Colloquium publications.

ANALYSIS SITUS.

117

first rk+l - Tk columns of Ck. They therefore appear in the
right-hand members of the congruences analogous to (K1) which
are determined by the matrix Ek+l.
The oriented k-circuits in the homologies (K5) correspond to
the last rk columns of Dk and also to the set of columns of Ck
which correspond to the coefficients of torsion tik and thus appear in the right-hand members of the congruences analogous to
(K2) which are determined by the matrix Ek+l.
30. All symbols (x1, x2,..., Xa,) in which the x's are integers
are linearly dependent on ak such symbols. Since the determinant of Dk is unity its columns are a set of linearly independent
symbols (xl, x2, * *, xa) the x's of no one of which have a common factor different from unity. Hence all symbols (xl, x2,., Xa,) are linearly expressible with integral coefficients in
terms of the columns of Dk. Of these, the ones which represent
sets of oriented k-circuits are expressible (~ 16) with integral
coefficients in terms of the last ak- rk columns of Dk. Hence
if k is any oriented k-circuit composed of oriented k-cells of Cn
it satisfies a homology,
C~k-rk
(1) rk ' -' aifrk+i,
i=l
in which the coefficients ai are integers and the rrk+l's are the
oriented k-circuits which appear in the congruences and homologies (K3), (K4), (K5). But by means of the homologies (K4)
and (K5) this reduces to
Pk-1 'k
(2) r k - airrk-i + E bFr ka-ki
i=1 i.=l
in which the ai's are integers and each bi is an integer whose
absolute value is less than the coefficient of torsion, tik.
With a slight change of notation this result may be stated as
follows: There exists a set of Pk - 1 k-circuits rk', Fk2,
rkPk+l and a set of rk sets of k-circuits, rkPk, rkPk+l,..., rkPk+k-t -
such that if k is any oriented k-circuit composed of oriented k-cells
of Cn it satisfies a homology,
Pk-1 'Tk
(3) rk a i airki + birk-'+
i=l i=1