Boundary parallel: Difference between revisions
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imported>Michael Hardy Changing short description from "When a closed manifold embeded in M has an isotopy onto a boundry component of M" to "When a closed manifold embeded in M has an isotopy onto a boundary component of M" |
imported>Orenburg1 m sp |
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{{Short description|When a closed manifold | {{Short description|When a closed manifold embedded in M has an isotopy onto a boundary component of M}} | ||
{{one source|date=June 2025}} | {{one source|date=June 2025}} | ||
In [[mathematics]], a '''boundary parallel''', '''∂-parallel''', or '''peripheral''' [[ | 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 (mathematics)|frontier]] of <math>P</math> in <math>M</math> and <math>P</math> is homeomorphic to <math>F \times [0, 1]</math>. | |||
< | |||
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==Context and applications== | ==Context and applications== | ||
{{expand section|date=June 2025}} | {{expand section|date=June 2025}} | ||
==See also== | ==See also== | ||
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==References== | ==References== | ||
{{reflist}} | {{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}} | {{DEFAULTSORT:Boundary Parallel}} | ||
[[Category:Geometric topology]] | [[Category:Geometric topology]] | ||
Latest revision as of 10:03, 17 November 2025
Template:Short description Script error: No such module "Unsubst". 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 .[1]
In general, if is a topologically embedded compact surface in a compact 3-manifold some more care is needed:Template:Sfn one needs to assume that admits a bicollar,[2] and then is boundary parallel if there exists a subset such that is the frontier of in and is homeomorphic to .
Context and applications
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See also
References
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