Editing Simplicial approximation theorem

{{short description|Continuous mappings can be approximated by ones that are piecewise simple}}
{{more sources needed|date=May 2024}}

In [[mathematics]], the '''simplicial approximation theorem''' is a foundational result for [[algebraic topology]], guaranteeing that [[continuous mapping]]s can be (by a slight deformation) approximated by ones that are [[piecewise]] of the simplest kind. It applies to mappings between spaces that are built up from [[simplex|simplices]]&mdash;that is, finite [[simplicial complex]]es. The general continuous mapping between such spaces can be represented approximately by the type of mapping that is (''affine''-) linear on each simplex into another simplex, at the cost (i) of sufficient [[barycentric subdivision]] of the simplices of the domain, and (ii) replacement of the actual mapping by a [[homotopic]] one.

This theorem was first proved by [[L.E.J. Brouwer]], by use of the [[Lebesgue covering theorem]] (a result based on [[compactness]]).{{Citation needed|date=August 2023}} It served to put the [[homology theory]] of the time&mdash;the first decade of the twentieth century&mdash;on a rigorous basis, since it showed that the topological effect (on [[homology group]]s) of continuous mappings could in a given case be expressed in a [[finitary]] way. This must be seen against the background of a realisation at the time that continuity was in general compatible with the [[pathological (mathematics)|pathological]], in some other areas. This initiated, one could say, the era of [[combinatorial topology]].

There is a further '''simplicial approximation theorem for homotopies''', stating that a [[homotopy]] between continuous mappings can likewise be approximated by a combinatorial version.

==Formal statement of the theorem==

Let <math> K </math> and <math> L </math> be two [[simplicial complex]]es. A [[simplicial map|simplicial mapping]] <math> f : K \to L </math> is called a simplicial approximation of a continuous function <math> F : |K| \to |L| </math> if for every point <math> x \in |K| </math>, <math> |f|(x) </math> belongs to the minimal closed simplex of <math> L </math> containing the point <math> F(x) </math>. If <math> f </math> is a simplicial approximation to a continuous map <math> F </math>, then the geometric realization of <math> f </math>, <math> |f| </math> is necessarily homotopic to <math> F </math>.{{Clarify|reason=The notations &#124;K&#124;, &#124;f&#124; should be defined here, to make the theorem statement clearer.|date=March 2023}}

The simplicial approximation theorem states that given any continuous map <math> F : |K| \to |L| </math> there exists a natural number <math> n_0 </math> such that for all <math> n \ge n_0 </math> there exists a simplicial approximation <math> f : \mathrm{Bd}^n K \to L </math> to <math> F </math> (where <math> \mathrm{Bd}\; K </math> denotes the [[barycentric subdivision]] of <math> K </math>, and <math> \mathrm{Bd}^n K </math> denotes the result of applying barycentric subdivision <math> n </math> times.), in other words, if <math>K</math> and <math>L</math> are simplicial complexes and <math>f:|K|\to |L|</math> is a continuous function, then there is a subdivision <math>K'</math> of <math>K</math> and a simplicial map <math>g:K'\to L</math> which is homotopic to <math>f</math>. Moreover, if <math>\epsilon:|L|\to\Bbb R</math> is a positive continuous map, then there are subdivisions <math>K',L'</math> of <math>K,L</math> and a simplicial map <math>g:K'\to L'</math> such that <math>g</math> is <math>\epsilon</math>-homotopic to <math>f</math>; that is, there is a homotopy <math>H:|K|\times[0,1]\to |L|</math> from <math>f</math> to <math>g</math> such that <math>\mathrm{diam}(H(x\times[0,1]))<\epsilon(f(x))</math> for all <math>x\in |K|</math>. So, we may consider the simplicial approximation theorem as a piecewise linear analog of [[Whitney approximation theorem]].

==References==
*{{Springer|id=Simplicial_complex|title=Simplicial complex}}

{{DEFAULTSORT:Simplicial Approximation Theorem}}
[[Category:Theory of continuous functions]]
[[Category:Simplicial sets]]
[[Category:Theorems in algebraic topology]]