Boundary parallel explained
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
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: one needs to assume that
admits a bicollar,
[2] and then
is boundary parallel if there exists a subset
such that
is the
frontier of
in
and
is homeomorphic to
.
See also
Notes and References
- cf. Definition 3.4.7 in Book: Schultens, Jennifer. Introduction to 3-manifolds. Introduction to 3-Manifolds. Jennifer Schultens. Graduate studies in mathematics. 151. 2014. 978-1-4704-1020-9. AMS.
- That is there exists a neighbourhood of in
which is homeomorphic to
