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 is an orientable closed surface smoothly embedded in the interior of an manifold with boundary then it is said to be boundary parallel if a connected component of is homeomorphic to .cf. Definition 3.4.7 in
In general, if is a topologically embedded compact surface in a compact 3-manifold some more care is needed: one needs to assume that admits a bicollar,That is there exists a neighbourhood of in which is homeomorphic to (plus the obvious boundary condition), which if is either orientable or 2-sided in is in practice always the case. and then is boundary parallel if there exists a subset such that is the frontier of in and is homeomorphic to .
Context and applications
See also Atoroidal Satellite knot
References