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

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

Context and applications

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References

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↑ cf. Definition 3.4.7 in Template:Cite book
↑ That is there exists a neighbourhood of $F$ in $M$ which is homeomorphic to $F \times] - 1, 1 [$ (plus the obvious boundary condition), which if $F$ is either orientable or 2-sided in $M$ is in practice always the case.

[1] . Definition 3.4.7 in Template:Cite book

[2] That is there exists a neighbourhood of $F$ in $M$ which is homeomorphic to $F \times] - 1, 1 [$ (plus the obvious boundary condition), which if $F$ is either orientable or 2-sided in $M$ is in practice always the case.

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Boundary parallel

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