Simplicial Structures in Topology

76 II Simplicial Complexes
If we do the alternate sum of the equalities (3) together with s(0)−β(0)=b(0),we
obtain
p
∑
n=0
(−1) n (s(n) − β(n)) = ±β (p);
since Cp+1(K)=0, we have that β(p)=0 and so the equality
holds true. The number
p
∑
n=0
(−1) n s(n)=
χ(K)=
p
∑
n=0
p
∑
n=0
(−1) n β(n)
(−1) n β (n)
is the Euler-Poincaré characteristic of K; this may be useful in determining the
homology of some finite simplicial complexes.
Let L be a simplicial subcomplex of a simplicial complex K; we now ask whether
it is possible to compare the homology of a subcomplex L ⊂ K with the homology of
K. The (positive) answer lies with the exact homology sequence of the pair (K,L).
Let us see how we may find this exact sequence. For every n ≥ 0, consider the
quotient of the chain groups Cn(K)/Cn(L) and define
by
∂ K,L
n : Cn(K)/Cn(L) → Cn−1(K)/Cn−1(L)
∂ K,L
n (c +Cn(L)) = (∂ K n (c)) +Cn−1(L).
This is a well-defined formula because, if c ′ is another representative of c +Cn(L),
then, c − c ′ ∈ Cn(L) and
∂ K n (c − c ′ )=∂ L n (c − c ′ ) ∈ Cn−1(L) ;
hence, ∂ K,L
n (c +Cn(L)) = ∂ K,L
n (c ′ +Cn(L)). The reader can easily verify that the
homomorphisms ∂ K,L
n are boundary homomorphisms and so that
C(K,L)={Cn(K)/Cn(L),∂ K,L
n }
is a chain complex whose homology groups Hn(K,L;Z) are the so-called relative
homology groups of the pair (K,L). We point out that
where
Hn(K,L;Z)=Zn(K,L)/Bn(K,L)
Zn(K,L)=ker∂ K,L
n
and Bn(K,L)=im∂ K,L
n+1 .
Let CCsim be the category whose objects are pairs (K,L),whereK is a simplicial
complex, L is one of its subcomplexes, and whose morphisms are pairs of simplicial
functions
(k,ℓ): (K,L) −→ (K ′ ,L ′ )