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