# 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

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

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 **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

Then

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 *b*_{0}(*S*) − *χ*(*S*), where *b*_{0}(*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 *b*_{0} 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 is a proper embedding of a compact surface in a 3-manifold. Then *S* is ** π_{1}-injective** (or

**algebraically incompressible**) if the induced map

on fundamental groups is injective.

In general, every *π*_{1}-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 *π*_{1}-injective.

However, if *S* is two-sided, the loop theorem implies Kneser's lemma, that if *S* is incompressible, then it is *π*_{1}-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 *π*_{1} injective in *S*^{3} − *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 *π*_{1} 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 *S*^{3} − *N*(*K*) splits as an HNN extension over *π*_{1}(*S*), which is a free group. The two maps from *π*_{1}(*S*) into *π*_{1}(*S*^{3} − *N*(*S*)) given by pushing loops off the surface to the positive or negative side of *N*(*S*) are both injections.

## 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.