nLab simplicial approximation theorem

Idea

The simplicial approximation theorem (also: simplicial approximation lemma) roughly says that if $X$ and $Y$ are simplicial complexes and $f \colon {|X|} \to {|Y|}$ is a continuous map between their topological realizations, then after further subdivisions $X'$, $Y'$ there is a simplicial map $g: X' \to Y'$ such that ${|g|}$ is homotopic to $f$.

Statement

Let $X$ be a simplicial complex, with vertex set $X_0$. We recall that the geometric realization of $X$ is a space whose points may be described as functions $\alpha: X_0 \to [0, 1]$ such that $\alpha^{-1}((0, 1])$ is a simplex of $X$ (in particular, finite) and $\sum_{v \in X_0} \alpha(v) = 1$. For a simplex $s$ of $X$, we let ${|s|}$ denote the closed (affine) simplex in the geometric realization ${|X|}$, defined by the formula

$${|s|} = \{\alpha \in {|X|}: \alpha(v) \gt 0 \; implies \; v \in s\}.$$

This may be identified with a standard affine simplex, and we topologize ${|s|}$ so that this identification is a homeomorphism. ${|X|}$ is then given the coherent topology: the largest topology rendering each inclusion ${|s|} \hookrightarrow {|X|}$ is continuous.

If $X, Y$ are simplicial complexes, then a function $f: X_0 \to {|Y|}$ can be extended linearly to a continuous map $\tilde{f}: {|X|} \to {|Y|}$ by the rule

$$\tilde{f}(\alpha) = \sum_{v \in X_0} \alpha(v)f(v),$$

provided that for every simplex $s$ of $X$, all convex combinations of elements of $f(s)$ lie in ${|Y|}$. Naturally this is the case if $f = j \circ \phi$ where $\phi: X_0 \to Y_0$ is a simplicial map and $j: Y_0 \to {|Y|}$ is the natural inclusion.

We may then define a general subdivision (not necessarily an iterated barycentric subdivision) of a simplicial complex $X$ to be a simplicial complex $X'$ such that

• Vertices of $X'$ are points of ${|X|}$,

• For every simplex $s'$ of $X'$, there is a simplex of $X$ such that $s' \subseteq {|s|}$,

• The linear map $\tilde{i}: {|X'|} \to {|X|}$ induced from the inclusion $i: X' \hookrightarrow {|X|}$ is a homeomorphism.

Often one treats such $\tilde{i}$ as an identity map.

Definition

Let $X, Y$ be simplicial complexes, and let $f: {|X|} \to {|Y|}$ be a continuous map. A simplicial map $\phi: X \to Y$ is a simplicial approximation to $f$ if $f(\alpha) \in {|s|}$ implies ${|\phi|}(\alpha) \in {|s|}$.

Proposition

Suppose $\phi: X \to Y$ is a simplicial approximation to a map $f: {|X|} \to {|Y|}$, and suppose $f = {|\phi|}$ on a subset $A \subseteq {|X|}$. Then there is a homotopy ${|\phi|} \simeq f \; rel \; A$.

Proof

Define the homotopy $H$ by $H(\alpha, t) = t f(\alpha) + (1-t){|\phi|}(\alpha)$. This makes sense since if $f(\alpha) \in {|s|}$, then ${|\phi|}(\alpha) \in {|s|}$ and ${|s|}$ is closed under convex combinations. Clearly $H(\alpha, t) = f(\alpha)$ for all $t$ on a set where $f$ and ${|\phi|}$ agree.

Theorem

Given simplicial pairs $(X, A)$, $(Y, B)$ and a map of pairs $f: ({|X|}, {|A|}) \to ({|Y|}, {|B|})$, there is a subdivision $i: (X', A') \hookrightarrow ({|X|}, {|A|})$ and a simplicial approximation $\phi: (X', A') \to (Y, B)$ to $f \circ \tilde{i}: ({|X'|}, {|A'|}) \to ({|Y|}, {|B|})$.

Moreover, if $(X, A)$ is a finite simplicial pair, we may choose the subdivision to be an iterated barycentric subdivision $(sd^n X, sd^n A)$ for any sufficiently large $n$ (given $f$).

Related concepts

• cellular approximation theorem

References

• Robert Switzer, Lemma 6.8, p. 76 in: Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975 (doi:10.1007/978-3-642-61923-6)

• Rudolf Fritsch, Renzo A. Piccinini, Section 2.4 in: Cellular structures in topology, Cambridge Studies in Advanced Mathematics 19, Cambridge University Press (1990) [doi:10.1017/CBO9780511983948, pdf]

• John F. Jardine, Simplicial approximation, Theory and Applications of Categories, 12 2 (2004) 34-72 [tac:12-02, pdf]