piecewise-linear triangulation
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In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangulable. Triangulation has various uses in different branches of mathematics, for instance in algebraic topology, in complex analysis or 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 rising 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 the piecewise-linear-topology (short PL- topology). Its main purpose is topological properties of simplicial complexes and its generalization, cell-complexes.


Simplicial complexes


Abstract simplicial complexes

An abstract simplicial complex above a set V is a system \mathcal \subset \mathcal (V) of non-empty subsets such that: * \ \subset \mathcal for each \mathcal \subset \mathcal (V) * if E \in \mathcal and \emptyset \neq F\subset E \Rightarrow F \in \mathcal. The elements of \mathcal are called ''simplices,'' the elements of V are called ''vertices.'' A simplex with n+1 has ''dimension'' n by definition. The dimension of an abstract simplicial complex is defined as \text(\mathcal)= \text\;\ \in \mathbb\cup \infty. Abstract simplicial complexes can be thought of as geometrical objects too. This requires the term of geometric simplex.


Geometric simplices

Let p_0,...p_n be n+1 affinely independent points in \mathbb^n, i.e. the vectors (p_1-p_0), (p_2-p_0),\dots (p_n-p_0)are
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
. The set \Delta = \Bigl\ is said to be the ''simplex spanned by p_0,...p_n''. It has ''dimension'' n by definition. The points p_0,...p_n are called the vertices of \Delta , the simplices spanned by n of the n+1 vertices are called faces and the boundary \partial \Delta is defined to be the union of its faces. The n''-dimensional standard-simplex'' is the simplex spanned by the
unit vectors In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vec ...
e_0,...e_n


Geometric simplicial complexes

A geometric simplicial complex \mathcal\subset \mathbb^n is a union of geometric simplices such that * If S is a simplex in \mathcal, then all its faces are in \mathcal. * If S, T are two distinct simplices in \mathcal, their inners are disjoint. The set \mathcal can be realized as a topological space , \mathcal, by choosing the closed sets to be \Bigl\ . It should be mentioned, that in general, the simplicial complex won't provide the natural topology of \mathbb^n . In the case that each point in the complex lies only in finetly many simplices, both topologies coincide Each geometric complex can be associated with an abstract complex by choosing as a ground set V the set of vertices that appear in any simplex of \mathcal and as system of subsets the subsets of V which correspond to vertex sets of simplices in \mathcal. A natural question is if vice versa, any abstract simplicial complex corresponds to a geometric complex. In general, the geometric construction as mentioned here is not flexible enough: Consider for instance abstract simplicial complex of infinite dimension. However, the following more abstract construction provides a topological space for any kind of abstract simplicial complex: Let \mathcal be an abstract simplicial complex above a set V . Choose a union of simplices (\Delta_F)_, but each in \mathbb ^N of dimension sufficiently large, such that the geometric simplex \Delta_F is of dimension n if the abstract geometric simplex F has dimension n. If E\subset F, \Delta_E\subset \mathbb^Ncan be identified with a face of \Delta_F\subset\mathbb^M and the resulting topological space is the
gluing Adhesive, also known as glue, cement, mucilage, or paste, is any non-metallic substance applied to one or both surfaces of two separate items that binds them together and resists their separation. The use of adhesives offers certain advant ...
\Delta_E \cup_\Delta_F Effectuating the gluing for each inclusion, one ends up with the desired topological space. As in the previous construction, by the topology induced by gluing, the closed sets in this space are the subsets being closed in the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
of each simplex \Delta_F. The simplicial complex \mathcal, which consists of all simplices \mathcal of dimension \leq n is called the n''-th skeleton'' of \mathcal. A natural neighborhood of a vertex V of a simplicial complex \mathcal is considered to be the star star(K)= \Big \ of a simplex, its boundary is the link lk(K)= \Big\.


Simplicial maps

The maps considered in this category are simplicial maps: Let \mathcal, \mathcal be abstract simplicial complexes above sets V_K, V_L. A simplicial map is a function f:V_K \rightarrow V_L which maps each simplex in \mathcal onto a simplex in \mathcal. By affine-linear extension on the simplices, f induces a map between the geometric realizations of the complexes.


Examples

* Let W =\ and let \mathcal = \Big\. The associated geometric complex is a star with center \. * Let V= \ and let \mathcal = \mathcal(V). Its geometric realization , \mathcal, is a
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
. * Let V as above and let \mathcal' =\; \mathcal(\mathcal)\setminus \. The geometric simplicial complex is the
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of a tetrahedron , \mathcal, = \partial , \mathcal, .


