Editing Triangulation (topology) (section)

=== 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.<ref name=":04"/> 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 <math>X</math> and <math>Y</math> are topological spaces that admit finite triangulations. A continuous map <math>f: X\rightarrow Y</math> induces homomorphisms '''<math>f_i: H_i(X,K)\rightarrow H_i(Y,K)</math>''' between its simplicial homology groups with coefficients in a field <math>K</math>. These are linear maps between <math>K </math>-vector spaces, so their trace <math>\operatorname{tr}_i</math> can be determined and their alternating sum

<math>L_K(f)= \sum_i(-1)^i\operatorname{tr}_i(f) \in K</math>

is called the ''Lefschetz number'' of <math>f</math>. If <math>f =\rm id</math>, this number is the Euler characteristic of <math>K</math>. The fixpoint theorem states that whenever <math>L_K(f)\neq 0</math>, <math>f</math> 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 <math>f:\mathbb{D}^n \rightarrow \mathbb{D}^n</math> is an endomorphism of the unit-ball. For <math>k \geq 1</math> all its homology groups <math>H_k(\mathbb{D}^n)</math> vanishes, and <math>f_0</math> is always the identity, so <math>L_K(f) =\operatorname{tr}_0(f) = 1 \neq 0</math>, so <math>f</math> has a fixpoint.<ref>{{citation|last=Bredon|first= Glen E.|publisher= Springer Verlag|title=Topology and Geometry|publication-place=Berlin/ Heidelberg/ New York|pages=254ff|isbn=3-540-97926-3|date=1993}}</ref>

==== Formula of Riemann-Hurwitz ====
{{Main|Riemann-Hurwitz formula}}
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.<ref>{{citation|surname1=Otto Forster|periodical=Heidelberger Taschenbücher|title=Kompakte Riemannsche Flächen|publisher=Springer Berlin Heidelberg|publication-place=Berlin, Heidelberg|at=pp.&nbsp;88–154|isbn=978-3-540-08034-3|date=1977|language=German
}}</ref>