Boundary parallel

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 ${\displaystyle F}$  is an orientable closed surface smoothly embedded in the interior of an manifold with boundary ${\displaystyle M}$  then it is said to be boundary parallel if a connected component of ${\displaystyle M\smallsetminus F}$  is homeomorphic to ${\displaystyle F\smallsetminus [0,1[}$ ^[1].

In general, if ${\displaystyle (F,\partial F)}$  is a topologically embedded compact surface in a compact 3-manifold ${\displaystyle (M,\partial M)}$  some more care is needed^[2]: one needs to assume that ${\displaystyle F}$  admits a bicollar^[3], and then ${\displaystyle F}$  is boundary parallel if there exists a subset ${\displaystyle P\subset M}$  such that ${\displaystyle F}$  is the frontier of ${\displaystyle P}$  in ${\displaystyle M}$  and ${\displaystyle P}$  is homeomorphic to ${\displaystyle F\times [0,1]}$ .

Context and applications

See also

• Atoroidal
• Satellite knot

References

1. cf. Definition 3.4.7 in Schultens, Jennifer (2014). Introduction to 3-manifolds. Graduate studies in mathematics. Vol. 151. AMS. ISBN 978-1-4704-1020-9.
2. Shalen 2002, p. 963.
3. That is there exists a neighbourhood of ${\displaystyle F}$  in ${\displaystyle M}$  which is homeomorphic to ${\displaystyle F\times \left]-1,1\right[}$  (plus the obvious boundary condition), which if ${\displaystyle F}$  is either orientable or 2-sided in ${\displaystyle M}$  is in practice always the case.

• Shalen, Peter B. (2002), "Representations of 3-manifold groups", in Daverman, R. J.; Sher, R. B. (eds.), Handbook of geometric topology, Amsterdam: Elsevier, pp. 955–1044