In mathematics, triangulation describes the replacement of topological spaces with simplicial complexes by the choice of an appropriate homeomorphism. A space that admits such a homeomorphism is called a triangulable space. Triangulations can also be used to define a piecewise linear structure for a space, if one exists. Triangulation has various applications both in and outside of mathematics, for instance in algebraic topology, in complex analysis, and in modeling.

Motivation

On the one hand, it is sometimes useful to forget about superfluous information of topological spaces: The replacement of the original spaces with simplicial complexes may help to recognize crucial properties and to gain a better understanding of the considered object.

On the other hand, simplicial complexes are objects of combinatorial character and therefore one can assign them quantities arising from their combinatorial pattern, for instance, the Euler characteristic. Triangulation allows one to assign such quantities to topological spaces.

Investigations concerning the existence and uniqueness of triangulations established a new branch in topology, namely piecewise linear topology (or PL topology). Its main purpose is to study the topological properties of simplicial complexes and their generalizations, cell-complexes.

Simplicial complexes

Abstract simplicial complexes

An abstract simplicial complex of non-empty subsets such that:

• ;

• .

are called simplices, are called vertices. vertices has dimension.

Abstract simplicial complexes can be realized as geometrical objects by associating each abstract simplex with a geometric simplex, defined below.

Geometric simplices

are linearly independent is said to be the . It has dimension is defined to be the union of the faces.

-dimensional standard-simplex is the simplex spanned by the unit vectors

Geometric simplicial complexes

is a collection of geometric simplices such that

• .

• , their interiors are disjoint.

is endowed with a topology by choosing the closed sets is closed for all. The topologies do coincide in the case that each point in the complex lies only in finitely many simplices.

.

and construct the geometric simplices as prescribed by the abstract simplicial complex. The idea is that you have enough space to build all the simplices.

In general, the geometric construction as mentioned here is not flexible enough: consider for instance an abstract simplicial complex of infinite dimension. However, the following more abstract construction provides a topological space for any kind of abstract simplicial complex:

and the resulting topological space is the gluing so it makes sense to talk about the

As in the previous construction, by the topology induced by gluing, the closed sets in this space are the subsets that are closed in the subspace topology in the complex.

-th skeleton.

A natural neighbourhood is considered to be given by the star.

Simplicial maps

induces a map between the geometric realizations of the complexes.

Examples

• .

• is a tetrahedron.

• . The geometric simplicial complex is the boundary.

Definition

is a homeomorphism is a simplicial complex. Topological spaces do not necessarily admit a triangulation and if they do, it is not necessarily unique.

Examples

• Simplicial complexes can be triangulated by identity.

• .

• The torus admits a triangulation. To see this, consider the torus as a square where the parallel faces are glued together. A triangulation of the square that respects the gluings, like that shown below, also defines a triangulation of the torus.

• The projective plane admits a triangulation (see CW-complexes)

• One can show that differentiable manifolds admit triangulations.

Invariants

Triangulations of spaces allow assigning combinatorial invariants rising from their dedicated simplicial complexes to spaces. These are characteristics that equal for complexes that are isomorphic via a simplicial map and thus have the same combinatorial pattern.

This data might be useful to classify topological spaces up to homeomorphism but only given that the characteristics are also topological invariants, meaning, they do not depend on the chosen triangulation. For the data listed here, this is the case. For details and the link to singular homology, see topological invariance.

Homology

Via triangulation, one can assign a chain complex to topological spaces that arise from its simplicial complex and compute its simplicial homology. Compact spaces always admit finite triangulations and therefore their homology groups are finitely generated and only finitely many of them do not vanish. Other data as Betti-numbers or Euler characteristic can be derived from homology.

Betti-numbers and Euler-characteristics

is defined to be the rank for instance represents the number of connected components. For a triangulated, closed orientable surfaces denotes the genus of the surface: Therefore its first Betti-number represents the doubled number of handles of the surface.

With the comments above, for compact spaces all Betti-numbers are finite and almost all are zero. Therefore, one can form their alternating sum

which is called the Euler characteristic of the complex, a catchy topological invariant.

Topological invariance

To use these invariants for the classification of topological spaces up to homeomorphism one needs invariance of the characteristics regarding homeomorphism.

A famous approach to the question was at the beginning of the 20th century the attempt to show that any two triangulations of the same topological space admit a common subdivision. This assumption is known as Hauptvermutung ( iff:

• and

• .

. Two complexes that correspond to another via piecewise linear bijection are said to be combinatorial isomorphic. In particular, two complexes that have a common refinement are combinatorially equivalent. Homology groups are invariant to combinatorial equivalence and therefore the Hauptvermutung would give the topological invariance of simplicial homology groups. In 1918, Alexander introduced the concept of singular homology. Henceforth, most of the invariants arising from triangulation were replaced by invariants arising from singular homology. For those new invariants, it can be shown that they were invariant regarding homeomorphism and even regarding homotopy equivalence. Furthermore it was shown that singular and simplicial homology groups coincide. This workaround has shown the invariance of the data to homeomorphism. Hauptvermutung lost in importance but it was initial for a new branch in topology: The piecewise linear topology (short PL-topology).

