Simplicial Structures in Topology

II.2 Abstract Simplicial Complexes 57
(II.2.14) Theorem. Every n-dimensional polyhedron |K| is homeomorphic to an
Euclidean simplicial complex.
Proof. Let N be the set of all points P i =(i,i 2 ,...,i 2n+1 ) ∈ R 2n+1 , for every
i ≥ 0. We claim that the set N has the following property: every 2n + 2 points
P i0,...,P i2n+1 are linearly independent. In fact, a linear combination
gives rise to the equations
2n+1
∑ α j(P
j=1
i j i0 − P )=0
2n+1
∑ α j = 0,
j=1
2n+1
∑ α ji
j=1
1 j = 0,
...
2n+1
∑ α ji
j=1
2n+1
j = 0;
because the determinant of the system of linear homogeneous equations defined by
the 2n + 2 equations written above is equal to ∏k> j(ik − i j) �= 0, the only solution
for the system is the trivial one, α1 = α2 = ...= α2n+1 = 0.
Assume that K =(X,Φ) with X = {a0,...,as} and s ≤ n. To each vertex
ai, we associate the point P i =(i 1 ,i 2 ,...,i 2n+1 ) ∈ R 2n+1 , and to each simplex
{a j0 ,aj1 ,...,a jp }∈Φ we associate the Euclidean simplex {P j 0,P j 1,...,P jp } (observe
that the points P ji with j = 0,...,p are linearly independent because p ≤
n < 2n + 1). Let K be the set of vertices and Euclidean simplexes obtained in
this way.
We begin by observing that K clearly satisfies condition 1 of the definition of
Euclidean simplicial complexes. Let us prove that condition 2 is also valid. Let σp
and σq be two Euclidean simplexes of K with r common vertices; altogether σp and
σq have p + q − r + 2 vertices. Because p + q − r + 2 ≤ 2n + 2, these vertices form
an Euclidean simplex of R 2n+1 having σp and σq as faces; hence, σp ∩ σq is either
empty (if r = 0) or a common face of σp and σq.
Therefore, K is an Euclidean simplicial complex homeomorphic to |K|. �
The reader could ask whether Theorem (II.2.14) is the best possible result or else,
whether it is possible to realize all n-dimensional simplicial complexes in Euclidean
spaces of dimension less than 2n + 1. Clearly, a complex of dimension n must be
immersed in a space of dimension at least n. We shall now give two examples of onedimensional
simplicial complexes (that is to say, graphs) that cannot be immersed
in R 2 .