Simplicial Structures in Topology

V.3 Poincaré Duality 191
(V.3.3) Theorem. For every p-simplex σ ∈ Φ, the block eσ is a (n − p)-block of
K (1) ; besides, the set defined by
is a block triangulation of K (1) .
Proof. We first have to prove that
e(K (1) )={eσ |σ ∈ Φ}
Hr(eσ , • eσ ;Z) ∼ =
� Z if r = n − p
0 ifr �= n − p .
As we have already seen, if σ is an n-simplex, eσ is a simplicial subcomplex of
K (1) having only the vertex b(σ) and no other simplex; so, • eσ = /0 and the statement
above is true. We then assume that dimσ ≤ n − 1. Since |K (1) | ∼ = |K| and
dimK (1) = n, the relative homology of the simplicial subcomplex S(b(σ)) of K (1)
(see the definition given in Theorem (II.2.10))is
(see Lemma (V.1.4)).
We now prove that
•
�Hr(S(b(σ));Z) ∼ =
� Z if r = n − 1
0 if r �= n − 1
S(b(σ)) = | (σ) (1) ∗ • eσ |
where (σ) (1) is the barycentric subdivision of the boundary of σ. In fact,
|{σ 1 ,...,σ r }| ∈ S(b(σ)) ⇐⇒ {σ 1 ,...,σ r }∈Φ (1) with σ ⊂ σ 1 ⊂ ...⊂ σ r
⇐⇒ (∃t) σ t ⊂ σ ⊂ σ t+1 with {σ 1 ,...,σ t }∈
•
On the other hand, | (σ) (1) | ∼ = Sp−1 and consequently
S(b(σ)) ∼ = S p−1 ∗| • eσ| .
•
•
(σ) (1) and {σ t+1 ,...,σ r }∈ •
Bσ .
Let us now go back a step to note that if L1,...,Lp are p simplicial complexes, each
of them having only two 0-simplexes (and no 1-simplex), then
|L1 ∗ ...∗ Lp| ∼ = S p−1
and we conclude that S(b(σ)) is homeomorphic to the pth suspension of | • eσ | (the
suspension of an abstract simplicial complex K is defined on p. 47 – it is easily seen
that the suspension of the sphere S n is homeomorphic to the sphere S n+1 ;thepth
suspension of K is defined by iteration). By Exercise 5 of Sect. II.4, we conclude
that
�Hq(S(b(σ));Z) ∼ = �Hq−p( • eσ ;Z)