Graphics Reference
In-Depth Information
Next, we define the incidence matrices with respect to arbitrary bases
{}
=
{
}
qq
q
q
c
c
,
c
,...,
c
12
n
q
for the free abelian groups C
q
(K). Since we have bases, there are unique integers h
ij
such that
n
q
(
)
=
Â
q
+
1
q
q
∂
c
h
c
.
q
+
1
j
ij
j
i
=
1
Definition.
The
qth incidence matrix
of K
with respect to the bases
{c
q+1
} and {c
q
} is
defined to be the (n
q
¥ n
q+1
)-matrix
+
LL
q
1
c
j
M
M
LL
M
Ê
ˆ
()
=
q
q
Á
Á
q
˜
˜
g
c
h
.
ij
i
ij
Ë
¯
M
Let c be an arbitrary (q + 1)-chain. If we express c with respect to the basis {c
q+1
},
then
n
q
+
Â
1
q
+
1
c
=
a
j
j
j
=
1
for some unique integers a
j
and
n
n
n
Ê
Á
ˆ
˜
q
+
1
q
q
+
1
(
)
=
Â
Â
Â
q
+
1
q
q
()
=
∂
c
a
∂
c
a
h
c
.
q
+
1
j
q
+
1
j
j
ij
i
j
=
1
i
=
1
j
=
1
This shows that the boundary homomorphisms ∂
q
of K, and hence the homology
groups of K, are completely determined once the incidence matrices are known with
respect to some bases. Our goal in the remainder of this section is to show how the
homology groups of K can be computed from knowledge of the basic incidence matri-
ces E
q
of K alone.
7.2.5.2. Lemma.
Choose a basis for each group C
q
(K). The matrix product of any
two successive incidence matrices with respect to any such choice of bases is the zero
matrix. Using the notation above, this means that for all q
(
)(
)
=
q
-
1
q
(
)
hh
ij
nn
¥
1
-matrix of zeros.
q
-
1
q
+
ij
Proof.
This is a straightforward consequence of Lemma 7.2.1.3(2).