Definition

A triangulation of a topological space X is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
t: , \mathcal, \rightarrow X where \mathcal 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 \mathcal, \mathcal be as in the examples seen above. The closed unit ball \mathbb^3 is homeomorphic to a tetraether so it admits a triangulation, namely the homeomorphism t:, \mathcal, \rightarrow \mathbb^3. Restricting t to , \mathcal', yields a homeomorphism t':, \mathcal', \rightarrow \mathbb^2. * The
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
\mathbb^2 = \mathbb^1 \times \mathbb^1 admits a triangulation. To see this, consider the torus as a square where the parallel faces are glued together. This square can be triangulated as shown below: * The
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
\mathbb^2 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 In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
, see topological invariance


Homology

Via triangulation, one can assign a
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
to topological spaces that arise from its simplicial complex and compute its ''
simplicial homology In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case ...
''.
Compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
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 , \mathcal, be a finite simplicial complex. The n- th Betti- number b_n(\mathcal) is defined to be the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
of the n- th simplicial homology- group of the spaces. These numbers encode geometric properties of the spaces: The Betti- Number b_0(\mathcal) for instance represents the number of
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
components. For a triangulated, closed
orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
surfaces A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. Surface or surfaces may also refer to: Mathematics *Surface (mathematics), a generalization of a plane which needs not be flat * Sur ...
F, b_1(F)= 2g holds where g denotes the
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 nom ...
of the surface: Therefore its first Betti- number represents the doubled number of
handles A handle is a part of, or attachment to, an object that allows it to be grasped and manipulated by hand. The design of each type of handle involves substantial ergonomic issues, even where these are dealt with intuitively or by following t ...
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 \sum_^ (-1)^b_k(\mathcal) which is called the ''Euler Charakteristik'' 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 , \mathcal, \subset \mathbb^N be a simplicial complex. A complex , \mathcal, \subset \mathbb^N is said to be a subdivision of \mathcal iff: * every simplex of \mathcal is contained in a simplex of \mathcal and * every simplex of \mathcal is a finite union of simplices in \mathcal . Those conditions ensure that subdivisions does not change the simplicial complex as a set or as a topological space. A map f: \mathcal \rightarrow \mathcal between simplicial complexes is said to be piecewise linear if there is a refinement \mathcal of \mathcal such that f is piecewise linear on each simplex of \mathcal. 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'' (short PL- topology) ''topology'' examines topological properties of topological spaces.


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 \leq 3 and for differentiable manifolds but it was disproved in general: An important tool to show that triangulations do not admit a common subdivision. i. e 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 (K,L) of CW- complexes: If L = \emptyset this characteristic will be a topological invariant but if L \neq \emptyset 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 4-sphere: Let p, q be natural numbers, such that p, q are coprime . The lens space L(p,q) is defined to be the orbit space of the free group action \Z/p\Z\times S^\to S^ (k,(z_1,z_2)) \mapsto (z_1 \cdot e^, z_2 \cdot e^ ). For different tuples (p, q), 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. Two Lens spaces L(p,q_1), L(p,q_2)are homeomorphic, if and only if q_1 \equiv \pm q_2^ \pmod . This is the case iff two Lens spaces are ''simple-homotopy-equivalent''. The fact can be used to construct counterexamples for the Hauptvermutung as follows. Suppose there are spaces L'_1, L'_2 derived from non-homeomorphic Lens spaces L(p,q_1), L(p,q_2)having different Reidemeister torsion. Suppose further that the modification into L'_1, L'_2 does not affect Reidemeister torsion but such that after modification L'_1 and L'_2 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 \leq 3 are always triangulable but there are non-triangulable manifolds for dimension n, for n arbitrary but greater than three. Further, differentiable manifolds always admit triangulations.


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: Let , X, be a simplicial complex such that every point admits an open neighborhood U such that there is a triangulation of U and a piecewise linear homeomorphism f: U \rightarrow \mathbb^n. Then , X, is said to be a ''piecewise linear (PL) manifold of dimension'' n and the triangulation together with the PL- atlas is said to be a ''PL- structure on'' , X, . An important lemma is the following: Let X be a topological space. It is equivalent # X is an n-dimensional manifold and admits a PL- structure. # There is a triangulation of X such that the link of each vertex is an n-1 sphere. # For each triangulation of X the link of each vertex is an n-1 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. One can show that differentiable manifolds admit a PL- structure as well as manifolds of dimension \leq 3. 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 n-2- dimensional PL- Homology-sphere X. The double suspension S^2X is a topological n-sphere. Choosing a triangulation t: , \mathcal, \rightarrow S^2 X obtained via the suspension operation on triangulations the resulting simplicial complex is not a PL- manifold, because there is a vertex v such that link(v) is not a n-1 sphere. A question arising with the definition is if PL-structures are always unique: Given two PL- structures for the same space Y, is there a there a homeomorphism F:Y\rightarrow Y 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

Pachner moves are a way to manipulate triangulations: Let \mathcal be a simplicial complex. For two simplices K, L the ''Join'' K*L= \Big\ are the points lying on straights between points in K and in L. Choose S \in \mathcal such that lk(S)= \partial K for any K lying not in \mathcal. A new complex \mathcal, can be obtained by replacing S * \partial K by \partial S * K. 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.


