Simplicial Structures in Topology

104 III Homology of Polyhedra
such that dim τ j+1 = dim τ j +1 and to which we could assign coefficients equal to
zero in the sum representing p). Consequently, we may assume that
and so
Therefore, if
σ 0 = {x0} ,
σ 1 = {x0,x1} ,
···
σ n = {x0,x1,...,xn}
� � � �
x0 + x1
x0 + x1 + ...+ xn
F(p)=α0x0 + α1 + ...+ αn
.
2
n + 1
and F(p) coincide, we have
and
q =
n
∑ βixi ∈|K|
i=0
β0 = α0 + α1/2 + ...+ αn/(n + 1)
β1 = α1/2 + ...+ αn/(n + 1)
···
βn = αn/(n + 1)
1 ≥ β0 ≥ β1 ≥ ...≥ βn ≥ 0 .
On the other hand, given a sequence of real numbers
the real numbers
1 ≥ β0 ≥ β1 ≥ ...≥ βn ≥ βn+1 = 0
αi =(i + 1)(βi − βi+1) , i = 0,1,...,n
satisfy the preceding equalities and so we see that the coefficients αi and βi are
determined by each other; this means that F is a bijection. The proof is completed
by recalling that F is a continuous bijection from a compact space to a Hausdorff
space. �
Although the spaces |K (r) | are homeomorphic, they may differ by the length of
the geometric realizations of their 1-simplexes. Indeed, let us consider |K (r) | as a
subspace of |K| (the topology on |K| being defined by the metric d) and define the
diameter of |K (r) | to be the maximum length of the geometric realizations of all
1-simplexes of K (r) ; the notation diam |K (r) | indicates the diameter of |K (r) |. We
note that diam |K| = √ 2. The next result shows that the diameter decreases with
the successive barycentric subdivisions.