Simplicial Structures in Topology

V.3 Poincaré Duality 193
therefore, β σ p (z) is an element of Cn−p−1( • eσ ) and thus
β σ p (z) ∈ Zn−p(eσ , • eσ ;Z).
The result (*) implies that β σ p (z) is the sum of all (n − p)-simplexes of Bσ with
coefficients ±1 and so, it is a cycle that generates Zn−p(eσ , • eσ ;Z).
We now recall that the set
e(K (1) )={eσ |σ ∈ Φ}
is a block triangulation of K (1) (see Theorem (V.3.3)); moreover, the block homology
of K (1) derives from the chain complex
C(e(K (1) )) = {Cn−p(e(K (1) ),d e(K(1)
n−p |n − p ≥ 0}
where Cn−p(B(K (1) )) is the free Abelian group defined by the generators
(see Sect. III.5). Thus, the map
β σ p (z) ∈ Zn−p(eσ , • eσ ;Z)
βp : Cp(K) → Cn−p(e(K (1) )) , βp(σ)=β σ p (z)
defined on all p-simplexes σ ∈ Φ is an isomorphism. It follows that there exists an
isomorphism
Exercises
H(βp): H p (K;Z) −→ Hn−p(e(K (1) );Z) ∼ = Hn−p(K;Z). �
1. Compute the Euler–Poincaré characteristic of the closed surfaces.
2. An n-dimensional connected compact manifold without boundary is called
a closed n-manifold. It is known that the closed 3-manifolds are triangulable
(E. Moise, 1952). Prove that if M and N are two orientable closed 3-manifolds such
that π1(M) ∼ = π1(N), thenHi(M;Z) ∼ = Hi(N;Z) for i = 0,1,2,3. 8
8 In particular, if π1(M)=0, then M has the same homology groups as the 3-sphere S 3 ; in 1904,
Poincaré asked the question: in this case, is it true that M is homeomorphic to S 3 ? It was only
recently that this famous “Poincaré conjecture” was proved affirmatively by Grigory Perelman,
who used methods of differential geometry, specially, the Ricci flow.