CW-complexes

A similar but more flexible construction than simplicial complexes is the one of CW-complexes. Its construction is as follows: An n- cell is the closed n- dimensional unit-ball B_n= ,1n, an open n-cell is its inner B_n= ,1n\setminus \mathbb^. Let X be a topological space, let f: \mathbb^\rightarrow X be a continuous map. The gluing X \cup_B_n is said to be ''obtained by gluing on an n-cell.'' A cell complex is a union X=\cup_ X_n of topological spaces such that * X_0 is a discrete set * each X_n is obtained from X_ by gluing on a family of n-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 \mathbb^2: 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 \mathbb^1. Moreover, surfaces, i.e. 2-manifolds, can be classified completely: Let S be a compact surface. * If S is orientable, it is homeomorphic to a 2-sphere with n tori of dimension 2 attached, for some n\geq 0. * If S is not orientable, it is homeomorphic to a Klein Bottle with n tori of dimension 2 attached, for some n\geq 0. To prove this theorem one constructs a fundamental polygon of the surface: This can be done by using the simplicial structure obtained by the triangulation.


Maps on simplicial complexes

Giving spaces the structure of a simplicial structure might help to understand maps defined on the spaces. The maps can often be assumed to be simplicial maps via the simplicial approximation theorem:


Simplicial approximation

Let \mathcal, \mathcal be abstract simplicial complexes above sets V_K, V_L. A simplicial map is a function f:V_K \rightarrow V_L which maps each simplex in \mathcal onto a simplex in \mathcal. By affin-linear extension on the simplices, f 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 f:\mathcal\rightarrow \mathcal ''.'' A simplicial map g:\mathcal\rightarrow \mathcal is said to be a ''simplicial approximation'' of f if and only if each x \in \mathcal is mapped by g onto the support of f(x) in \mathcal. If such an approximation exists, one can construct a homotopy H transforming f into g by defining it on each simplex; there it always exists, because simplices are contractible. The simplicial approximation theorem guarantees for every continuous function f:V_K \rightarrow V_L the existence of a simplicial approximation at least after refinement of \mathcal, for instance by replacing \mathcal 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'' is a useful tool to find out whether a continuous function admits fixed-points. This data is computed as follows: Suppose that X and Y are topological spaces that admit finite triangulations. A continous map f: X\rightarrow Y induces homomorphisms f_i: H_i(X,K)\rightarrow H_i(Y,K) between its simplicial homology groups with coefficients in a field K. These are linear maps between K - vectorspaces, so their trace tr_i can be determined and their alternating sum L_K(f)= \sum_i(-1)^itr_i(f) \in K is called the ''Lefschetz number'' of f. If f = id, this number is the Euler characteristic of K. The fixpoint theorem states that whenever L_K(f)\neq 0, f has a fixed-point. In the proof this is first shown only for simplicial maps and then generalized for any continuous functions via the approximation theorem. Brouwer's fixpoint theorem treats the case where f:\mathbb^n \rightarrow \mathbb^n is an endomorphism of the unit-ball. For k \geq 1 all its homology groups H_k(\mathbb^n) vanishes, and f_0 is always the identity, so L_K(f) = tr_0(f) = 1 \neq 0, so f has a fixpoint.


Formula of Riemann-Hurwitz

The Riemann- Hurwitz formula allows to determine the gender of a compact, connected Riemann surface X without using explicit triangulation. The proof needs the existence of triangulations for surfaces in an abstract sense: Let F:X \rightarrow Y be a non-constant holomorphic function on a surface with known gender. The relation between the gender g of the surfaces X and Y is 2g(X)-2= deg(F)(2g(Y)-2) \sum_(ord(F)-1) where deg(F) 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.{{citation, surname1=Otto Forster, periodical=Heidelberger Taschenbücher, title=Kompakte Riemannsche Flächen, publisher=Springer Berlin Heidelberg, publication-place=Berlin, Heidelberg, at=pp. 88–154, isbn=978-3-540-08034-3, date=1977, language=German


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 An International Standard Serial Number (ISSN) is an eight-digit serial number used to uniquely identify a serial publication, such as a magazine. The ISSN is especially helpful in distinguishing between serials with the same title. ISSNs ...
0072-5285, doi:10.1007/978-1-4684-9372-6. Topology Algebraic topology Geometric topology Structures on manifolds Triangulation (geometry)