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

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

Context and applications

See also

  • 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)