In
low-dimensional topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot the ...
, a boundary-incompressible surface is a two-dimensional surface within a three-dimensional
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
whose topology cannot be made simpler by a certain type of operation known as boundary compression.
Suppose ''M'' is a
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds ...
with boundary. Suppose also that ''S'' is a
compact surface
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example ...
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
and
are arcs in
, with
,
, and
is an essential arc in ''S'' (
does not cobound a disk in ''S'' with another arc in
).
The surface ''S'' is said to be boundary-compressible if either ''S'' is a disk that cobounds a ball with a disk in
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
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example ...
(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
). 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
or there exists a boundary-compressing disk for ''S'' in ''M''. Otherwise, ''S'' is boundary-incompressible.
For instance, if ''K'' is a
trefoil knot
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest kn ...
embedded in the boundary of a solid torus ''V'' and ''S'' is the closure of a small annular neighborhood of ''K'' in
, 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
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 In mathematics, an incompressible surface is a surface properly embedded in a 3-manifold, which, in intuitive terms, is a "nontrivial" surface that cannot be simplified. In non-mathematical terms, the surface of a suitcase is compressible, because ...
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. {{MR, 1192734.
Manifolds