Simplicial approximation theorem
In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings 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 simplices—that is, finite simplicial complexes. 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] It served to put the homology theory of the time—the first decade of the twentieth century—on a rigorous basis, since it showed that the topological effect (on homology groups) 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, 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]\displaystyle{ K }[/math] and [math]\displaystyle{ L }[/math] be two simplicial complexes. A simplicial mapping [math]\displaystyle{ f : K \to L }[/math] is called a simplicial approximation of a continuous function [math]\displaystyle{ F : |K| \to |L| }[/math] if for every point [math]\displaystyle{ x \in |K| }[/math], [math]\displaystyle{ |f|(x) }[/math] belongs to the minimal closed simplex of [math]\displaystyle{ L }[/math] containing the point [math]\displaystyle{ F(x) }[/math]. If [math]\displaystyle{ f }[/math] is a simplicial approximation to a continuous map [math]\displaystyle{ F }[/math], then the geometric realization of [math]\displaystyle{ f }[/math], [math]\displaystyle{ |f| }[/math] is necessarily homotopic to [math]\displaystyle{ F }[/math].[clarification needed]
The simplicial approximation theorem states that given any continuous map [math]\displaystyle{ F : |K| \to |L| }[/math] there exists a natural number [math]\displaystyle{ n_0 }[/math] such that for all [math]\displaystyle{ n \ge n_0 }[/math] there exists a simplicial approximation [math]\displaystyle{ f : \mathrm{Bd}^n K \to L }[/math] to [math]\displaystyle{ F }[/math] (where [math]\displaystyle{ \mathrm{Bd}\; K }[/math] denotes the barycentric subdivision of [math]\displaystyle{ K }[/math], and [math]\displaystyle{ \mathrm{Bd}^n K }[/math] denotes the result of applying barycentric subdivision [math]\displaystyle{ n }[/math] times.), in other words, if [math]\displaystyle{ K }[/math] and [math]\displaystyle{ L }[/math] are simplicial complexes and [math]\displaystyle{ f:|K|\to |L| }[/math] is a continuous function, then there is a subdivision [math]\displaystyle{ K' }[/math] of [math]\displaystyle{ K }[/math] and a simplicial map [math]\displaystyle{ g:K'\to L }[/math] which is homotopic to [math]\displaystyle{ f }[/math]. Moreover, if [math]\displaystyle{ \epsilon:|L|\to\Bbb R }[/math] is a positive continuous map, then there are subdivisions [math]\displaystyle{ K',L' }[/math] of [math]\displaystyle{ K,L }[/math] and a simplicial map [math]\displaystyle{ g:K'\to L' }[/math] such that [math]\displaystyle{ g }[/math] is [math]\displaystyle{ \epsilon }[/math]-homotopic to [math]\displaystyle{ f }[/math]; that is, there is a homotopy [math]\displaystyle{ H:|K|\times[0,1]\to |L| }[/math] from [math]\displaystyle{ f }[/math] to [math]\displaystyle{ g }[/math] such that [math]\displaystyle{ \mathrm{diam}(H(x\times[0,1]))\lt \epsilon(f(x)) }[/math] for all [math]\displaystyle{ x\in |K| }[/math]. So, we may consider the simplicial approximation theorem as a piecewise linear analog of Whitney approximation theorem.
References
- Hazewinkel, Michiel, ed. (2001), "Simplicial complex", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Simplicial_complex
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