Boundary parallel
Template:Short description Template:One source 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 <math>F</math> is an orientable closed surface smoothly embedded in the interior of an manifold with boundary <math>M</math> then it is said to be boundary parallel if a connected component of <math>M \smallsetminus F</math> is homeomorphic to <math>F \smallsetminus [0, 1[</math>.<ref>cf. Definition 3.4.7 in Template:Cite book</ref>
In general, if <math>(F, \partial F)</math> is a topologically embedded compact surface in a compact 3-manifold <math>(M, \partial M)</math> some more care is needed:Template:Sfn one needs to assume that <math>F</math> admits a bicollar,<ref>That is there exists a neighbourhood of <matH>F</math> in <math>M</math> which is homeomorphic to <math>F \times \left]-1, 1\right[</math> (plus the obvious boundary condition), which if <math>F</math> is either orientable or 2-sided in <math>M</math> is in practice always the case.</ref> and then <math>F</math> is boundary parallel if there exists a subset <math>P \subset M</math> such that <math>F</math> is the frontier of <math>P</math> in <math>M</math> and <math>P</math> is homeomorphic to <math>F \times [0, 1]</math>.