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