Triangulation (topology)

thumb|250px|A triangulated torus thumb|250px|Another triangulation of the torusthumb|250px|A triangulated dolphin shape

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 now 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 above a set <math>V</math> is a system <math>\mathcal{T} \subset \mathcal{P}(V)</math> of non-empty subsets such that:

<math>\{v_0\} \in \mathcal{T}</math> for each <math>v_0\in V</math>;
if <math>E \in \mathcal{T}</math> and <math>\emptyset \neq F\subset E,</math> then <math>F \in \mathcal{T}</math>.

The elements of <math>\mathcal{T}</math> are called simplices, the elements of <math>V</math> are called vertices. A simplex with <math>n+1</math> vertices has dimension <math>n</math> by definition. The dimension of an abstract simplicial complex is defined as <math>\text{dim}(\mathcal{T})= \text{sup}\;\{\text{dim}(F):F \in \mathcal{T}\} \in \mathbb{N}\cup \infty</math>.

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

thumb|200px|Geometric simplices in dimension 1, 2 and 3

Geometric simplices

Let <math>p_0,...p_n</math> be <math>n+1

</math> affinely independent points in <math>\mathbb{R}^n</math>; i.e. the vectors <math>(p_1-p_0), (p_2-p_0),\dots (p_n-p_0)</math> are linearly independent. The set <math display=inline>\Delta = \{ \sum_{i=0}^n t_ip_i \,|\, \text{each}\, t_i \in [0,1] \,\text{and}\, \sum_{i=0}^n t_i = 1\}</math> is said to be the simplex spanned by <math>p_0,...p_n</math>. It has dimension <math>n</math> by definition. The points <math>p_0,...p_n</math> are called the vertices of <math> \Delta </math>, the simplices spanned by <math>n</math> of the <math>n+1</math> vertices are called faces, and the boundary <math>\partial \Delta</math> is defined to be the union of the faces.

The <math>n</math>-dimensional standard-simplex is the simplex spanned by the unit vectors <math> e_0,...e_n</math>

Geometric simplicial complexes

A geometric simplicial complex <math>\mathcal{S}\subseteq\mathcal{P}(\mathbb{R}^n)</math> is a collection of geometric simplices such that

If <math>S</math> is a simplex in <math>\mathcal{S}</math>, then all its faces are in <math>\mathcal{S}</math>.
If <math>S, T</math> are two distinct simplices in <math>\mathcal{S}</math>, their interiors are disjoint.

The union of all the simplices in <math>\mathcal{S}</math> gives the set of points of <math>\mathcal{S}</math>, denoted <math display=inline>|\mathcal{S}|=\bigcup_{S \in \mathcal{S S.</math> This set <math>|\mathcal{S}|</math> is endowed with a topology by choosing the closed sets to be <math>\{A \subseteq |\mathcal{S}| \;\mid\; A \cap \Delta </math> is closed for all <math> \Delta \in \mathcal{S}\}</math>. Note that, in general, this topology is not the same as the subspace topology that <math>|\mathcal{S}|</math> inherits from <math>\mathbb{R}^n</math>. The topologies do coincide in the case that each point in the complex lies only in finitely many simplices.

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

Let <math>|\mathcal{S}|</math> be a finite simplicial complex. The <math>n</math>-th Betti-number <math>b_n(\mathcal{S})</math> is defined to be the rank of the <math>n</math>-th simplicial homology group of the spaces. These numbers encode geometric properties of the spaces: The Betti-number <math>b_0(\mathcal{S})</math> for instance represents the number of connected components. For a triangulated, closed orientable surfaces <math>F</math>, <math>b_1(F)= 2g</math> holds where <math>g</math> 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

: <math>\sum_{k=0}^{\infty} (-1)^{k}b_k(\mathcal{S})</math>

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 ( German: Main assumption). Let <math>|\mathcal{L}|\subset \mathbb{R}^N </math> be a simplicial complex. A complex <math> |\mathcal{L'}|\subset \mathbb{R}^N</math> is said to be a subdivision of <math>\mathcal{L}</math> iff:

every simplex of <math>\mathcal{L'} </math> is contained in a simplex of <math>\mathcal{L} </math> and
every simplex of <math>\mathcal{L} </math> is a finite union of simplices in <math>\mathcal{L'} </math> . Furthermore it was shown that singular and simplicial homology groups coincide.

Hauptvermutung

The Hauptvermutung (German for main conjecture) states that two triangulations always admit a common subdivision. Originally, its purpose was to prove invariance of combinatorial invariants regarding homeomorphisms. The assumption that such subdivisions exist in general is intuitive, as subdivision are easy to construct for simple spaces, for instance for low dimensional manifolds. Indeed the assumption was proven for manifolds of dimension <math>\leq 3</math> 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 <math>(K,L)</math> of CW-complexes: If <math>L = \emptyset</math> this characteristic will be a topological invariant but if <math>L \neq \emptyset</math> 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: This is the case if and only if two lens spaces are simple homotopy equivalent. The fact can be used to construct counterexamples for the Hauptvermutung as follows. Suppose there are spaces <math>L'_1, L'_2</math> derived from non-homeomorphic lens spaces <math>L(p,q_1), L(p,q_2)</math> having different Reidemeister torsion. Suppose further that the modification into <math>L'_1, L'_2</math> does not affect Reidemeister torsion but such that after modification <math>L'_1</math> and <math>L'_2</math> are homeomorphic. The resulting spaces will disprove the Hauptvermutung.

