Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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108 III Homology of Polyhedra<br />
We now def<strong>in</strong>e<br />
as follows:<br />
Hn : Mn × I −→ Mn<br />
1. For every σ ∈ Φ �Ψ, the restriction of Hn onto |σ|×I × I co<strong>in</strong>cides with Hσ<br />
2. For every (x,t) ∈ Mn−1 × I, Hn(x,t)=x<br />
The function<br />
rn = Hn(−,1): Mn −→ Mn−1<br />
is a retraction; if dimK = m, then Mm = |K|×I and the composite function<br />
r = r0 ···rm is a retraction of |K|×I onto � |K|. �<br />
III.2 Homology of Polyhedra<br />
We beg<strong>in</strong> this section by recall<strong>in</strong>g that the geometric realizations of a polyhedron<br />
and any one of its barycentric subdivisions are homeomorphic; one of our objectives<br />
is to prove that, given two simplicial complexes K and L, andamap f : |K|→|L|,<br />
there exists a simplicial function from a barycentric subdivision K (r) to L whose<br />
geometric realization is homotopic to the composite of f and the homeomorphism<br />
F : |K (r) |→|K|. Once this “simplicial approximation” of f has been obta<strong>in</strong>ed, we<br />
may def<strong>in</strong>e a functor<br />
H∗(−,Z): P → Ab Z<br />
from the category of polyhedra and cont<strong>in</strong>uous functions to that of graded Abelian<br />
groups. Here is an overview of how this is done. We associate the graded Abelian<br />
group H∗(K,Z) with a polyhedron |K|;themapf<strong>in</strong>duces a homomorphism, among<br />
the correspond<strong>in</strong>g graded groups, which derives from the “approximation” of f .<br />
With all this, it is not surpris<strong>in</strong>g that we may also give the concept of homotopy<br />
among cha<strong>in</strong> morphisms, <strong>in</strong> parallel to the homotopy of maps among spaces. We<br />
shall see later some consequences of this important result.<br />
Once aga<strong>in</strong> we rem<strong>in</strong>d the reader that an oriented simplicial complex K def<strong>in</strong>es<br />
a positive free augmented cha<strong>in</strong> complex (C(K),∂ K ) with augmentation homomorphism<br />
ε : C0(K) → Z. In particular, if the simplicial complex K is acyclic (for <strong>in</strong>stance,<br />
K is the simplicial complex σ generated by a simplex σ, a simplicial n-cone,<br />
or an abstract cone), then the cha<strong>in</strong> complex (C(K),∂ K ) is acyclic.<br />
We def<strong>in</strong>e the functor<br />
Hn(−,Z): P → Ab Z<br />
on the objects |K|∈P simply as<br />
Hn(|K|,Z)=Hn(K,Z) .