mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a 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 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 com ...

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 In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...

, 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 c ...

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 i ...

, 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