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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 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.

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.

An abstract simplicial complex above a set is a system of non-empty subsets such that:

- for each ;
- if and .

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 .^{[1]}

Abstract simplicial complexes can be thought of as geometrical objects too. This requires the term of geometric simplex.

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 its faces.

The *-dimensional standard-simplex* is the simplex spanned by the unit vectors ^{[2]}

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.^{[2]}

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. 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.

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 .

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.^{[2]}

- 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 .

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.

- 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. This square can be triangulated as shown below:
- The projective plane admits a triangulation (see CW-complexes)
- One can show that differentiable manifolds admit triangulations.
^{[3]}

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.^{[4]} For details and the link to singular homology, see topological invariance.

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.

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.^{[5]}

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.

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 .
^{[2]}

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.^{[6]} Furthermore it was shown that singular and simplicial homology groups coincide.^{[6]} 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).^{[7]}

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 and for differentiable manifolds but it was disproved in general:^{[8]} 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.

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:^{[8]}

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.

Two lens spaces are homeomorphic, if and only if .^{[9]} 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 derived from non-homeomorphic lens spaces having different Reidemeister torsion. Suppose further that the modification into does not affect Reidemeister torsion but such that after modification and are homeomorphic. The resulting spaces will disprove the Hauptvermutung.

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 are always triangulable^{[10]}^{[11]}^{[1]} but there are non-triangulable manifolds for dimension , for arbitrary but greater than three.^{[12]}^{[13]} Further, differentiable manifolds always admit triangulations.^{[3]}

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 be a simplicial complex such that every point admits an open neighborhood such that there is a triangulation of and a piecewise linear homeomorphism . Then 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:

Let be a topological space. It is equivalent

- is an -dimensional manifold and admits a PL-structure.
- There is a triangulation of such that the link of each vertex is an sphere.
- For each triangulation of the link of each vertex is an 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.^{[14]} One can show that differentiable manifolds admit a PL-structure as well as manifolds of dimension .^{[15]} 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 -dimensional PL-homology-sphere . The double suspension is a topological -sphere. Choosing a triangulation obtained via the suspension operation on triangulations the resulting simplicial complex is not a PL-manifold, because there is a vertex such that is not a sphere.^{[16]}

A question arising with the definition is if PL-structures are always unique: Given two PL-structures for the same space , is there a there a homeomorphism 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 are a way to manipulate triangulations: Let be a simplicial complex. For two simplices the *Join*

are the points lying on straights between points in and in . Choose such that for any lying not in . A new complex , can be obtained by replacing by . 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.^{[17]}

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

An -cell is the closed -dimensional unit-ball , an open -cell is its inner . Let be a topological space, let be a continuous map. The gluing is said to be *obtained by gluing on an -cell.*

A cell complex is a union of topological spaces such that

- is a discrete set
- each is obtained from by gluing on a family of -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.^{[18]} 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 : Its construction as a CW-complex needs three cells, whereas its simplicial complex consists of 54 simplices.

By triangulating 1-dimensional manifolds, one can show that they are always homeomorphic to disjoint copies of the real line and the unit sphere . Moreover, surfaces, i.e. 2-manifolds, can be classified completely: Let be a compact surface.

- If is orientable, it is homeomorphic to a 2-sphere with tori of dimension attached, for some .
- If is not orientable, it is homeomorphic to a Klein Bottle with tori of dimension attached, for some .

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.^{[19]}

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:

Let , be abstract simplicial complexes above sets , . A simplicial map is a function which maps each simplex in onto a simplex in . By affin-linear extension on the simplices, 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 *.* A simplicial map is said to be a *simplicial approximation* of if and only if each is mapped by onto the support of in . If such an approximation exists, one can construct a homotopy transforming into by defining it on each simplex; there it always exists, because simplices are contractible.

The simplicial approximation theorem guarantees for every continuous function the existence of a simplicial approximation at least after refinement of , for instance by replacing by its iterated barycentric subdivision.^{[2]} 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.*

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 and are topological spaces that admit finite triangulations. A continuous map induces homomorphisms ** ** between its simplicial homology groups with coefficients in a field . These are linear maps between -vector spaces, so their trace can be determined and their alternating sum

is called the *Lefschetz number* of . If , this number is the Euler characteristic of . The fixpoint theorem states that whenever , 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 is an endomorphism of the unit-ball. For all its homology groups vanishes, and is always the identity, so , so has a fixpoint.^{[20]}

