View source for Boundary-incompressible surface

In [[Low-dimensional topology|low-dimensional topology]], a '''boundary-incompressible surface''' is a two-dimensional surface within a three-dimensional [[Manifold|manifold]] whose topology cannot be made simpler by a certain type of operation known as '''boundary compression'''.
Suppose ''M'' is a [[3-manifold|3-manifold]] with boundary. Suppose also that ''S'' is a compact surface with boundary that is [[Embedding#Differential topology|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 <math> D \cap S = \alpha </math> and <math> D \cap \partial M = \beta </math> are arcs in <math> \partial D </math>, with <math> \alpha \cup \beta = \partial D </math>, <math> \alpha \cap \beta = \partial \alpha = \partial \beta </math>, and <math> \alpha </math> is an essential arc in ''S'' (<math> \alpha </math> does not cobound a disk in ''S'' with another arc in <math> \partial S </math>).

The surface ''S'' is said to be '''boundary-compressible''' if either ''S'' is a disk that cobounds a ball with a disk in <math> \partial M</math> 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 <math> \partial S </math>).  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 <math> \partial M</math> 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 <math> \partial V </math>, 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 <math> \partial V </math> 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&ndash;674.  {{MR|1192734}}.

[[Category:Manifolds]]

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