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In-Depth Information
One can see from the examples above that homology groups of a simplicial com-
plex K do give us important information about the holes in the underlying space ΩKΩ.
A point, which has no holes at all, had zero homology groups above dimension zero.
In Examples 7.2.1.5 and 7.2.1.7 the generators of H
1
(K) were associated in a natural
way with one-dimensional holes. The two-dimensional hole in the sphere and the
torus corresponded to the generator of H
2
(K). The fact that the group H
0
(K) was non-
zero in all of our examples (it was isomorphic to
Z
) also makes sense if one recalls
that 0-dimensional holes intuitively correspond to imbedded 0-spheres and
S
0
con-
sists of two points. Exercise 7.2.1.2 asks you to prove that the rank of H
0
(K) equals
the number of connected components of ΩKΩ. The spaces in our examples were all con-
nected and had a single connected component.
Working through our examples should also have brought out another point, namely,
the
algebraic
nature of homology theory. Although our original motivation was to
detect
geometric
“spherical” holes (perhaps even imbedded spheres), the “homological
holes,” or k-cycles, are more general. For example, the results in the case of the projec-
tive plane may have been somewhat unexpected for someone new to homology (twice
the generator of H
1
(K) was zero and there was no 2-cycle) but they all make sense once
one understands this algebraic nature of homology a little better. A k-cycle for a simpli-
cial complex cannot always be represented as an imbedded k-sphere in ΩKΩ and an
imbedded k-sphere in ΩKΩwhose corresponding k-cycle is homologous to zero does not
necessarily bound a (k + 1)-disk in ΩKΩ. The general question of when k-cycles can be rep-
resented by imbedded k-spheres and when imbedded k-spheres bound (k + 1)-disks is
extremely interesting but often difficult to answer, even for manifolds. For example, it is
already nontrivial to determine those 1-cycles that can be represented by imbedded
circles in the simple case of the torus. The fact is that homology theory is really associ-
ated to
abstract
simplicial complexes because all that it needs is an appropriate opera-
tor ∂
q
on linear combinations of formal symbols of the form [
v
0
v
1
···
v
q
]. We may have
used some geometric intuition to motivate our proofs, but the proofs themselves were
independent of it. This is the reason that the study of topology that deals with simplicial
complexes is called combinatorial topology. We shall see in Section 7.2.5 how a com-
puter can compute homology groups. However we look at it though, we should be
excited by the prospect that we have a theory that detects geometric invariants.
We move on and introduce some more standard terminology. Let K be a simpli-
cial complex. The group C
q
(K) is by definition a finitely generated free abelian group
with the q-simplices of K forming a set of generators. Since subgroups and quotient
groups of finitely generated abelian groups are again finitely generated, we conclude
that H
q
(K) is finitely generated. It follows from the fundamental theorem about such
groups (Theorem B.5.7) that
()
ª≈
HK F T
q
q
q
where F
q
is a free group and T
q
is the torsion subgroup of H
q
(K).
Definition.
The rank of F
q
, which is also the rank of H
q
(K), is called the
qth Betti
number
of K and will be denoted by b
q
(K). The torsion coefficients of T
q
are called
the
qth torsion coefficients
of K.
Clearly, once one knows the Betti numbers and torsion coefficients, one knows
the homology groups. The Betti numbers and torsion coefficients for Examples
7.2.1.4-8 above are easily determined.
