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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
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