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.
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 \smallsetminus F is homeomorphic to F \smallsetminus [0, 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: one needs to assume that F admits a bicollar, 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
See also
Atoroidal Satellite knot
References
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