Simplicial Structures in Topology

III.1 The Category of Polyhedra 103
then, the vertices of K (1) are represented abstractly by the elements of the set
X (1) = {b(σ)|σ ∈ Φ}
(in other words, X (1) contains all vertices of K together with those vertices represented
by barycenters of all simplexes of K that are not vertices). From this point of
view, a set {b(σi 0 ),...,b(σin )} of n + 1 vertices of K(1) is an n-simplex if and only
if σi 0 ⊂ σi 1 ⊂ ...⊂ σin .
From the simplicial complexes point of view, the elements of the sequence
{K (r) |r ≥ 0} are all different; but, from the geometric point of view, they are not
distinct; actually, we have the following result:
(III.1.4) Theorem. Let K =(X,Φ) be a simplicial complex and r any positive
integer. Then the polyhedra |K| and |K (r) | are homeomorphic.
Proof. It is sufficient to prove the result for r = 1. For instance, Fig. III.2 shows
{0, 1}
{0,1,2}
{0}
{0, 2}
{1} {2}
{1, 2}
Fig. III.2
the barycentric subdivision of a 2-simplex. The function that associates with each
σ ∈ K (1) the barycentre b(σ) may be extended linearly to a continuous function
F : |K (1) �
n
|−→|K| , F ∑ αiσ
i=0
i
�
n
= ∑ αib(σ
i=0
i )
(all linear functions are continuous: cf. Theorem (II.2.8)). We wish to prove that
the function F is a bijection. Let p = ∑ n i=0 αiσ i be an arbitrary point of |K (1) |; note
that σ 0 ,...σ n are simplexes of K such that
σ 0 ⊂ σ 1 ⊂ ...⊂ σ n .
Let us suppose that dim σ 0 = r; we may assume that dim σ 1 = r + 1 (otherwise,
we could take intermediate simplexes
σ 0 = τ 0 ⊂ τ 1 ⊂ ...⊂ τ k = σ 1