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