Simplicial Structures in Topology

136 III Homology of Polyhedra
since e (p−1) contains no p-simplex, we conclude that
c ′ p = ∑ i
miβ i p
and so cp is homologous to ∑i miβ i p. �
This lemma has an important consequence, essential to the definition of block
homology:
(III.5.8) Theorem. The following results hold for any polyhedron |K| with a block
triangulation:
(1) Every p-cycle zp ∈ Zp(K) is homologous to a cycle of p-blocks ∑i miβ i p .
(2) If ∑i miβ i p is a boundary, there exists a chain of (p + 1)-blocks ∑ j n jβ j
p+1 such
that
∂p+1
�
∑ j
n jβ j
p+1
�
= ∑miβ i
i p ;
(3) ∂p(β i p ) is a chain of (p − 1)-blocks like ∑ j mi j
jβ p−1 .
Proof. (1) Let zp be a p-cycle of K; then∂p(zp) =0andso∂p(zp) ⊂ e (p−1) ;we
conclude from Lemma (III.5.7) that there is a chain of p- blocksc ′ p = ∑i miβ i p,
which is homologous to zp; then, c ′ p is a cycle because ∂p(c ′ p )=∂p(zp)=0.
(2) Let us suppose that ∑i miβ i p = ∂p+1(cp+1), wherecp+1∈Cp+1(K); since
∂p+1(cp+1) ⊂ e (p) ,the(p + 1)-chain cp+1 is homologous to a chain of (p + 1)blocks
∑ j n jβ j
p+1 ; hence
∑ i
miβ i p
= ∂p+1(cp+1)=∂p+1
�
∑ j
n j β j
p+1
(3) By definition, β i p is a generator of Hp(e i p, • e i p) ∼ = Z; hence
∂p(β i p) ⊂ • e i p ⊂ e (p−1) ;
�
.
we now proceed as we did for the second part of Lemma (III.5.7); we substitute
∂p(β i p) for c ′ p and p − 1forp. �
Let |K| be a polyhedron with a block triangulation e(K). We now construct the
chain complex
��
� �
C(e(K)) =
| n ≥ 0
Cn(e(K)),∂ e(K)
n
as follows: for each n ≥ 0, Cn(e(K)) is the free Abelian group generated by β i n ;
the boundary operators ∂ e(K)
n are defined on generators according to part (3) of
Theorem (III.5.8):
∂ e(K)
p (β i p )=∂p(β i p )=∑m j
i jβ j
p−1 ;