Simplicial Structures in Topology

II.4 Simplicial Homology 77
such that k : K → K ′ and ℓ: L → L ′ is the restriction of k to L. The reader can easily
verify that the relative homology determines a covariant functor
H(−,−;Z): CCsim −→ Ab Z .
The next result, which is an immediate application of the Long Exact Sequence
Theorem (II.3.1), is called Long Exact Homology Sequence Theorem; it relates
the homology groups of L, K, and(K,L) to each other.
(II.4.1) Theorem. Let (K,L) be a pair of simplicial complexes. For every n > 0,
there is a homomorphism
λn : Hn(K,L;Z) → Hn−1(L;Z)
(connecting homomorphism) that causes the following sequence of homology
groups
...→ Hn(L;Z) Hn(i)
−→ Hn(K;Z) q∗(n)
−→ Hn(K,L;Z) λn
−→ Hn−1(L;Z) → ... ,
to be exact; here, Hn(i) is the homomorphism induced by the inclusion i: L → K and
q∗(n) is the homomorphism induced by the quotient homomorphism qn : Cn(K) →
Cn(K)/Cn(L).
Proof. For every n > 0, let
qn : Cn(K) → Cn(K)/Cn(L)
be the quotient homomorphism. With the given definitions, it is easily proved that
and therefore,
(∀n ≥ 0) ∂ K,L
n qn = qn−1∂ K n
q = {qn}: C(K) → C(K,L)
is a homomorphism of chain complexes. We note furthermore that for each n ≥ 0,
the sequence of Abelian groups
Cn(L) ��
Cn(i)
��
Cn(K)
qn ��
��
Cn(K)/Cn(L)
is a short exact sequence and therefore, we have a short exact sequence of chain
complexes
C(L) �
C(i)
�
��
C(K)
q
��
��
C(K,L);
the result follows from Theorem (II.3.1). �
The exact sequence of homology groups described in the statement of Theorem
(II.4.1) is the exact homology sequence of the pair (K,L).