Graphics Reference
In-Depth Information
7.2.3
Applications of Homology Theory
Before describing some applications, it is worthwhile to briefly pause and summarize
what we have accomplished so far; otherwise, it is easy to lose sight of the global
picture and get lost in a sequence of lemmas and theorems. The main results can be
summarized by the following:
Fact 1.
For every simplicial complex K and every integer q there is an abelian group
H
q
(K) called the qth homology group of K.
Fact 2.
For every continuous map f :ΩKΩÆΩLΩbetween the underlying spaces of two
simplicial complexes K and L there are a homomorphisms
()
Æ
()
f
:
HKHL
*
q
q
q
whose natural properties are best summarized by the commutative diagram
|L|
g
f , f ¢
f
f ¢
1
|K|
|K|
|K|
|M|
H
q
(L)
=
f
*q
g
*q
(f ¢)
*q
=
1
H
q
(K)
H
q
(M)
H
q
(K)
H
q
(K)
(g
f)
*q
(1
|K|
)
*q
°
The top line in the diagram deals with simplicial complexes and maps and
the bottom lines deal with groups and homomorphisms.
For our purposes, Facts 1 and 2 contain essentially everything that we need to know
about homology groups and induced maps. Many of our applications will follow in a
purely formal way from Facts 1 and 2 with the geometry being irrelevant. Actual defini-
tions are only needed for a few specific computations. There is one caveat though. We
would really like to have well-defined homology groups and induced maps associated to
polyhedra and their continuous maps. Singular homology theory (see Section 7.6)
accomplishes that, but we shall at times pretend that we have this here also. To avoid
such pretense and restore rigor we could pick a fixed triangulation for each polyhedron
and translate continuous maps between them to maps between the underlying spaces of
the simplicial complexes. This would validate our arguments but the messy details
would obscure geometric ideas. By the way, homology theory can be described axiomat-
ically by means of the so-called
Eilenberg-Steenrod axioms
. Facts 1 and 2 correspond to
a subset of these axioms. Statements made in Chapter 6 about algebraic topology asso-
ciating “algebraic invariants” to spaces should make a lot more sense now.
7.2.3.1. Theorem.
Two simplicial complexes K and L with homotopy equivalent
underlying spaces have isomorphic homology groups.
