Colloquium publications.

ANALYSIS SITUS.

77

By precisely the same reasoning as that used in the 0- and 1 -dimensional cases (cf. ~ 28, Chap. II) the boundary of a Ck is a
(k - 1)-dimensional circuit or a set of (k - 1)-dimensional
circuits having at most a (k - 2)-dimensional complex in common.
From this reasoning it also follows that every bounding (k - 1)circuit is a sum (mod. 2) of a set of (k - 1)-circuits which
bound k-cells, i.e., which are represented by columns of Hk.
Hence all bounding (k - 1)-circuits are linearly expressible in
terms of those corresponding to a linearly independent set of
Pk columns of Hk, where Pk is the rank of Hk.
8. As in the 0-, 1-, and 2-dimensional cases (cf. ~ 24, Chap. 1),
7rilkX +i2k + + i2aX2 + '* + kXak
is 1 or 0 according as there are an odd or an even number of k-cells
incident with the (k - 1)-cell aik-. Hence if
X11 Yi
X2 i Vy2
(1) HI.. -
Yak-1
(Y1, y2, ''', ya,-) represents the boundary of (x1, x2, Xak).
As a corollary it follows that the k-circuits are the solutions of
the equations
(Hk) ijk= 0 (i= 1, 2, * *, ak-1)
j=1
Since the columns of the matrix Hk represent (k - 1)-circuits they
represent solutions of the equations
Oak-1
(H -i) E k-~1X = 0 (i = 1, 2,.., ak-2)
j=l
and hence
(2) Hk-1_Hk = 0 (k = 1, 2,..., n).
The Connectivities Ri
9. If pk denotes the rank of Hk (mod. 2) the number of solutions of the linear homogeneous equations (Hk) in a complete