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Observe that, for s < t, each term [
v
0
···
ˆ
s
···
ˆ
t
···
v
q
] appears twice in the sum above.
It appears once with coefficient (-1)
i+j-1
when i = s and j = t, and a second time with
coefficient (-1)
i+j
when j = s and i = t. It follows that the terms in the sum cancel pair-
wise, so that the sum is zero. The lemma is proved.
We shall see that the second part of Lemma 7.2.1.3 is fundamental to the whole
theory of homology groups. Here are some more basic definitions.
Definition.
Let c ΠC
q
(K). If ∂
q
(c) = 0, then we shall call c a
q-cycle
of K. If c =∂
q+1
(d)
for some d ΠC
q+1
(K), then we shall call c a
q-boundary
of K. The set of q-cycles and
q-boundaries of K will be denoted by Z
q
(K) and B
q
(K), respectively.
Clearly,
()
=
()
=
Z
K
ker
∂
and
B
K
im
∂
,
q
q
q
q
+
1
so that we are dealing with subgroups of C
q
(K). Furthermore, by Lemma 7.2.1.3(2),
the group B
q
(K) is actually a subgroup of Z
q
(K). It follows that we have inclusions
()
Ã
()
Ã
()
BK ZK CK
q
q
q
and it makes sense to talk about the quotient group of q-cycles modulo q-boundaries.
Definition.
The
q-th homology group
of K, H
q
(K), is defined by
()
()
.
ZK
BK
q
q
()
=
HK
q
As in the case of C
q
(K), it is notationally convenient to have the groups B
q
(K),
Z
q
(K), and H
q
(K) defined for all values of q. Of course, only values of q satisfying 0 £
q £ dim K are interesting. Negative values of q will in particular always be ignored in
computations.
With the definition of the homology groups we have arrived at some important
algebraic invariants for polyhedra, although we shall have to establish quite a few
other facts before we will be ready to prove this. We could have used other regular
figures, such as q-dimensional cubes, as building blocks for spaces to define these
groups but there are at least two reasons for the choice of simplices, namely, with
other figures both the orientation and the important maps ∂
q
would have been more
complicated to define.
7.2.1.4. Example.
To compute the homology groups of K = {
v
0
}.
Solution.
Clearly,
q
()
=
C
K
0
,
if q
>
0
,
()
=
()
=
C
K
Zv
,
and
0
0
BK
0
,
0
so that
