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 by pinching off tubes. They are useful for decomposition of Haken manifolds, normal surface theory, and studying fundamental groups of 3-manifolds.

Formal definition

For an incompressible surface S, every compressing disk D bounds a disk D′ in S. Together, D and D′ form a 2-sphere. This sphere need not bound a ball unless M is irreducible.

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

${\displaystyle D\cap S=\partial D}$

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

Compressing a surface S along a disk D results in a surface S', which is obtained by removing the annulus boundary of N(D) from S and adding in the two disk boundaries of N(D).

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 compressing S along D. 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

${\displaystyle (D\times [-1,1])\cap S=\partial D\times [-1,1].}$

Then

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

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 ${\displaystyle \iota :S\rightarrow M}$ is a proper embedding of a compact surface in a 3-manifold. Then S is π₁-injective (or algebraically incompressible) if the induced map

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

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.

See also

• Haken manifold
• Virtually Haken conjecture
• Thurston norm
• Boundary-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.
• 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.