Simplicial Structures in Topology

II.4 Simplicial Homology 87
We now define the homomorphisms
as follows:
j : C(K1 ∨ K2) → C(K1) and h: C(K1 ∨ K2,K1;Z) → C(K1 ∨ K2)
(∀σ 1 n ×{x2 0 }∈Φ1 ×{x 2 0 }) jn(σ 1 n ×{x2 0 })=σ 1 n
(∀{x 1 0}×σ 2 n ∈{x 1 0}×Φ2) jn({x 1 0}×σ 2 n )=0
(∀σ 2 n ∈ Φ2) hn(σ 2 n )={x1 0 }×σ 2 n .
Morphisms i,k, j, andh are induced by simplicial functions and so they commute
with boundary operators. Moreover, ji = 1 C(K1) and kh = 1 C(K2), a property
that extends to the respective homomorphisms regarding homology groups.
Hence, for each q ≥ 1, we have a splitting short exact sequence of homology
groups
Hq(K1;Z) ��
Hq(i)
��
Hq(K1 ∨ K2;Z) Hq(k)
II.4.1 Reduced Homology
��
��
Hq(K2;Z).
It is sometimes an advantage to introduce a little change to the simplicial homology,
named reduced homology; the only difference between the two homologies lies on
the group H0(−;Z). To obtain the reduced homology �H∗(K;Z) of a simplicial
complex K, we consider the chain complex
�C(K,Z)={ �Cn(K), � dn}
where
⎧
⎨Cn(K)
, n ≥ 0
�Cn(K)= Z , n = −1
⎩
0 , n ≤−2
and define the boundary homomorphism
⎧
⎨ ∂n , n ≥ 1
�dn = ε : C0(K) → Z , n = 0
⎩
0 , n ≤−1
where ε is the augmentation homomorphism (see Lemma (II.4.5)). We only need to
verify that � d0 � d1 = 0; but this follows directly from the definition of ε.
We leave to the reader, as an exercise, to prove that if K is a connected simplicial
complex, then
(∀n �= 0) �Hn(K;Z) ∼ = Hn(K;Z)
and
�H0(K;Z) ∼ = 0.
