Simplicial Structures in Topology

II.4 Simplicial Homology 81
� �
k
k
∑ gi{xi}− ∑ gi {x} = ∂1(c1).
i=0
i=0
Therefore, it is clear that c0 ∈ ker ε implies c0 ∈ B0(K).
3 ⇒ 2: Given two homological 0-cycles z0 and z ′ 0 , it follows from the property
z0 − z ′ 0 ∈ B0(K) that ε(z0)=ε(z ′ 0 ) and so we may define the homomorphism
θ : H0(K;Z) → Z , z0 + B0(K) ↦→ ε(z0)
which is easily seen (by hypothesis 3) to be injective. The surjectivity of θ follows
immediately; in fact, for every g ∈ Z, wehaveθ(g{x} + B0(K)) = g, wherex ∈ X
is a fixed vertex.
2 ⇒ 1: Let K = K1 ⊔ K2 ⊔ ...⊔ Kk be the decomposition of K into its connected
components. We obtain
H0(K;Z) �
k
∑
i=1
H0(Ki;Z) �
from the given definitions and from what we have proved so far; however, since
H0(K;Z) � Z, wemusthavek = 1, which means that K is connected. �
The next three examples are examples of abstract simplicial complexes called
acyclic because they induce chain complexes which are acyclic (see Sect. II.3).
Homology of σ –Letσbe the simplicial complex generated by a simplex σ =
{x0,x1,...,xn}. Sinceσis connected, Lemma (II.4.5) ensures that H0(σ,Z) =Z.
We wish to prove that Hi(σ,Z)=0foreveryi > 0. With this in mind, we begin to
order the set of vertices, assuming that x0 is the first element. Then, for any integer
0 < j < n and any ordered simplex {xi0 ,...,xi j },wedefine
k j({xi0 ,...,xi
�
{x0,xi0
j } =
,...,xi j } for i0 > 0
0 fori0 = 0
and linearly extend it to all j-chain of σ and therefore, to a homomorphism
k j : Cj(σ) −→ Cj+1(σ).
A simple computation (on the simplexes of σ) shows that for every chain c ∈ Cj(σ)
∂ j+1k j(c)+k j−1∂ j(c)=c
and so any z j ∈ Z j(σ) is a boundary, that is to say, Hj(σ,Z) ∼ = 0. Regarding
Hn(σ,Z), we note that σ, being the only n-simplex of σ, cannot be a cycle; consequently,
Zn(σ) ∼ = 0.
Homology of a simplicial cone –Sinceσ = {x0,x1,...,xn+1}, we call the sim-
plicial complex
k
∑
i=1
C(σ)= •
σ � {x1,...,xn+1},
Z