In mathematics, an incompressible surface is a

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...

properly embedded in 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 ...

, 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."An Introduction to Knot Theory", W. B. Raymond Lickorish, p. 38, Springer, 1997, Incompressible surfaces are used for

decomposition Decomposition or rot is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. The process is a part of the nutrient cycle and is ...

Haken manifold 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 wh ...

s, in normal surface theory, and in the study of the

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

s of 3-manifolds.

Formal definition

Compressing disk in an incompressible surface

Let ''S'' be 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 ...

properly embedded in a smooth or PL 3-manifold ''M''. A compressing disk ''D'' 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 A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...

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 Three-ball (or "3-ball", colloquially) is a folk game of pool played with any three standard pool and . The game is frequently gambled upon. The goal is to () the three object balls in as few shots as possible.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 In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle. The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the ...

of ''D'' whose closure is an embedding of ''D'' × 1,1with ''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 \)

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 In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...

(the number of connected components) and ''χ''(''S'') is the

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...

. 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

\iota: S \rightarrow M

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 :

\iota_\star: \pi_1(S) \rightarrow \pi_1(M)

s is

injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...

. In general, every ''π''₁-injective surface is incompressible, but the reverse implication is not always true. For instance, the

Lens space A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions. In the 3-manifold case, a lens space can be visualiz ...

''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

Seifert surface In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link. Such surfaces can be used to study the properties of the associated knot or link. For exampl ...

''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

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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...

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 Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...

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 In mathematics ( differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition ...

of the knot complement, which can be certified with a taut

sutured manifold hierarchy Suture, literally meaning "seam", may refer to: Arts, entertainment, and media * ''Suture'' (album), a 2000 album by American Industrial rock band Chemlab * ''Suture'' (film), a 1993 film directed by Scott McGehee and David Siegel * Suture (band ...

. Given an incompressible Seifert surface ''S'' for a knot ''K'', then the

of ''S''³ − ''N''(''K'') splits as an

HNN extension In mathematics, the HNN extension is an important construction of combinatorial group theory. Introduced in a 1949 paper ''Embedding Theorems for Groups'' by Graham Higman, Bernhard Neumann, and Hanna Neumann, it embeds a given group ''G'' into ano ...

over ''π''₁(''S''), which is a

free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''− ...

. 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. {{DEFAULTSORT:Incompressible Surface 3-manifolds

Formal definition

Compression

Algebraically incompressible surfaces

Seifert surfaces

See also

References