystyle M}$ and ${\displaystyle P}$ is homeomorphic to ${\displaystyle F\times [0,1]}$.

Context and applications

See also

• Atoroidal
• Satellite knot