Simplicial Structures in Topology

II.4 Simplicial Homology 79
define simplicial functions
iα : K1 ∩ K2 −→ Kα , jα : Kα −→ K1 ∪ K2 , α = 1,2
which, in turn, define the homomorphisms
ĩ(n): Cn(K1 ∩ K2) → Cn(K1) ⊕Cn(K2)
c ↦→ (Cn(i1)(c),Cn(i2)(c)) ,
˜j(n): Cn(K1) ⊕Cn(K2) → Cn(K1 ∪ K2)
(c,c ′ ) ↦→ Cn( j1)(c) −Cn( j2)(c ′ ).
These homomorphisms have the following properties:
1. ĩ(n) is injective;
2. ˜j(n) is surjective;
3. im ĩ(n)=ker ˜j(n);
4. (∂ K 1
n ⊕ ∂ K 2
5. ˜j(n − 1)(∂ K 1
n ⊕ ∂ K 2
n )ĩ(n)=ĩ(n − 1)∂ K1∩K2 n ;
˜j(n).
n )=∂ K1∪K2 n
In this way, the chain complex sequence
ĩ
0 → C(K1 ∩ K2) −→ C(K1) ⊕C(K2) −→ C(K1 ∪ K2) → 0
is short exact.
Theorem (II.3.1) enables us to state the next theorem, known as Mayer–Vietoris
Theorem:
(II.4.3) Theorem. For every n ∈ Z, there is a homomorphism
λn : Hn(K1 ∪ K2;Z) −→ Hn−1(K1 ∩ K2;Z)
such that the infinite sequence of homology groups
is exact.
...→ Hn(K1 ∩ K2;Z) Hn(ĩ)
−→ Hn(K1;Z) ⊕ Hn(K2;Z)
Hn(˜j)
−→ Hn(K1 ∪ K2;Z) λn
−→ Hn−1(K1 ∩ K2;Z) → ...
We now give some results on the homology of certain simplicial complexes.
Given a simplicial complex K =(X,Φ), we say that two vertices x,y ∈ X are connected
if there is a sequence of 1-simplexes
{{x i 0 ,xi1 }∈Φ,i = 0,...,n}
where x0 0 = x,xn 1 = y,andxi 1 = xi+1
0 ; we then have an equivalence relation on the set
X, braking it down into a union of disjoint subsets X = X1 ⊔ X2 ⊔ ...⊔ Xk. Thesets
˜j