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

where 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.^{[21]}

- ^
^{a}^{b}John M. Lee (2000), Springer Verlag (ed.),*Introduction to Topological manifolds*(in German), New York/Berlin/Heidelberg: Springer Verlag, p. 92, ISBN 0-387-98759-2 - ^
^{a}^{b}^{c}^{d}^{e}James R. Munkres (1984),*Elements of algebraic topology*(in German), vol. 1984, Menlo Park, California: Addison Wesley, p. 83, ISBN 0-201-04586-9 - ^
^{a}^{b}J. H. C. Whitehead (1940), "On C1-Complexes",*Annals of Mathematics*(in German), vol. 41, no. 4, pp. 809–824, doi:10.2307/1968861, ISSN 0003-486X, JSTOR 1968861 **^**J. W. Alexander (1926), "Combinatorial Analysis Situs",*Transactions of the American Mathematical Society*(in German), vol. 28, no. 2, pp. 301–329, doi:10.1090/S0002-9947-1926-1501346-5, ISSN 0002-9947, JSTOR 1989117**^**R. Stöcker, H. Zieschang (1994),*Algebraische Topologie*(in German) (2. überarbeitete ed.), Stuttgart: B.G.Teubner, p. 270, ISBN 3-519-12226-X- ^
^{a}^{b}Allen Hatcher (2006),*Algebraic Topologie*(in German), Cambridge/New York/Melbourne: Cambridge University Press, p. 110, ISBN 0-521-79160--X **^**A.A.Ranicki. "One the Hauptvermutung" (PDF).*The Hauptvermutung Book*.- ^
^{a}^{b}John Milnor (1961), "Two Complexes Which are Homeomorphic But Combinatorially Distinct",*The Annals of Mathematics*(in German), vol. 74, no. 3, p. 575, doi:10.2307/1970299, ISSN 0003-486X, JSTOR 1970299 **^**Marshall M. Cohen (1973), "A Course in Simple-Homotopy Theory",*Graduate Texts in Mathematics*, Graduate Texts in Mathematics (in German), vol. 10, doi:10.1007/978-1-4684-9372-6, ISBN 978-0-387-90055-1, ISSN 0072-5285**^**Edwin Moise (1977),*Geometric Topology in Dimensions 2 and 3*(in German), New York: Springer Verlag**^**Tibor Rado. "Über den Begriff der Riemannschen Fläche" (PDF).**^**R. C. Kirby, L. C. Siebenmann (1977-12-31), "Annex B. On The Triangulation of Manifolds and the Hauptvermutung",*Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. (AM-88)*(in German), Princeton University Press, pp. 299–306**^**"Chapter IV; Casson's Invariant for Oriented Homology 3-spheres",*Casson's Invariant for Oriented Homology Three-Spheres*(in German), Princeton University Press, pp. 63–79, 1990-12-31**^**Toenniessen, Fridtjof (2017),*Topologie*(PDF) (in German), doi:10.1007/978-3-662-54964-3, ISBN 978-3-662-54963-6, retrieved 2022-04-20**^**Edwin E. Moise (1952), "Affine Structures in 3-Manifolds: V. The Triangulation Theorem and Hauptvermutung",*The Annals of Mathematics*(in German), vol. 56, no. 1, p. 96, doi:10.2307/1969769, ISSN 0003-486X, JSTOR 1969769**^**Robert D. Edwards (2006-10-18), "Suspensions of homology spheres",*arXiv:math/0610573*(in German), arXiv:math/0610573, Bibcode:2006math.....10573E**^**W B R Lickorish (1999-11-20), "Simplicial moves on complexes and manifolds",*Proceedings of the Kirbyfest*(in German), Mathematical Sciences Publishers, arXiv:math/9911256, doi:10.2140/gtm.1999.2.299, S2CID 9765634**^**Toenniessen, Fridtjof (2017),*Topologie*(PDF) (in German), p. 315, doi:10.1007/978-3-662-54964-3, ISBN 978-3-662-54963-6, retrieved 2022-04-20**^**Seifert, H. (Herbert), 1907-1996. (2003),*Lehrbuch der Topologie*(in German), AMS Chelsea Pub., ISBN 0-8218-3595-5`{{citation}}`

: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)**^**Bredon, Glen E. (1993), Springer Verlag (ed.),*Topology and Geometry*(in German), Berlin/ Heidelberg/ New York, pp. 254 f, ISBN 3-540-97926-3`{{citation}}`

: CS1 maint: location missing publisher (link)**^**Otto Forster (1977), "Kompakte Riemannsche Flächen",*Heidelberger Taschenbücher*(in German), Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 88–154, ISBN 978-3-540-08034-3

- 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.