Editing Triangulation (topology) (section)

==== 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.''