{{Short description|When a closed manifold embedded in M has an isotopy onto a boundary component of M}} {{one source|date=June 2025}} 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 {{cite book|title=Introduction to 3-manifolds|title-link=Introduction to 3-Manifolds|last=Schultens|first=Jennifer|author-link=Jennifer Schultens|series=Graduate studies in mathematics|volume=151|year=2014|ISBN=978-1-4704-1020-9|publisher=AMS}}</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:{{sfn|Shalen|2002|p=963}} 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>.
==Context and applications== {{expand section|date=June 2025}}
==See also== *Atoroidal *Satellite knot
==References== {{reflist}}
*{{Citation | last = Shalen | first = Peter B. | editor-last = Daverman | editor-first = R. J. | editor-last2 = Sher | editor-first2 = R. B. | contribution = Representations of 3-manifold groups | title = Handbook of geometric topology | date = 2002 | pages = 955–1044 | publisher = Amsterdam: Elsevier }}
{{DEFAULTSORT:Boundary Parallel}} Category:Geometric topology