Colloquium publications.

ANALYSIS SITUS.

81

Pk-1 + Pk-1 - 1 columns of Bk-i. Let this be done for all
values of k from 1 to n.
14. By this process it is brought about that the matrices Ak
are identical with the matrices Bk except for a permutation of
columns. The columns of each matrix Bk fall into three blocks.
The first pk columns represent k-dimensional complexes bounded
by sets of (k - 1)-circuits. Each of the next Rk - 1 columns
represents a non-bounding k-circuit. The last pk-1 columns
represent sets of bounding k-circuits. Thus the reduction of the
incidence matrices to normal form affords an explicit method of
determining the bounding and non-bounding circuits of all
dimensionalities.
Congruences and Homologies, Modulo 2
15. The definition of congruences and homologies modulo 2
which was made in ~.~ 37, 38, Chap. II, applies without change to
the n-dimensional case. Thus
(1) Ck -Ck- (mod. 2)
means that Ck-_ is the boundary of Ck; and with reference to a
complex Cn
(2) Ck-i - 0 (mod. 2)
means that there exists a complex Ck on Cn which satisfies the
congruence (1). The remarks about linear combination of congruences and complexes made in Chap. II apply here without
change.
All the relations stated above by means of the matrices Hk
can also be expressed in terms of congruences and homologies.
For if we let ak (j= 1, 2,..., ak; k = 1, 2, *, n) represent
the cell ajk and its boundary, instead of the cell alone as in the
notation heretofore used, we have the congruences*
ak-1
(3) ak -- ikaik-1 (mod. 2)
i=I
* We are here making the obvious convention that 7 aik-1 = aik-l if 77 = 1
and vaik-1 = 0 if X/ = 0.