Simplicial Structures in Topology

72 II Simplicial Complexes
(II.3.7) Corollary. Let (C,∂) and (C ′ ,∂ ′ ) be two positive augmented chain complexes;
assume (C,∂) to be free and (C ′ ,∂ ′ ) to be acyclic. If f : (C,∂ ) → (C ′ ,∂ ′ ) is
an extension of the trivial homomorphism 0:Z → Z, then f is chain null-homotopic.
The next result gives a good criterion to check if a positive, free, augmented
chain complex is acyclic.
(II.3.8) Lemma. Let (C,∂) be a positive, free, augmented chain complex with augmentation
homomorphism
ε : C0 −→ Z.
Then, (C,∂) is acyclic if and only if the following conditions hold true:
I. There exists a function η : Z → C0 such that εη = 1.
II. There exists a chain homotopy s: C → C such that
1. ∂1s0 = 1 − ηε,
2. (∀n ≥ 1) ∂n+1sn + sn−1∂n = 1.
Proof. Suppose that C is acyclic. Since ε : C0 → Z is a surjection, there exists
x ∈ C0 such that ε(x)=1. Define
η : Z → C0 , n ↦→ nx.
Clearly εη = 1.
The homomorphisms 1: Z → Z and 0: Z → Z can be extended trivially to the
chain homomorphisms 1,0: (C,∂) → (C,∂); because of Theorem (II.3.6), there
exists a chain homotopy s: C → C satisfying conditions 1 and 2.
Conversely, assume that there exists a chain homotopy s: C → C ′ , and a
homomrophism η with properties 1. and 2.; because of Noether’s Homomorphism
Theorem,
C0/kerε ∼ = Z;
since ε∂1 = 0, we have that im∂1 ⊂ kerε and, from ∂1s0 = 1−ηε, we conclude that
every x ∈ C0 can be written as
x = ∂1s0(x)+ηε(x).
Hence for every x ∈ kerε,wehavethatx = ∂1s0(x), that is to say, kerε ⊂ im∂1,and
therefore
H0(C) ∼ = Z.
Finally, because
∂n+1sn + sn−1∂n = 1
in all positive dimensions, it follows that Hn(C)=0foreveryn > 0. Thus, (C,∂) is
acyclic. �
Theorem (II.3.6) and Corollary (II.3.7) require (C ′ ,∂ ′ ) to be acyclic; this requirement
can be replaced by a more interesting condition within the framework of the socalled
acyclic carriers. The following definition is needed: given (C,∂),(C ′ ,∂ ′ )∈ C,