Simplicial Structures in Topology

III.2 Homology of Polyhedra 111
where b(σn) is the barycenter of σn (hence, a vertex of K (1) )and
ℵn−1(∂ K n (σn)) ∗ b(σn)
represents the n-chain obtained by taking the join of each component of the (n − 1)chain
ℵn−1(∂ K n (σn)) and b(σn) (abstract cone with vertex b(σn)). The only important
fact to be proved here is that ℵn−1∂ K n = ∂ K(1)
n ℵn. But this is immediate:
∂ K(1)
n (ℵn(σn)) = ℵn−1(∂ K n (σn)) − ∂ K(1)
n−1 (ℵn−1(∂ K n (σn)) ∗ b(σn))
= ℵn−1(∂ K n (σn))
since ∂ K(1)
n−1 ℵn−1 = ℵn−2∂ K n−1 . We call ℵ: C(K) −→ C(K(1) ) barycentric subdivision
homomorphism.
Arriving to the equality C(π)ℵ = 1C(K) is a straightforward procedure. We now
prove that 1C(K (1) ) − ℵC(π) is chain null-homotopic and so, by Proposition (II.3.4),
we have H∗(ℵ)H∗(π) =1. As a matter of simplicity, we shall indicate 1C(K (1) )
with 1, K (1) with K ′ , ∂ K(1)
n with ∂ ′ n ,andalln-simplexes of K′ with the generic
expression σ ′ n .
Let ε : C0(K ′ ) → Z be the augmentation homomorphism of the chain complex
(C(K ′ ),∂ ′ );since
ε((1 − ℵ0C0(π))(σn)=ε(σn −{xn})=0,
1 − ℵC(π)) is a trivial extension of the homomorphism Z on itself.
On the other hand, to each generator
σ ′ n = {σ 0 ,...,σ n }
of Cn(K ′ ), we associate the chain complex S(σ ′ n) ≤ C(K ′ ) defined by the free groups
S(σ ′ n )i
�
Ci((σ
=
n ) (1) ) , i ≥ 0
0 , i < 0 .
The simplicial complex (σ n ) (1) comes from the first barycentric subdivision of σ n
and is, therefore, an acyclic complex since it may be interpreted as the abstract
cone with vertex at the barycenter of σ n relative to the boundary of (σ n ) (1) (see
Sect. II.4); hence, the chain complex
S(σ ′ n ) ≤ C(K′ )
just defined is acyclic. We have thus obtained an acyclic carrier of C(K ′ ) on itself.
We maintain that S is an acyclic carrier of 1 − ℵC(π). Indeed, for any vertex σn =
{x0,...,xn} of K ′ ,
(1 − ℵ0C0(π))(σn)=σn −{xn}∈C0(σ (1)
n )=S(σ ′ n )0 ;