Boundary parallel

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

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

Context and applications

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See also

• Atoroidal
• Satellite knot

References

• Shalen, Peter B. (2002), "Representations of 3-manifold groups", in Daverman, R. J.; Sher, R. B. (eds.), Handbook of geometric topology, Amsterdam: Elsevier, pp. 955–1044{{citation}}: CS1 maint: publisher location (link)