Simplicial Structures in Topology

74 II Simplicial Complexes
(II.3.10) Corollary. Let (C,∂) and (C ′ ,∂ ′ ) be given positive augmented chain complexes
and let (C,∂ ) be free. Then any chain homomorphism f : (C,∂) → (C ′ ,∂ ′ ),
extending the trivial homomorphism 0: Z → Z and having an acyclic carrier S, is
null-homotopic.
We prove now an important result known as Five Lemma.
(II.3.11) Lemma. Let the diagram of Abelian groups and homomorphisms
α
A
��
A ′
f
f ′
��
β
B
��
��
′
B
g
g ′
��
C
γ
��
��
′
C
h ��
h ′
D
δ
��
��
′
D
k ��
be commutative and with exact lines. If the homomorphisms α, β, δ, and ε are
isomorphisms, so is γ.
Proof. Let c ∈ C be such that γ(c) =0; then δ h(c) =h ′ γ(c) =0 and because δ
is an isomorphism, h(c)=0. In view of the exactness condition, there is a b ∈ B
with g(b)=c and g ′ β(b)=γg(b)=0; thus, there exists a ′ ∈ A ′ such that f ′ (a ′ )=
β (b).But
c = g(b)=gβ −1 f ′ (a ′ )=gfα −1 (a ′ )=0
and so γ is injective. For an arbitrary c ′ ∈ C ′ ,
kδ −1 h ′ (c ′ )=ε −1 k ′ h ′ (c ′ )=0
and, hence, there exists c ∈ C such that h(c)=δ −1 h ′ (c ′ ). Moreover,
h ′ (c ′ − γ(c)) = h ′ (c ′ ) − δδ −1 h ′ (c ′ )=0
k ′
ε
E
��
��
′
E
and hence, there exists b ′ ∈ B ′ such that g ′ (b ′ )=c ′ − γ(c). It follows that
γ(c + gβ −1 (b ′ )) = γ(c)+g ′ ββ −1 (b ′ )=c ′
and so γ is also surjective. �
Exercises
1. A short exact sequence of Abelian groups
Gn+1 ��
fn+1 ��
Gn
fn ��
��
Gn−1