In mathematics, a

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

''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''

embedded Embedded or embedding (alternatively imbedded or imbedding) may refer to: Science * Embedding, in mathematics, one instance of some mathematical object contained within another instance ** Graph embedding * Embedded generation, a distributed ge ...

in an (''n'' + 1)-manifold ''M'' is boundary parallel (or ∂-parallel, or peripheral) if there is an isotopy of ''N'' onto a

boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...

component Circuit Component may refer to: •Are devices that perform functions when they are connected in a circuit. In engineering, science, and technology Generic systems * System components, an entity with discrete structure, such as an assem ...

of ''M''.

An example

Consider the

annulus Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to: Human anatomy * ''Anulus fibrosus disci intervertebralis'', spinal structure * Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus co ...

I \times S^1

. Let π 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 π restricted to ''S'' is a bijection, then ''S'' is boundary parallel. (The

converse Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a definite or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical ...

is not true.) If, on the other hand, a circle ''S'' is embedded into the annulus so that π restricted to ''S'' is not

surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...

, then ''S'' is not boundary parallel. (Again, the converse is not true.) Image:Annulus.circle.pi 1-injective.png, An example wherein π is not bijective on ''S'', but ''S'' is ∂-parallel anyway. Image:Annulus.circle.bijective-projection.png, An example wherein π is bijective on ''S''. Image:Annulus.circle.nulhomotopic.png, An example wherein π is not surjective on ''S''. {{DEFAULTSORT:Boundary Parallel Geometric topology