• Source: Boundary-incompressible surface

In low-dimensional topology, a boundary-incompressible surface is a two-dimensional surface within a three-dimensional manifold whose topology cannot be made simpler by a certain type of operation known as boundary compression.
Suppose M is a 3-manifold with boundary. Suppose also that S is a compact surface with boundary that is properly embedded in M,
meaning that the boundary of S is a subset of the boundary of M and the interior points of S are a subset of the interior points of M.
A boundary-compressing disk for S in M is defined to be a disk D in M such that



D
∩
S
=
α


{\displaystyle D\cap S=\alpha }

and



D
∩
∂
M
=
β


{\displaystyle D\cap \partial M=\beta }

are arcs in



∂
D


{\displaystyle \partial D}

, with



α
∪
β
=
∂
D


{\displaystyle \alpha \cup \beta =\partial D}

,



α
∩
β
=
∂
α
=
∂
β


{\displaystyle \alpha \cap \beta =\partial \alpha =\partial \beta }

, and



α


{\displaystyle \alpha }

is an essential arc in S (



α


{\displaystyle \alpha }

does not cobound a disk in S with another arc in



∂
(
S


{\displaystyle \partial (S}

).
The surface S is said to be boundary-compressible if either S is a disk that cobounds a ball with a disk in



∂
M


{\displaystyle \partial M}

or there exists a boundary-compressing disk for S in M. Otherwise, S is boundary-incompressible.
Alternatively, one can relax this definition by dropping the requirement that the surface be properly embedded. Suppose now that S is a compact surface (with boundary) embedded in the boundary of a 3-manifold M. Suppose further that D is a properly embedded disk in M such that D intersects S in an essential arc (one that does not cobound a disk in S with another arc in



∂
S


{\displaystyle \partial S}

). Then D is called a boundary-compressing disk for S in M. As above, S is said to be boundary-compressible if either S is a disk in



∂
M


{\displaystyle \partial M}

or there exists a boundary-compressing disk for S in M. Otherwise, S is boundary-incompressible.
For instance, if K is a trefoil knot embedded in the boundary of a solid torus V and S is the closure of a small annular neighborhood of K in



∂
V


{\displaystyle \partial V}

, then S is not properly embedded in V since the interior of S is not contained in the interior of V. However, S is embedded in



∂
V


{\displaystyle \partial V}

and there does not exist a boundary-compressing disk for S in V, so S is boundary-incompressible by the second definition.




See also


Incompressible surface




References


W. Jaco, Lectures on Three-Manifold Topology, volume 43 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, R.I., 1980.
T. Kobayashi, A construction of 3-manifolds whose homeomorphism classes of Heegaard splittings have polynomial growth, Osaka J. Math. 29 (1992), no. 4, 653–674. MR1192734.

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