Incompressible surface

Last updated January 21, 2022

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.^[1]

Formal definition

Let S be a compact surface properly embedded in a smooth or PL 3-manifold M. A compressing diskD is a disk embedded in M such that

D\cap S=\partial D

and the intersection is transverse. If the curve ∂D does not bound a disk inside of S, then D is called a nontrivial compressing disk. If S has a nontrivial compressing disk, then we call S a compressible surface in M.

If S 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.

Compression

Given a compressible surface S with a compressing disk D that we may assume lies in the interior of M and intersects S transversely, one may perform embedded 1-surgery on S to get a surface that is obtained by compressingSalongD. There is a tubular neighborhood of D whose closure is an embedding of D× [-1,1] with D× 0 being identified with D and with

(D\times [-1,1])\cap S=\partial D\times [-1,1].

Then

(S-\partial D\times (-1,1))\cup (D\times \{-1,1\})

is a new properly embedded surface obtained by compressing S along D.

A non-negative complexity measure on compact surfaces without 2-sphere components is b₀(S) −χ(S), where b₀(S) is the zeroth Betti number (the number of connected components) and χ(S) is the Euler characteristic. When compressing a compressible surface along a nontrivial compressing disk, the Euler characteristic increases by two, while b₀ 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 S be compressible. If D were to bound a disk inside S (which is always the case if S is incompressible, for example), then compressing S along D would result in a disjoint union of a sphere and a surface homeomorphic to S. The resulting surface with the sphere deleted might or might not be isotopic to S, and it will be if S is incompressible and M is irreducible.

Algebraically incompressible surfaces

There is also an algebraic version of incompressibility. Suppose $\iota :S\rightarrow M$ is a proper embedding of a compact surface in a 3-manifold. Then S is π₁-injective (or algebraically incompressible) if the induced map

\iota _{\star }:\pi _{1}(S)\rightarrow \pi _{1}(M)

on fundamental groups is injective.

In general, every π₁-injective surface is incompressible, but the reverse implication is not always true. For instance, the Lens space L(4,1) contains an incompressible Klein bottle that is not π₁-injective.

However, if S is two-sided, the loop theorem implies Kneser's lemma, that if S is incompressible, then it is π₁-injective.

Seifert surfaces

A Seifert surface S for an oriented link L is an oriented surface whose boundary is L with the same induced orientation. If S is not π₁ injective in S³−N(L), where N(L) is a tubular neighborhood of L, then the loop theorem gives a compressing disk that one may use to compress S 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 π₁ alone cannot certify the genus of a link. 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 S for a knot K, then the fundamental group of S³−N(K) splits as an HNN extension over π₁(S), which is a free group. The two maps from π₁(S) into π₁(S³−N(S)) given by pushing loops off the surface to the positive or negative side of N(S) are both injections.

Related Research Articles

In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.

In mathematics, genus has a few different, but closely related, meanings. Probably the fastest, easiest and most intuitive way to introduce genus is that it is the number of "holes" of a surface. A sphere has genus 0, and a torus has genus 1.

Torus Doughnut-shaped surface of revolution

In geometry, a torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.

In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries . In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.

In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface.

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 theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

In the mathematical field of geometric topology, a Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.

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 look like our universe does to a small enough observer. This is made more precise in the definition below.

Allen Edward Hatcher is an American topologist.

In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries.

In mathematics, a Seifert surface is an orientable surface whose boundary is a given knot or link.

In mathematics, the knot complement of a tame knot K is the space where the knot isn't. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that K is a knot in a three-manifold M. Let N be a tubular neighborhood of K; so N is a solid torus. The knot complement is then the complement of N,

A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a $-bundle over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for all compact oriented manifolds in 6 of the 8 Thurston geometries of the geometrization conjecture.$

In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem.

In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold.

In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential is everywhere injective. Explicitly, f : M → N is an immersion if

In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include composite knots, cable knots, and Whitehead doubles. A satellite link is one that orbits a companion knot K in the sense that it lies inside a regular neighborhood of the companion.

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.

References

↑ "An Introduction to Knot Theory", W. B. Raymond Lickorish, p. 38, Springer, 1997, ISBN 0-387-98254-X

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.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] "An Introduction to Knot Theory", W. B. Raymond Lickorish, p. 38, Springer, 1997, ISBN 0-387-98254-X

[1]