Simplicial Structures in Topology

192 V Triangulable Manifolds
and therefore
�Hr( • eσ ;Z) ∼ =
� Z if r = n − p − 1
0 ifr �= n − p − 1 .
This result and the fact that the (reduced) homology of eσ is trivial (for eσ is
contractible) applied to the exact sequence of the (reduced) homology of the pair
(eσ , • eσ ) enable us to conclude that
(∀σ ∈ Φ , dimσ = p) eσ is an (n − p)-block of K (1) .
We only need to prove that the set {eσ |σ ∈ Φ} is a block triangulation of K (1) .In
fact, let �σ = {σ 0 ,...,σ r } be any simplex of K (1) ; by the definition of K (1) ,wehave
σ 0 ⊂ ...⊂ σ r and so
�σ ⊂ e σ 0 � • e σ 0 .
Finally, for every σ ∈ Φ,
•
eσ = ∪τ⊂σ,τ�=σeτ . �
We now have all that is needed to prove the Poincaré Duality Theorem.
(V.3.4) Theorem. Let V be an oriented triangulable n-manifold, with triangulation
given by the simplicial complex K =(X,Φ). Then, for every integer p with 0 ≤
p ≤ n,
H p (V;Z) ∼ = Hn−p(V;Z) .
Proof. To prove this theorem, we must consider the first barycentric subdivision
K (1) of K and use both the projection π : K (1) → K and the homomorphism of chain
complexes ℵ: C(K) → C(K (1) ) (review Sect. III.2). Let z be the cycle of dimension
n defined as the formal sum of all n-simplexes of K; the homology class [z] is a
generator of Hn(|K|;Z) ∼ = Z (see the proof of Theorem (V.3.2)).
For each generator cσ ∈ C p (K;Z) 7 ,wedefinethemap
β σ p : C p (K;Z) −→ Cn−p(K (1) ;Z)
β σ p (cσ ) := cσCp(π) ∩ ℵn(z) .
Since d K(1)
n ℵn = ℵn−1d K n and z is a cycle, the n-chain ℵn(z) is a cycle: in fact, it
is the sum of all oriented n-simplexes of K (1) in agreement with the fact that |K (1) |
is an oriented n-manifold. For any term of this sum, in other words, an n-simplex
{σ0,...,σ n } of K (1) ,wehave
(*) cσCp(π) ∩{σ0,...,σ n } = ±{σ0,...,σ n }
and so β σ p (z) ∈ Cn−p(Bσ ).
By Theorem (IV.3.3),
d K(1)
n−p (cσCp(π)∩ℵn(z)) = (−1) p (cσCp(π)∩d K(1)
n (ℵn(z))−d K p (cσCp(π))∩ℵn(z))
=(−1) p (cσCp(π) ∩ d K(1)
n (ℵn(z)));
7 We remind the reader that cσ is the map that takes the p-simplex σ to 1 ∈ Z and all other
p-simplexes to 0 ∈ Z.