Hauptvermutung

The Hauptvermutung (German for main conjecture and for differentiable manifolds but it was disproved in general: An important tool to show that triangulations do not admit a common subdivision, that is, their underlying complexes are not combinatorially isomorphic is the combinatorial invariant of Reidemeister torsion.

Reidemeister torsion

To disprove the Hauptvermutung it is helpful to use combinatorial invariants which are not topological invariants. A famous example is Reidemeister torsion. It can be assigned to a tuple in general not. An approach to Hauptvermutung was to find homeomorphic spaces with different values of Reidemeister torsion. This invariant was used initially to classify lens-spaces and first counterexamples to the Hauptvermutung were built based on lens-spaces:

Classification of lens spaces

In its original formulation, lens spaces is defined to be the orbit space of the free group action

1. .

, lens spaces will be homotopy equivalent but not homeomorphic. Therefore they can't be distinguished with the help of classical invariants as the fundamental group but by the use of Reidemeister torsion.

. This is the case if and only if two lens spaces are simple homotopy equivalent are homeomorphic. The resulting spaces will disprove the Hauptvermutung.

Existence of triangulation

are always triangulable arbitrary but greater than three. Further, differentiable manifolds always admit triangulations.

Piecewise linear structures PL-Structures

Manifolds are an important class of spaces. It is natural to require them not only to be triangulable but moreover to admit a piecewise linear atlas, a PL-structure:

is said to be a piecewise linear (PL) manifold of dimension and the triangulation together with the PL-atlas is said to be a PL-structure on.

An important lemma is the following:

be a topological space. Then the following statements are equivalent:

1. -dimensional manifold and admits a PL-structure.

2. sphere.

3. sphere.

The equivalence of the second and the third statement is because that the link of a vertex is independent of the chosen triangulation up to combinatorial isomorphism.. Counterexamples for the triangulation conjecture are counterexamples for the conjecture of the existence of PL-structure of course.

sphere.

which is piecewise linear with respect to both PL-structures? The assumption is similar to the Hauptvermutung and indeed there are spaces which have different PL-structures which are not equivalent. Triangulation of PL-equivalent spaces can be transformed into one another via Pachner moves:

Pachner Moves

the Join. This replacement is called a Pachner move. The theorem of Pachner states that whenever two triangulated manifolds are PL-equivalent, there is a series of Pachner moves transforming both into another.

Cellular complexes CW-complexes

A similar but more flexible construction than simplicial complexes is the one of cellular complexes (or CW-complexes). Its construction is as follows:

is said to be -cell.

of topological spaces such that

• is a discrete set

• -cells.

Each simplicial complex is a CW-complex, the inverse is not true. The construction of CW-complexes can be used to define cellular homology and one can show that cellular homology and simplicial homology coincide.: Its construction as a CW-complex needs three cells, whereas its simplicial complex consists of 54 simplices.

Other applications

Classification of manifolds

By triangulating 1-dimensional manifolds, one can show that they are always homeomorphic to disjoint copies of the real line and the unit sphere. The classification of closed surfaces, i.e. compact 2-manifolds, can also be proven by using triangulations. This is done by showing any such surface can be triangulated and then using the triangulation to construct a fundamental polygon for the surface.

Maps on simplicial complexes

Giving spaces simplicial structures can help to understand continuous maps defined on the spaces. The maps can often be assumed to be simplicial maps via the simplicial approximation theorem:

Simplicial approximation

induces a map between the geometric realizations of the complexes. Each point in a geometric complex lies in the inner of exactly one simplex, its support. Consider now a continuous. is said to be a simplicial approximation by defining it on each simplex; there it always exists, because simplices are contractible.

The simplicial approximation theorem guarantees for every continuous function by its iterated barycentric subdivision. The theorem plays an important role for certain statements in algebraic topology in order to reduce the behavior of continuous maps on those of simplicial maps, for instance in Lefschetz's fixed-point theorem.

Lefschetz's fixed-point theorem

The Lefschetz number induces homomorphisms can be determined and their alternating sum

is called the Lefschetz number has a fixpoint.

Formula of Riemann-Hurwitz

The Riemann-Hurwitz formula allows to determine the genus of a compact, connected Riemann surface is

denotes the degree of the map. The sum is well defined as it counts only the ramifying points of the function.

The background of this formula is that holomorphic functions on Riemann surfaces are ramified coverings. The formula can be found by examining the image of the simplicial structure near to ramifiying points.

Citations

See also

• Triangulation (geometry)

• Triangle mesh

Literature

• Allen Hatcher: Algebraic Topology, Cambridge University Press, Cambridge/New York/Melbourne 2006, ISBN 0-521-79160-X

• James R. Munkres: . Band 1984. Addison Wesley, Menlo Park, California 1984, ISBN 0-201-04586-9

• Marshall M. Cohen: A course in Simple-Homotopy Theory. In: Graduate Texts in Mathematics. 1973, ISSN 0072-5285, doi:10.1007/978-1-4684-9372-6.