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 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[2]: one needs to assume that admits a bicollar[3], and then is boundary parallel if there exists a subset such that is the frontier of in and is homeomorphic to .

Context and applications


See also

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 in which is homeomorphic to (plus the obvious boundary condition), which if is either orientable or 2-sided in 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{{citation}}: CS1 maint: publisher location (link)
Prefix: a b c d e f g h i j k l m n o p q r s t u v w x y z 0 1 2 3 4 5 6 7 8 9

Portal di Ensiklopedia Dunia

Kembali kehalaman sebelumnya