Colloquium publications.

ANALYSIS SITUS.

115

is also written
(6) ri - i rn1
and expressed in words by saying that rn-1 is homologous to
rn-1'. Since a homology can always be reduced to a congruence
it follows that homologies can be combined linearly according
to the rules that hold for the linear combination of congruences.
Since the boundary of an oriented k-dimensional complex is a
set of oriented k-circuits, the homology (5) implies the congruence
(7) rk —i 0.
It should be noted that these definitions do not permit the
operation of dividing the terms of a homology by an integer
which is a common factor of the coefficients. In other words,
(8) prk-1 0
does not necessarily imply (7). Thus we are dealing with what
Poincare calls " homologies without division."
The Fundamental Congruences and Homologies
27. The relations between the k-cells and the (k - 1)-cells
given by the matrix Ek are equivalent to the system of congruences
Ogk —1
i=1
The matrix of this system of congruences is obtained from Ek by
interchanging rows and columns. The symbol (xI, x2, '* *x, Xak)
was used in ~ 9 to denote an oriented k-dimensional complex
XlOlk + X2O2k + * * + XakOakk
Hence any matrix equation,
(9) Ek- xi p yi
x2 Y2
x<^k \ yaki1