Simplicial Structures in Topology

II.4 Simplicial Homology 75
splits (or is split) if there exists a homomorphism hn−1 : Gn−1 → Gn such that
fnhn−1 = 1Gn−1 (or if there exists a homomorphism kn : Gn → Gn+1 such that
). Prove that if the short exact sequence
kn fn+1 = 1Gn+1
splits, then
2. Prove Theorem (II.3.3).
II.4 Simplicial Homology
Gn+1 ��
fn+1 ��
Gn
Gn ∼ = Gn+1 ⊕ Gn−1.
fn ��
��
Gn−1
In this section, we give some results which allow us to study more in depth the
homology of a simplicial complex. We begin with some important remarks on the
homology of a simplicial complex K. The groups Cn(K) of the n-chains are free,
with rank equal to the (finite) number of n-simplexes of K; hence, also the subgroups
Zn(K) and Bn(K) of Cn(K) are free, with a finite number of generators. Finally,
the homology groups Hn(K) are Abelian and finitely generated; therefore, by the
decomposition theorem for finitely generated Abelian groups, they are isomorphic
to direct sums
Z β (n) ⊕ Z n(1) ⊕ ...⊕ Z n(k)
where Z n(i) is cyclic of order n(i). The number β(n) – equal to the rank of the
Abelian group Hn(K) –isthenth-Betti number of the complex K.
Let p be the dimension of the simplicial complex K; for each 0 ≤ n ≤ p,lets(n)
be the number of n-simplexes of K (remember that K is finite). Hence, the rank
of the free Abelian group Cn(K) is s(n). We indicate with z(n) and b(n) the ranks of
the groups Zn(K) and Bn(K), respectively, where n = 0,...,p. Since the boundary
homomorphism ∂n : Cn(K) → Cn−1(K) is a surjection on Bn−1(K),byNöther’s Homomorphism
Theorem, for each n ≥ 1
(1) s(n) − z(n)=b(n − 1);
if n = 0, we have s(0)=z(0) because C0(K)=Z0(K) and B−1(K)=0; on the other
hand, Hn(K,Z)=Zn(K)/Bn(K) and so
(2) β(n)=z(n) − b(n)
for n ≥ 0. Subtracting (2) from (1) (when n ≥ 1) it follows that
(3) s(n) − β (n)=b(n) − b(n − 1).