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 above a set is a system of non-empty subsets such that:
for each ; if and then .
The elements of are called simplices, the elements of are called vertices. A simplex with vertices has dimension by definition. The dimension of an abstract simplicial complex is defined as .
Abstract simplicial complexes can be realized as geometrical objects by associating each abstract simplex with a geometric simplex, defined below.
Geometric simplices Let be affinely independent points in ; i.e. the vectors are linearly independent. The set is said to be the simplex spanned by . It has dimension by definition. The points are called the vertices of , the simplices spanned by of the vertices are called faces, and the boundary is defined to be the union of the faces.
The -dimensional standard-simplex is the simplex spanned by the unit vectors
Geometric simplicial complexes A geometric simplicial complex is a collection of geometric simplices such that
If is a simplex in , then all its faces are in . If are two distinct simplices in , their interiors are disjoint. The union of all the simplices in gives the set of points of , denoted This set is endowed with a topology by choosing the closed sets to be is closed for all . Note that, in general, this topology is not the same as the subspace topology that inherits from . The topologies do coincide in the case that each point in the complex lies only in finitely many simplices.
Each geometric complex can be associated with an abstract complex by choosing as a ground set the set of vertices that appear in any simplex of and as system of subsets the subsets of which correspond to vertex sets of simplices in .
A natural question is if vice versa, any abstract simplicial complex corresponds to a geometric complex. The answer is positive when the abstract complex is finite, i.e. the set is finite. If its cardinality is , take affinely independent points in 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:
Let be an abstract simplicial complex above a set . Choose a union of simplices , but each in of dimension sufficiently large, such that the geometric simplex is of dimension if the abstract geometric simplex has dimension . If , can be identified with a face of and the resulting topological space is the gluing Effectuating the gluing for each inclusion, one ends up with the desired topological space. This space is actually unique up to homeomorphism for each choice of so it makes sense to talk about the geometric realization of
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 of every simplex in the complex.
The simplicial complex which consists of all simplices of dimension is called the -th skeleton of .
A natural neighbourhood of a vertex in a simplicial complex is considered to be given by the star of a simplex, whose boundary is the link .
Simplicial maps The maps considered in this category are simplicial maps: Let , be abstract simplicial complexes above sets , . A simplicial map is a function which maps each simplex in onto a simplex in . By affine-linear extension on the simplices, induces a map between the geometric realizations of the complexes.
Examples Let and let . The associated geometric complex is a star with center . Let and let . Its geometric realization is a tetrahedron. Let as above and let . The geometric simplicial complex is the boundary of a tetrahedron .
Definition A triangulation of a topological space is a homeomorphism where 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. Let be as in the examples seen above. The closed unit ball is homeomorphic to a tetrahedron so it admits a triangulation, namely the homeomorphism . Restricting to yields a homeomorphism . 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 Let be a finite simplicial complex. The -th Betti-number is defined to be the rank of the -th simplicial homology group of the spaces. These numbers encode geometric properties of the spaces: The Betti-number for instance represents the number of connected components. For a triangulated, closed orientable surfaces , holds where 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 ( German: Main assumption). Let be a simplicial complex. A complex is said to be a subdivision of iff:
every simplex of is contained in a simplex of and every simplex of is a finite union of simplices in .
Those conditions ensure that subdivisions does not change the simplicial complex as a set or as a topological space. A map between simplicial complexes is said to be piecewise linear if there is a refinement of such that is piecewise linear on each simplex of . 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) 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 subdivisions are easy to construct for simple spaces, for instance for low dimensional manifolds. Indeed the assumption was proven for manifolds of dimension 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 of CW-complexes: If this characteristic will be a topological invariant but if 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 are 3-manifolds, constructed as quotient spaces of the 3-sphere: Let be natural numbers, such that are coprime. The lens space is defined to be the orbit space of the free group action
.
For different tuples , 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.