Graphics Reference
In-Depth Information
Assertion A
k
:
It is possible to choose bases for the groups C
q
(K) so that with
respect to these bases the incidence matrices are
0
*
-
k
1
k
k
+
1
n
-
1
NNNE
,...,
,
,
,...,
E
.
*
(All other incidence matrices are uninteresting since they are zero.)
Note the following important property of the matrices N
k
in assertion A
k
that
follows easily from the fact that, by Lemma 7.2.5.2, the product N
k-1
N
k
is the zero
matrix.
The last g
k-1
rows of the matrix N
k
in assertion A
k
are zero.
Claim 3.
We shall use induction on k to prove the assertions A
k
above. Assertion A
1
follows
from Claims 1 and 2. These claims show that E
0
can be transformed into the nor-
malized form N
*
by changing the basis of C
0
(K) and C
1
(K) appropriately. Although
the 1st incidence matrix may have changed, the ith incidence matrices for i ≥ 2 have
not. One can check that the proof actually shows that d
i
= 1, for 1 £ i £g
0
.
Assume inductively that assertion A
k
is true for some k > 0. One can show that N
k
can be transformed into a normalized form such as is required for Theorem 7.2.5.3
by a sequence of matrix operations of the type described in Claim 1. This fact is a
special case of a normalization theorem for matrices that is not hard but too long to
reproduce here. A proof can, for example, be found in [Cair68].
Using Claim 3 we may assume that only the first n
k
-g
k-1
rows of N
k
will be affected
as we transform the matrix to its normalized form. Translating the changes we make
in the matrix into the corresponding changes in the basis for C
k
(K) and their effect
on N
k-1
, we can easily see that only the first n
k
-g
k-1
columns of N
k-1
are manipu-
lated. Since these consist entirely of zeros, the matrix N
k-1
is left unchanged. Of
course, the matrix E
k+1
will certainly have changed, but we have established assertion
A
k+1
. By induction, assertion A
k
is true for all k > 0. Assertion A
n
proves Theorem
7.2.5.3.
A detailed version of the qth incidence matrix (7.3) of Theorem 7.2.5.3:
Recall
that the rows and columns of the incidence matrices are indexed by the chains in the
chosen basis of the appropriate chain groups of K. Therefore, assume that 0 £ q £ n
and that the basis elements of C
q
(K) corresponding to the rows of N
*
have been labeled
as follows:
The first g
q
basis elements are labeled as A
i
's (note that g
-1
=g
n
= 0),
the last g
q-1
are labeled as C
i
's,
the remaining b=n
q
-g
q
-g
q-1
basis elements, if there are any, are labeled as B
i
's,
and
if the d
i
are the elements shown in the normalized matrix (7.3), then the integer
r
q
is defined by
(
{
}
)
q
r
=
max
0
,
i
d
>
1
.
q
i
With this notation, matrix (7.3) can be rewritten as