Simplicial Structures in Topology

II.3 Homological Algebra 69
The connecting homomorphisms λn are natural in the following sense:
(II.3.3) Theorem. Let
(C,∂) ��
h
��
( ¯C, d) ¯ ��
f
¯f
��
(C ′ ,∂ ′ )
k
��
��
( C ¯′ , d ¯′ )
g
¯g
��
��
(C ′′ ,∂ ′′ )
ℓ
��
��
��
( C ¯′′
, d ¯′′
)
be a commutative diagram of chain complexes in which the horizontal lines are
short exact sequences. Then, for every n ∈ Z, the next diagram commutes.
Hn(C ′′ )
Hn(ℓ)
��
Hn( ¯ C ′′ )
λn ��
¯λn
Hn−1(C)
Hn−1(h)
��
��
Hn−1( ¯C)
The proof of this theorem is easy and is left to the reader.
Let f ,g: (C,∂) → (C ′ ,∂ ′ ) be chain complex morphisms. We say that f and g
are chain homotopic if there is a graded group morphism of degree + 1, s: C → C ′
such that f − g = d ′ s + sd; more precisely
(∀n ∈ Z) fn − gn = ∂ ′ n+1 sn + sn−1∂n .
The morphism s: C → C ′ is a chain homotopy between f and g (or from f to g).
Notice that the chain homotopy relation just defined is an equivalence relation in
the set
C((C,∂),(C ′ ,∂ ′ )) .
In particular, a morphism f ∈ C((C,∂),(C ′ ,∂ ′ )) is chain null-homotopic if there
exists a chain homotopy s such that f = d ′ s + sd (it follows that f and g are chain
homotopic if and only if f − g is chain null-homotopic).
(II.3.4) Proposition. If f ,g ∈ C((C,∂),(C ′ ,∂ ′ )) are chain homotopic, then
(∀n) Hn( f )=Hn(g): HnC → HnC ′ .
Proof. For any cycle z ∈ ZnC,wehavethat
Hn f [z]=[fn(z)] = [gn(z)] + [∂ ′ n+1 sn(z)] + [sn−1∂n(z)] = Hng[z];
we now notice that ∂nz = 0andthat∂ ′ n+1 sn(z) is a boundary and thus, homologous
to zero. �