Colloquium publications.

94

THE CAMBRIDGE COLLOQUIUM.

are the same as those among the O-cells, 1-cells and 2-cells of a
two-dimensional manifold. Hence a complex C3"' can be defined
whose cells coincide with those of C3" and which satisfies the
definition of a generalized manifold.
C3.' will be a manifold in the narrow sense only in the case
where each of the groups associated with each vertex ak~ has
the incidence relations of the cells of a sphere.
34. Since the boundary of any complex consists of one or more
circuits, it consists of one or more generalized manifolds any or
all of which may be singular.
Bounding and Non-bounding k-circuits
35. Let us now take up the problem: Given a k-circuit, Ck on
a complex Cn, to determine whether or not there exists a (k + 1)dimensional complex, singular or not, on Cn which is bounded by Ck.
This is the problem solved in Chap. II (cf. ~ 35) for the case
where n = 2 and k = 1. As the problem is now formulated k
may be less than, equal to, or greater than n, and Ck may have
singularities of any degree of complexity compatible with the
definition in ~ 3.
The solution of the problem in the simplest case is contained
in the following obvious theorem which is a direct generalization
of that given in ~ 36, Chap. II: Any sphere of k dimensions on an
n-cell an is the boundary of a (k + 1)-cell on an. The (k + 1)cell can be constructed by joining an arbitrary point, P, of an
to all the points of the k-dimensional sphere by straight 1-cells
or, in case of points of the sphere which coincide with P, by
singular 1-cells coincident with P. The solution of our problem
for the general case which we shall now develop is entirely parallel
to that carried out in ~~ 39 to 46, Chap. II.
36. Let Ki be an i-dimensional complex on Cn. Let Cn be a
regular sub-division of Cn. Let a definition of distance and
straightness be introduced relative to Cn and let all references to
distance and straightness in the rest of this argument be understood to refer to this definition. Let Cn be a regular subdivision of Cn. By simple continuity considerations it can be