Existence of triangulation

Besides the question of concrete triangulations for computational issues, there are statements about spaces that are easier to prove given that they are simplicial complexes. Especially manifolds are of interest. Topological manifolds of dimension <math>\leq 3</math> are always triangulable Further, differentiable manifolds always admit triangulations. One can show that differentiable manifolds admit a PL-structure as well as manifolds of dimension <math>\leq 3</math>. Counterexamples for the triangulation conjecture are counterexamples for the conjecture of the existence of PL-structure of course.

Moreover, there are examples for triangulated spaces which do not admit a PL-structure. Consider an <math>n-2</math>-dimensional PL-homology-sphere <math>X</math>. The double suspension <math>S^2X</math> is a topological <math>n</math>-sphere. Choosing a triangulation <math>t: |\mathcal{S}| \rightarrow S^2 X</math> obtained via the suspension operation on triangulations the resulting simplicial complex is not a PL-manifold, because there is a vertex <math>v</math> such that <math>link(v)</math> is not a <math>n-1</math> sphere.

A question arising with the definition is if PL-structures are always unique: Given two PL-structures for the same space <math>Y</math>, is there a there a homeomorphism <math>F:Y\rightarrow Y</math> 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

thumb|241x241px|One Pachner-move replaces two tetrahedra by three tetrahedra

Pachner moves are a way to manipulate triangulations: Let <math>\mathcal{S} </math> be a simplicial complex. For two simplices <math>K, L,</math> the Join <math display=inline>K*L = \{ (1-t)k+tl\;|\; k \in K, l \in L, t \in [0,1]\}</math> is the set of points that lie on straights between points in <math>K</math> and in <math>L</math>. Choose <math>S \in \mathcal{S}</math> such that <math>lk(S)= \partial K</math> for any <math>K</math> lying not in <math>\mathcal{S}</math>. A new complex <math>\mathcal{S'}</math>, can be obtained by replacing <math>S * \partial K</math> by <math>\partial S * K</math>. 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

thumb|The real projective plane as a simplicial complex and as CW-complex. As CW-complex it can be obtained by gluing first <math>\mathbb{D}^0</math> and <math>\mathbb{D}^1</math> to get the 1-sphere and then attaching the disc <math>\mathbb{D}^2</math> by the map <math>g: \mathbb{S}^1 \rightarrow \mathbb{S}^1, e^{ix} \mapsto e^{2ix}</math>.

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

An <math>n</math>-cell is the closed <math>n</math>-dimensional unit-ball <math>B_n= [0,1]^n</math>, an open <math>n</math>-cell is its inner <math>B_n= [0,1]^n\setminus \mathbb{S}^{n-1}</math>. Let <math>X</math> be a topological space, let <math>f: \mathbb{S}^{n-1}\rightarrow X</math> be a continuous map. The gluing <math>X \cup_{f}B_n</math> is said to be obtained by gluing on an <math>n</math>-cell.

A cell complex is a union <math>X=\cup_{n\geq 0} X_n</math> of topological spaces such that

<math>X_0</math> is a discrete set
each <math>X_n</math> is obtained from <math>X_{n-1}</math> by gluing on a family of <math>n</math>-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. For computational issues, it is sometimes easier to assume spaces to be CW-complexes and determine their homology via cellular decomposition, an example is the projective plane <math>\mathbb{P}^2</math>: 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 <math>\mathbb{S}^1</math>. 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

Let <math>\mathcal{K}</math>, <math>\mathcal{L}</math> be abstract simplicial complexes above sets <math>V_K</math>, <math>V_L</math>. A simplicial map is a function <math>f:V_K \rightarrow V_L</math> which maps each simplex in <math>\mathcal{K}</math> onto a simplex in <math>\mathcal{L}</math>. By affin-linear extension on the simplices, <math>f </math> 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 map <math>f:\mathcal{K}\rightarrow \mathcal{L} </math>. A simplicial map <math>g:\mathcal{K}\rightarrow \mathcal{L} </math> is said to be a simplicial approximation of <math>f</math> if and only if each <math>x \in \mathcal{K}</math> is mapped by <math>g</math> onto the support of <math>f(x)</math> in <math>\mathcal{L}</math>. If such an approximation exists, one can construct a homotopy <math>H</math> transforming <math>f </math> into <math>g</math> by defining it on each simplex; there it always exists, because simplices are contractible.

The simplicial approximation theorem guarantees for every continuous function <math>f:V_K \rightarrow V_L</math> the existence of a simplicial approximation at least after refinement of <math>\mathcal{K}</math>, for instance by replacing <math>\mathcal{K}</math> by its iterated barycentric subdivision.

Formula of Riemann-Hurwitz

The Riemann-Hurwitz formula allows to determine the genus of a compact, connected Riemann surface <math>X </math> without using explicit triangulation. The proof needs the existence of triangulations for surfaces in an abstract sense: Let <math>F:X \rightarrow Y </math> be a non-constant holomorphic function on a surface with known genus. The relation between the genus <math>g </math> of the surfaces <math>X </math> and <math>Y </math> is

<math>2g(X)-2=\deg(F)(2g(Y)-2)+\sum_{x\in X}(\operatorname{ord}(F)-1)</math>

where <math>\deg(F)</math> 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

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.