Simplicial Structures in Topology

II.4 Simplicial Homology 85
defined on the n-simplexes of K by the conditions
such that
�
0 if σn ∈ Φ �Ψ
βn(σn)=
σn if σn ∈ Ψ
αn : Cn(K,L) → Cn(K) , σn ∈ Φ �Ψ ↦→ σn
μn : Cn(K) → Cn(K,L)
�
σn if σn ∈ Φ �Ψ
μn(σn)=
0 if σn ∈ Ψ .
It is easy to check that βnCn(i)=1, μnαn = 1, μnCn(i)=0, and Cn(i)βn + αnμn = 1
for each n ≥ 0. Hence, for every n ≥ 0, we have a short exact sequence
Cn(L) ��
Cn(i)
��
Cn(K)
μn ��
��
Cn(K,L).
We now consider the boundary homomorphism ∂n : Cn(K) → Cn−1(K) and define
∂ n : Cn(K,L) → Cn−1(K,L)
as the composite homomorphism ∂ n = μn−1∂nαn. We note that
∂ n−1∂ n =(μn−2∂n−1αn−1)(μn−1∂nαn)
=(μn−2∂n−1)(1 −Cn−1(i)βn−1)∂nαn
= μn−2∂n−1∂nαn − μn−2∂n−1Cn−1βn−1 = 0
since the factor ∂n−1∂n = 0 appears in the first term and also because the second term
is null on all (n − 1)-simplex of K. The graded Abelian group {Cn(K,L) | n ∈ Z},
where Cn(K,L)=0foreveryn < 0, has a boundary homomorphism {∂ n | n ∈ Z}
with ∂ n = 0forn ≤ 0; let
H∗(K,L;Z)={Hn(K,L;Z)}
be its homology. Let θn : Cn(K,L) → Cn(K) be the linear homomorphism defined
on an n-simplex σn ∈ Φ �Ψ by θn(σn)=σn +Cn(L) (if n < 0, we define θn = 0).
We note that θn commutes with the boundary homomorphisms; it is sufficient to
verify this statement for an n-simplex σn ∈ Φ �Ψ:
∂ K,L
n θn(σn)=∂n(σn)+Cn−1(L)= ∑ (−1)
σn−1,i∈Φ�Φ
i σn−1,i +Cn(L);
θn−1∂ n(σn)=θn−1(μn−1∑ i
(−1) i σn−1,i)= ∑ (−1)
σi,n−1∈Φ�Φ
i σn−1,i +Cn(L).