Simplicial Structures in Topology

134 III Homology of Polyhedra
For the real projective plane, with the previously described triangulation we have
the block triangulation given by:
1. e0 = {0}
2. e1 = {{1},{2},{0,1},{1,2},{2,0}}
3. e2 = P � (e0 ∪ e1)
Let |K| be a polyhedron with a block triangulation e(K)={e i p}; for every integer
q such that 0 ≤ q ≤ dimK,wedefine
e (q) = {e i p ∈ e(K)|p ≤ q} .
(III.5.6) Proposition. Let A be any union of q-blocks. Then, e (q) � A is a simplicial
subcomplex of K.
Proof. Let σ be any simplex of e (q) � A; thenσ is in a certain block e i p of e(q) � A,
with p ≤ q. Suppose σ ′ ⊂ σ; then either e(σ ′ )=e i p or σ ′ ∈ • e i p; in the latter case,
e(σ ′ ) is an r-block with r < p; in any case, σ ′ ∈ e (q) � A. �
At this point, we can define the “block homology” of |K| with block triangulation
e(K): in a nutshell, it comes from the chain complex C(e(K)) = {Cn(e(K))}
defined as
Cn(e(K)) = Hn(e (n) ,e (n−1) ;Z) ∼ ℓn�
= Hn(e j n , • e j n ;Z)
where {e1 n ,...,eℓn n } is the set of n-blocks. The boundary operator ∂ e(K)
n will be given
by suitable sum of compositions
Hn(en, • en;Z) λn
−→ Hn−1( • en;Z) Hn−1(i) −→ Hn−1(en−1;Z) q∗(n−1)
−→ Hn−1(en−1, • en−1;Z)
where λn, q∗(n−1) are the appropriate homomorphisms of the long exact sequences
for (en, • en), (en−1, • en−1), respectively, and Hn−1(i) is the homomorphism arising
from the inclusion • en ⊂ en−1. The problem is also to relate this homology to
H∗(|K|;Z). For this, we proceed along the lines of [17, 3.8], and we will define
∂ e(K)
∗ in the process.
We begin with the following key result:
(III.5.7) Lemma. Let cp be a p-chain of K such that ∂p(cp) ⊂ e (p−1) ; then there
exists a chain of p-blocks ∑i miβ i p homologous to cp.
Proof. Suppose that cp ⊂ e (q) , with q > p. Let us first prove the existence of a
p-chain c ′ p ∈ Cp(K) homologous to cp, wherec ′ p ⊂ e (p) (this means that we may
remove cp from the blocks with dimension strictly larger than p). To this end, it
is enough to prove the existence of a p-chain fp ∈ Cp(K) homologous to cp, with
fp ⊂ e (q−1) : in fact, since cp and fp are homologous, ∂p(cp) =∂p( fp); wemay
therefore conclude that ∂p( fp) ⊂ e (p−1) from the hypothesis ∂p(cp) ⊂ e (p−1) ;if
q − 1 > p, we apply the preceding argument on fp, and so forth.
j=1