mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a boundary parallel, ∂-parallel, or peripheral closed ''n''-

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

''N'' embedded in an (''n'' + 1)-manifold ''M'' is one for which there is an isotopy of ''N'' onto a boundary

component Component may refer to: In engineering, science, and technology Generic systems *System components, an entity with discrete structure, such as an assembly or software module, within a system considered at a particular level of analysis * Lumped e ...

of ''M''.Definition 3.4.7 in

An example

Consider the annulus

I \times S^1

. Let denote the projection map :

\pi\colon I \times S^1 \rightarrow S^1,\quad (x, z) \mapsto z.

If a circle ''S'' is embedded into the annulus so that restricted to ''S'' is a

bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

, then ''S'' is boundary parallel. (The converse is not true.) If, on the other hand, a circle ''S'' is embedded into the annulus so that restricted to ''S'' is not

surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...

, then ''S'' is not boundary parallel. (Again, the converse is not true.) Image:Annulus.circle.pi 1-injective.png, An example in which is not bijective on ''S'', but ''S'' is ∂-parallel anyway. Image:Annulus.circle.bijective-projection.png, An example in which is bijective on ''S''. Image:Annulus.circle.nulhomotopic.png, An example in which is neither surjective nor injective on ''S''.

Context and applications

References

{{DEFAULTSORT:Boundary Parallel Geometric topology

An example

Context and applications

Further reading

See also

References