Simplicial Structures in Topology

102 III Homology of Polyhedra
where αi is the sum of the coefficients ζr of the elements zr whose first coordinate
equals xi; now, we can easily see that ∑ m i=0 αi = 1. After making similar remarks
for |pr2|(u), we conclude that
ψφ = 1 |K×L| .
Since |K × L| is compact, φ is a homeomorphism. �
(III.1.2) Example. The diagram in Fig. III.1 illustrates the case in which K is
(0, 1)
(0, 0)
(2, 1)
(2, 0)
(1, 1)
(1, 0)
Fig. III.1
a 2-simplex with vertices {0,1,2} and L is the 1-simplex with vertices {0,1}.
The resulting prism consists of the three 3-simplexes {(0,0),(1,0),(2,0),(2,1)},
{(0,0),(1,0), (1,1),(2,1)} and {(0,0),(0,1),(1,1),(2,1)}, respectively.
Let K =(X, Φ) be a simplicial complex; we construct an infinite sequence of
simplicial complexes {K (r) |r ≥ 0} as follows:
1. K (0) = K
2. K (1) =(X (1) ,Φ (1) ) is the simplicial complex defined by:
(i) X (1) = Φ
(ii) Φ (1) is the set of all nonempty subsets of Φ such that
3. K (r) is defined iteratively:
{σi0 ,...,σin }∈Φ(1) ⇔ σi0 ⊂ ...⊂ σin ;
K (r) =(K (r−1) ) (1) .
The simplicial complex K (r) is the rth- barycentric subdivision of K.
(III.1.3) Remark. The definition we gave of the first barycentric subdivision K (1)
of a simplicial complex is, perhaps, a little convoluted, and so we give here another,
which is equivalent but refers to the geometric realization |K|. Foreveryn-simplex
σ = {x0,...,xn} of K,wedefinethebarycenter of σ to be the point
b(σ)=
n
∑
i=0
1
n + 1 xi ;