Simplicial Structures in Topology

112 III Homology of Polyhedra
the given definitions ensure that, for every n ≥ 1,
(1 − ℵnCn(π))(σ ′ n ) ∈ S(σ ′ n ) .
We complete the proof by using Corollary (II.3.10). �
Theorem (III.2.2) may be extended by iteration: Let K (r) be the rth-barycentric
subdivision of K and let
π r : K (r) → K
be the composition of projections
Then for every n ≥ 0,
K (r) π
−→ K (r−1) π
−→ ... π
−→ K .
Hn(π r ,Z): Hn(K (r) ,Z) → Hn(K,Z)
is an isomorphism whose inverse is induced by the (not simplicial) homomorphism
ℵ r n : Cn(K) −→ Cn(K (r) )
obtained from ℵn by iteration.
We precede the important Simplicial Approximation Theorem by some remarks
toward the characterization of the simplexes of a simplicial complex K =(X,Φ) by
working with the topology on |K|. For each vertex x of K, letA(x) be the set of all
points p ∈|K| such that p(x) > 0; in addition, we define the function
δx : |K|−→R , δx(p)=p(x) .
This function is continuous since, for every q ∈|K|,
|δx(p) − δx(q)| < d(p,q) ;
since A(x)=δ −1
x (0,∞), we conclude that A(x) is open in |K|. Note that in this way
we obtain an open covering of |K|.
(III.2.3) Lemma. Given x0,...,xn ∈ X arbitrarily,
Proof. ⇒: ifσ ∈ Φ, its barycenter
n�
σ = {x0,...,xn}∈Φ ⇐⇒ A(xi) �= /0 .
i=0
n
1
b(σ)= ∑
i=0 n + 1 xi
n�
∈ A(xi) .
i=0
⇐: ifp ∈ � n i=0 A(xi), thenp(xi) > 0foreveryi = 0,...,n; hence
{x0,...,xn}⊂s(p) ∈ Φ . �