Colloquium publications.

114

THE CAMBRIDGE COLLOQUIUM.

Congruences and Homologies
25. The results obtained from the reduction of the matrices E k
to normal form will perhaps be clearer if they are restated in
terms of another notation. Following Poincare, we shall say
that an oriented n-dimensional complex, r,, is congruent to a
set of oriented (n- 1)-circuits, rn,_, if nP,_ is the boundary of F,
and shall denote this relation by the symbols
(1) TFn = rn-1
In case Fn has no boundary (i.e., is a set of n-circuits) Fn is said
to be congruent to zero, and this is indicated by
(2) r - 0.
The expressions (1) and (2) are called congruences and (2) is
regarded as a special case of (1).
From ~ 10 it is evident that the sum of the left hand members
of the two congruences is congruent to the sum of the right
hand members. Moreover if both members of a congruence
are multiplied by an integer, m, the resulting congruence,
(3) mr - mrn-_
has a meaning and is a consequence of (1) if we understand that
mrn is an oriented complex which covers rn m times. If we
understand that - rn stands for the oriented complex obtained
from Fn by reversing the orientation of each of its cells, this
statement can be extended to cover the cases in which m is
negative. Hence any congruence derived from a set of valid
congruences of the same dimensionality by forming a linear homogeneous combination of them with integral coefficients is a valid
congruence.
26. Whenever the congruence
(4) r k = rk-1
is satisfied by an oriented complex rk on Cn, rk-1 is said to be
homologous to zero,
(5) rk-1 ~ 0.
The relation
rn-1 - rn-1' ' 0