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Boundary parallel
When a closed manifold embeded in M has an isotopy onto a boundary component of M From Wikipedia, the free encyclopedia
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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.
This article relies largely or entirely on a single source. (June 2025) |
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[2]: one needs to assume that admits a bicollar[3], and then is boundary parallel if there exists a subset such that is the frontier of in and is homeomorphic to .
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Context and applications
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