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 we could cut the handle and shrink it into the surface. But a Conway sphere (a sphere with four holes) is incompressible, because there are essential parts of a knot or link both inside and out, so there is no way to move the entire knot or link to one side of the punctured sphere. The mathematical definition is as follows. There are two cases to consider. A sphere is incompressible if both inside and outside the sphere there are some obstructions that prevent the sphere from shrinking to a point and also prevent the sphere from expanding to encompass all of space. A surface other than a sphere is incompressible if any disk with its boundary on the surface spans a disk in the surface.<ref>"An Introduction to Knot Theory", W. B. Raymond Lickorish, p. 38, Springer, 1997, Template:ISBN</ref>
Incompressible surfaces are used for decomposition of Haken manifolds, in normal surface theory, and in the study of the fundamental groups of 3-manifolds.
Formal definitionEdit
Let Template:Math be a compact surface properly embedded in a smooth or PL 3-manifold Template:Math. A compressing disk Template:Math is a disk embedded in Template:Math such that
- <math>D \cap S = \partial D</math>
and the intersection is transverse. If the curve Template:Math does not bound a disk inside of Template:Math, then Template:Math is called a nontrivial compressing disk. If Template:Math has a nontrivial compressing disk, then we call Template:Math a compressible surface in Template:Math.
If Template:Math is neither the 2-sphere nor a compressible surface, then we call the surface (geometrically) incompressible.
Note that 2-spheres are excluded since they have no nontrivial compressing disks by the Jordan-Schoenflies theorem, and 3-manifolds have abundant embedded 2-spheres. Sometimes one alters the definition so that an incompressible sphere is a 2-sphere embedded in a 3-manifold that does not bound an embedded 3-ball. Such spheres arise exactly when a 3-manifold is not irreducible. Since this notion of incompressibility for a sphere is quite different from the above definition for surfaces, often an incompressible sphere is instead referred to as an essential sphere or a reducing sphere.
CompressionEdit
Given a compressible surface Template:Math with a compressing disk Template:Math that we may assume lies in the interior of Template:Math and intersects Template:Math transversely, one may perform embedded 1-surgery on Template:Math to get a surface that is obtained by compressing Template:Math along Template:Math. There is a tubular neighborhood of Template:Math whose closure is an embedding of D × [-1,1] with D × 0 being identified with D and with
- <math>(D\times [-1,1])\cap S=\partial D\times [-1,1].</math>
Then
- <math>(S-\partial D\times(-1,1))\cup (D\times \{-1,1\})</math>
is a new properly embedded surface obtained by compressing Template:Math along Template:Math.
A non-negative complexity measure on compact surfaces without 2-sphere components is Template:Math, where Template:Math is the zeroth Betti number (the number of connected components) and Template:Math is the Euler characteristic of Template:Math. When compressing a compressible surface along a nontrivial compressing disk, the Euler characteristic increases by two, while Template:Math might remain the same or increase by 1. Thus, every properly embedded compact surface without 2-sphere components is related to an incompressible surface through a sequence of compressions.
Sometimes we drop the condition that Template:Math be compressible. If Template:Math were to bound a disk inside Template:Math (which is always the case if Template:Math is incompressible, for example), then compressing Template:Math along Template:Math would result in a disjoint union of a sphere and a surface homeomorphic to Template:Math. The resulting surface with the sphere deleted might or might not be isotopic to Template:Math, and it will be if Template:Math is incompressible and Template:Math is irreducible.
Algebraically incompressible surfacesEdit
There is also an algebraic version of incompressibility. Suppose <math>\iota: S \rightarrow M</math> is a proper embedding of a compact surface in a 3-manifold. Then Template:Math is Template:Math-injective (or algebraically incompressible) if the induced map
- <math>\iota_\star: \pi_1(S) \rightarrow \pi_1(M)</math>
on fundamental groups is injective.
In general, every Template:Math-injective surface is incompressible, but the reverse implication is not always true. For instance, the Lens space Template:Math contains an incompressible Klein bottle that is not Template:Math-injective.
However, if Template:Math is two-sided, the loop theorem implies Kneser's lemma, that if Template:Math is incompressible, then it is Template:Math-injective.
Seifert surfacesEdit
A Seifert surface Template:Math for an oriented link Template:Math is an oriented surface whose boundary is Template:Math with the same induced orientation. If Template:Math is not Template:Math-injective in Template:Math, where Template:Math is a tubular neighborhood of L, then the loop theorem gives a compressing disk that one may use to compress Template:Math along, providing another Seifert surface of reduced complexity. Hence, there are incompressible Seifert surfaces.
Every Seifert surface of a link is related to one another through compressions in the sense that the equivalence relation generated by compression has one equivalence class. The inverse of a compression is sometimes called embedded arc surgery (an embedded 0-surgery).
The genus of a link is the minimal genus of all Seifert surfaces of a link. A Seifert surface of minimal genus is incompressible. However, it is not in general the case that an incompressible Seifert surface is of minimal genus, so Template:Math alone cannot certify the genus of a link. David Gabai proved in particular that a genus-minimizing Seifert surface is a leaf of some taut, transversely oriented foliation of the knot complement, which can be certified with a taut sutured manifold hierarchy.
Given an incompressible Seifert surface Template:Math' for a knot Template:Math, then the fundamental group of Template:Math splits as an HNN extension over Template:Math, which is a free group. The two maps from Template:Math into Template:Math given by pushing loops off the surface to the positive or negative side of Template:Math are both injections.
See alsoEdit
ReferencesEdit
- W. Jaco, Lectures on Three-Manifold Topology, volume 43 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, R.I., 1980.
- http://users.monash.edu/~jpurcell/book/08Essential.pdf
- https://homepages.warwick.ac.uk/~masgar/Articles/Lackenby/thrmans3.pdf
- D. Gabai, "Foliations and the topology of 3-manifolds." Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 1, 77–80.