Simplicial Structures in Topology

70 II Simplicial Complexes
Please notice that if f is chain null-homotopic, then Hn( f )=0foreveryn.
A chain complex (C,∂) is free if all of its groups are free Abelian; it is positive
if Cn = 0foreveryn < 0. A positive chain complex (C,∂ ) is augmented (to Z) if
there exists an epimorphism
ε : C0 → Z
such that ε∂1 = 0. The homomorphism ε is the augmentation (homomorphism).
(II.3.5) Remark. The chain complex C(K) associated with a simplicial complex K
is free and positive. Moreover, the function
is an augmentation.
ε : C0(K) → Z, Σ n i=1 ai{xi} ↦→ Σ n i=1 ai
A chain complex (C,∂) is acyclic if, for every n ∈ Z, ker∂n = im∂n+1, thatisto
say, if the sequence
···
��
Cn+1
∂n+1 ��
Cn
∂n ��
Cn−1
��
···
is exact. A positive chain complex (C,∂ ) with augmentation is acyclic if the
sequence
···
��
Cn
∂n ��
···
∂1 ��
C0
ε ��
��
Z
is exact.
Let (C,∂) and (C ′ ,∂ ′ ) be two positive augmented chain complexes. A morphism
f ∈ C((C,∂),(C ′ ,∂ ′ )) is an extension of a homomorphism ¯f : Z → Z if the next
diagram commutes.
···
···
��
C1
f1
��
��
′
C 1
∂1 ��
f0
C0
��
∂ ′ 1 ��
′
C 0
ε ��
��
ε ′
Z
��
��
��
Z
(II.3.6) Theorem. Let (C,∂) and (C ′ ,∂ ′ ) be positive augmented chain complexes;
assume that (C,∂) is free and (C ′ ,∂ ′ ) is acyclic. Then any homomorphism
¯f : Z → Z admits an extension f : (C,∂) → (C ′ ,∂ ′ ), unique up to chain
homotopy.
Proof. Since the augmentation ε ′ : C ′ 0 → Z is surjective, for every basis element x0
of C0, we choose an element of C ′ 0 which is taken onto ¯f ε(x0) by ε ′ ;inthisway,we
obtain a homomorphism f0 : C0 → C ′ 0 such that ¯f ε = ε ′ f0. We now take an arbitrary
basis element x1 of C1; because
¯f