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.

Statement

Let ${\displaystyle K}$ and ${\displaystyle L}$ be two simplicial complexes. A simplicial mapping ${\displaystyle s:K\to L}$ is called a simplicial approximation of a continuous function ${\displaystyle f:|K|\to |L|}$ if for every point ${\displaystyle x}$ in ${\displaystyle |K|}$, ${\displaystyle |s|(x)}$ belongs to the minimal closed simplex of ${\displaystyle L}$ containing ${\displaystyle f(x)}$. If ${\displaystyle s}$ is a simplicial approximation to a continuous map ${\displaystyle f}$, then the geometric realization ${\displaystyle |s|}$ of ${\displaystyle s}$ is necessarily homotopic to ${\displaystyle f}$. ^{[1] [ clarification needed ]}

The simplicial approximation theorem states that given any continuous map ${\displaystyle f:|K|\to |L|}$ there exists a natural number ${\displaystyle n_{0}}$ such that for all ${\displaystyle n\geq n_{0}}$ there exists a simplicial approximation ${\displaystyle s:\mathrm {Bd} ^{n}K\to L}$ to ${\displaystyle f}$ (where ${\displaystyle \mathrm {Bd} \;K}$ denotes the barycentric subdivision of ${\displaystyle K}$, and ${\displaystyle \mathrm {Bd} ^{n}K}$ denotes the result of applying barycentric subdivision ${\displaystyle n}$ times.), in other words, if ${\displaystyle K}$ and ${\displaystyle L}$ are simplicial complexes and ${\displaystyle f:|K|\to |L|}$ is a continuous function, then there is a subdivision ${\displaystyle K'}$ of ${\displaystyle K}$ and a simplicial map ${\displaystyle s:K'\to L}$ whose geometric realization is homotopic to ${\displaystyle f}$. ^[2]

Moreover, if ${\displaystyle \epsilon$ :|L|\to {\mathbb {R}}} is a positive continuous map, then there are subdivisions ${\displaystyle K',L'}$ of ${\displaystyle K,L}$ and a simplicial map ${\displaystyle s:K'\to L'}$ such that ${\displaystyle g=|s|}$ is ${\displaystyle \epsilon }$-homotopic to ${\displaystyle f}$; that is, there is a homotopy ${\displaystyle H:|K|\times [0,1]\to |L|}$ from ${\displaystyle f}$ to ${\displaystyle g}$ such that ${\displaystyle \mathrm {diam} (H(x\times [0,1]))<\epsilon (f(x))}$ for all ${\displaystyle x\in |K|}$.^{[ citation needed ]} So, we may consider the simplicial approximation theorem as a piecewise linear analog of Whitney approximation theorem.

Sketch of proof ^[3]

The idea of the proof is quite intuitive; if there are sufficiently many vertices, then on each simplex, a continuous map can be approximated by a piecewise-linear map; thus globally so.

Precisely, let ${\displaystyle \operatorname {St} ^{0}(w)}$ denote the open star of ${\displaystyle w}$; i.e., the union of all relatively-open simplexes containing ${\displaystyle w}$ in the closure. Note ${\displaystyle \operatorname {St} ^{0}(w)}$ is the complement of the union of all simplexes disjoint from ${\displaystyle \operatorname {St} ^{0}(w)}$; in particular, is an open subset and thus ${\displaystyle U_{w}:=f^{-1}(\operatorname {St} ^{0}(w))}$, ${\displaystyle w}$ a vertex, is an open cover of ${\displaystyle |K|}$. Let ${\displaystyle \delta }$ be the Lebesgue number of this open cover; i.e., a positive real number such that if ${\displaystyle A\subset |K|}$ is a subset of diameter ${\displaystyle <\delta }$, then ${\displaystyle A}$ is contained in some open set in the cover.

Now, let ${\displaystyle K'}$ be some refinement of ${\displaystyle K}$ with the property that the diameter of each simplex in ${\displaystyle K'}$ is less then ${\displaystyle \delta /2}$. Then the diameter of ${\displaystyle \operatorname {St} ^{0}(v)}$ is less than ${\displaystyle \delta }$, since ${\displaystyle |x-y|\leq |x-v|+|v-y|<\delta }$ for each ${\displaystyle x,y}$ in ${\displaystyle \operatorname {St} ^{0}(v)}$. Thus, for each vertex ${\displaystyle v}$, we have ${\displaystyle \operatorname {St} ^{0}(v)\subset f^{-1}(\operatorname {St} ^{0}(w))}$ or

${\displaystyle f(\operatorname {St} ^{0}(v))\subset \operatorname {St} ^{0}(w)}$

for some vertex ${\displaystyle w}$. Let ${\displaystyle g(v)}$ denote some such ${\displaystyle w}$. Then ${\displaystyle g}$ is a map between the sets of the vertices and one can show it extends to a simplicial map ${\displaystyle g:|K|\to |L|}$ (this involves a matter of how to define a simplicial map). Now, for each ${\displaystyle x}$ in ${\displaystyle |K|}$, ${\displaystyle f(x)}$ belongs to a unique relatively-open simplex ${\displaystyle \tau }$ in ${\displaystyle L}$. Let ${\displaystyle x=\sum \lambda _{i}v_{i}}$ be a convex combination with nonzero coefficients for some vertices ${\displaystyle v_{i}}$ in ${\displaystyle K.}$ Then ${\displaystyle g(x)=\sum \lambda _{i}g(v_{i})}$. Let ${\displaystyle w_{i}=g(v_{i})}$. For each ${\displaystyle i}$, we have ${\displaystyle x\in \operatorname {St} ^{0}(v_{i})}$ and so

${\displaystyle f(x)\in f(\operatorname {St} ^{0}(v_{i}))\subset \operatorname {St} ^{0}(w_{i}).}$

Thus, ${\displaystyle f(x)}$ belongs to some ${\displaystyle \tau '}$ whose closure contains ${\displaystyle w_{i}}$ and by uniqueness, ${\displaystyle \tau '=\tau }$. Then we have

${\displaystyle g(x)\in \sum \lambda _{i}{\overline {\tau }}\subset {\overline {\tau }}.}$

Hence, ${\displaystyle f(x),g(x)}$ belongs to the same simplex ${\displaystyle {\overline {\tau }}}$. So, if we let ${\displaystyle h_{t}=(1-t)f+g}$, then ${\displaystyle h_{t}}$ is a homotopy ${\displaystyle f\sim g}$. ${\displaystyle \square }$

References

1. Spanier, Edwin H. (2012). Algebraic Topology. Springer-Verlag. ISBN   9781468493221.^{: Section 3.4, Theorem 3}
2. Hatcher, Allen (2002). Algebraic topology. Cambridge ; New York: Cambridge University Press. ISBN   978-0521795401.^{: Theorem 2C.1}
3. Bredon 2013 , Proof of Theorem 22.10.

• "Simplicial complex", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
• Bredon, G.E. (2013). Topology and Geometry. Graduate Texts in Mathematics. Vol. 139. Springer Science & Business Media.