Triangulation (topology)
In mathematics, triangulation describes the replacement of topological spaces with simplicial complexes by the choice of an appropriate homeomorphism.
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.
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.
Its main purpose is to study the topological properties of simplicial complexes and their generalizations, cell-complexes.
The topologies do coincide in the case that each point in the complex lies only in finitely many simplices.
In general, the geometric construction as mentioned here is not flexible enough: consider for instance an abstract simplicial complex of infinite dimension.
These are characteristics that equal for complexes that are isomorphic via a simplicial map and thus have the same combinatorial pattern.
Therefore, one can form their alternating sum which is called the Euler characteristic of the complex, a catchy topological invariant.
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.
iff: Those conditions ensure that subdivisions does not change the simplicial complex as a set or as a topological space.
Two complexes that correspond to another via piecewise linear bijection are said to be combinatorial isomorphic.
[7] The Hauptvermutung (German for main conjecture) states that two triangulations always admit a common subdivision.
and for differentiable manifolds but it was disproved in general:[8] An important tool to show that triangulations do not admit a common subdivision.
An approach to Hauptvermutung was to find homeomorphic spaces with different values of Reidemeister torsion.
is defined to be the orbit space of the free group action For different tuples
Therefore they can't be distinguished with the help of classical invariants as the fundamental group but by the use of Reidemeister torsion.
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.
It is natural to require them not only to be triangulable but moreover to admit a piecewise linear atlas, a PL-structure: Let
obtained via the suspension operation on triangulations the resulting simplicial complex is not a PL-manifold, because there is a vertex
of topological spaces such that Each simplicial complex is a CW-complex, the inverse is not true.
[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
[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.
In the proof this is first shown only for simplicial maps and then generalized for any continuous functions via the approximation theorem.
[20] The Riemann-Hurwitz formula allows to determine the genus of a compact, connected Riemann surface
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.
