nLab simplicial approximation theorem




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The simplicial approximation theorem (also: simplicial approximation lemma) roughly says that if XX and YY are simplicial complexes and f:|X||Y|f \colon {|X|} \to {|Y|} is a continuous map between their topological 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.


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.


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|}.


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.


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.


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).


Last revised on August 7, 2022 at 12:25:07. See the history of this page for a list of all contributions to it.