Simplicial Structures in Topology

108 III Homology of Polyhedra
We now define
as follows:
Hn : Mn × I −→ Mn
1. For every σ ∈ Φ �Ψ, the restriction of Hn onto |σ|×I × I coincides with Hσ
2. For every (x,t) ∈ Mn−1 × I, Hn(x,t)=x
The function
rn = Hn(−,1): Mn −→ Mn−1
is a retraction; if dimK = m, then Mm = |K|×I and the composite function
r = r0 ···rm is a retraction of |K|×I onto � |K|. �
III.2 Homology of Polyhedra
We begin this section by recalling that the geometric realizations of a polyhedron
and any one of its barycentric subdivisions are homeomorphic; one of our objectives
is to prove that, given two simplicial complexes K and L, andamap f : |K|→|L|,
there exists a simplicial function from a barycentric subdivision K (r) to L whose
geometric realization is homotopic to the composite of f and the homeomorphism
F : |K (r) |→|K|. Once this “simplicial approximation” of f has been obtained, we
may define a functor
H∗(−,Z): P → Ab Z
from the category of polyhedra and continuous functions to that of graded Abelian
groups. Here is an overview of how this is done. We associate the graded Abelian
group H∗(K,Z) with a polyhedron |K|;themapfinduces a homomorphism, among
the corresponding graded groups, which derives from the “approximation” of f .
With all this, it is not surprising that we may also give the concept of homotopy
among chain morphisms, in parallel to the homotopy of maps among spaces. We
shall see later some consequences of this important result.
Once again we remind the reader that an oriented simplicial complex K defines
a positive free augmented chain complex (C(K),∂ K ) with augmentation homomorphism
ε : C0(K) → Z. In particular, if the simplicial complex K is acyclic (for instance,
K is the simplicial complex σ generated by a simplex σ, a simplicial n-cone,
or an abstract cone), then the chain complex (C(K),∂ K ) is acyclic.
We define the functor
Hn(−,Z): P → Ab Z
on the objects |K|∈P simply as
Hn(|K|,Z)=Hn(K,Z) .