Editing Triangulation (topology) (section)

==== 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>