Colloquium publications.

60

THE CAMBRIDGE COLLOQUIUM.

determine three 2-cells bi2, bj2, b12. These 2-cells are incident
by pairs with the 1-cells joining the three vertices of kp2 to their
correspondents under the correspondence A. The vertices of C2
to which the vertices of kp2 correspond are either the three vertices of a 2-cell Cq2 of C2 or the two ends of a 1-cell of C2 or a
single 0-cell of C2. In the first case the 2-cells, kp2, bi2, bj2, b12
and Cq2 are the 2-cells of a sphere; in the second and third cases
the 2-cells kp2, b?2, bj2, and b12 are the 2-cells of a sphere. Let the
sphere which is thus in every case determined by kp2 be denoted
by S2P
A 2-cell bi2 is in an odd number of these spheres if and only if
it is incident with a 1-cell kil of the boundary of K2. Hence the
result of adding the spheres S2P to K2 (mod. 2) is either zero or a
complex K2' the 2-cells of which are either 2-cells of C2 or 2-cells
bi2 determined by the 1-cells of the boundary of K2. In particular, if K2 is a 2-circuit, either K2 is the sum (mod. 2) of the
spheres S2P or K2' is composed entirely of cells of C2.
45. If K2 has a boundary, so that
(4) K2 K1 (mod. 2),
the result of the last section is that by adding a number of
congruences,
(5) S2P = 0 (mod. 2),
to (4) we obtain a congruence,
(6) K2'= K1 (mod. 2),
such that all 2-cells of K2' are either 2-cells of C2 or 2-cells bi2
determined by the boundary K1 of K2'. The complex B2" composed of the latter 2-cells and their boundaries is such that
(7) B2" - K1 + K1" (mod. 2)
where K1" is composed of 0-cells and 1-cells of C2. On adding
(6) and (7) we obtain a congruence
(8) K2' + B2" K1" (mod. 2)
in which the left-hand member represents a complex composed
only of cells of C2.