nLab
simplicial approximation theorem

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

The simplicial approximation theorem roughly says that if XX and YY are simplicial complexes and f:|X||Y|f: {|X|} \to {|Y|} is a continuous map between their geometric realizations, then after further subdivisions XX', YY' there is a simplicial map g:XYg: X' \to Y' such that |g|{|g|} is homotopic to ff.

Statement

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

|s|={α|X|:α(v)>0impliesvs}.{|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|{|s|} so that this identification is a homeomorphism. |X|{|X|} is then given the coherent topology: the largest topology rendering each inclusion |s||X|{|s|} \hookrightarrow {|X|} is continuous.

If X,YX, Y are simplicial complexes, then a function f:X 0|Y|f: X_0 \to {|Y|} can be extended linearly to a continuous map f˜:|X||Y|\tilde{f}: {|X|} \to {|Y|} by the rule

f˜(α)= vX 0α(v)f(v),\tilde{f}(\alpha) = \sum_{v \in X_0} \alpha(v)f(v),

provided that for every simplex ss of XX, all convex combinations of elements of f(s)f(s) lie in |Y|{|Y|}. Naturally this is the case if f=jϕf = j \circ \phi where ϕ:X 0Y 0\phi: X_0 \to Y_0 is a simplicial map and j:Y 0|Y|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 XX to be a simplicial complex XX' such that

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

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

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

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

Definition

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

Proposition

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

Proof

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

Theorem

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

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

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)

Last revised on January 28, 2021 at 00:11:40. See the history of this page for a list of all contributions to it.