Simplicial Structures in Topology

III.2 Homology of Polyhedra 109
Now the problem is to define Hn(−,Z) on the morphisms of P, inotherwords,
on continuous functions between polyhedra. To solve this problem, we make here
some initial remarks. Two simplicial functions f ,g: K =(X,Φ) → L =(Y,Ψ) are
contiguous if for every σ ∈ Φ there exists τ ∈ Ψ such that f (σ) ⊂ τ and g(σ) ⊂ τ;
in symbols,
(∀σ ∈ Φ)(∃τ ∈ Ψ) f (σ) ⊂ τ , g(σ) ⊂ τ .
Suppose that f and g are contiguous; for each n-simplex σ ∈ Φ (a generator of
Cn(K)), let τ be the smallest simplex of L which contains both simplexes f (σ) and
g(σ). We define the chain complex (S(σ),∂ σ ) as follows:
�
Cn(τ) , n ≥ 0
S(σ)n =
0 , n < 0
(recall that τ is the simplicial complex (τ,℘(τ) � /0)). Since τ is an acyclic simplicial
complex, it follows that S is an acyclic carrier between C(K) and C(L). Moreover,
S is an acyclic carrier of C( f ) −C(g): in fact, if x is a vertex of K such that
f (x) �= g(x), the fact that f and g are contiguous ensures the existence of a simplex
τ of L, which contains both vertices f (x) and g(x); therefore,
(C0( f ) −C0(g))(x) ∈ C0(τ)=S(x)0 .
It is easy to prove that, for any generator σ of C(K),
(C( f ) −C(g))(σ) ⊂ S(σ)
that is to say, S is an acyclic carrier for the chain homomorphism C( f ) −C(g).
(III.2.1) Theorem. If f ,g: K → L are contiguous, Hn( f ,Z) =Hn(g,Z) for every
n ∈ Z.
Proof. We first prove that the chain homomorphism
C( f ) −C(g): C(K) → C(L)
extends the homomorphism 0: Z → Z. Indeed, let x beavertexofK;if f (x)=g(x),
then (C0( f ) −C0(g))(x)=0; otherwise, if f (x) �= g(x),
ε(C0( f ) −C0(g))(x)=ε( f (x) − g(x)) = 0
and so, C( f )−C(g) extends 0 (recall that ε : C0(K) → Z is the augmentation homomorphism).
By Corollary (II.3.10), we conclude that C( f ) −C(g) is homologically
null. We complete the proof with the help of Theorem (II.3.4). �
Theorem (III.1.4) shows that the geometric realizations of a simplicial complex
and one of its barycentric subdivisions are homeomorphic; we now show that the
homology groups of simplicial complexes remain unchanged (up to isomorphism)
under barycentric subdivisions.