Colloquium publications.

66

THE CAMBRIDGE COLLOQUIUM.

It is impossible that r2 should be less than a2- 1 because this
would imply that at least two of the columns of E2 were expressible
linearly in terms of the others and hence on reducing modulo 2,
that the same statement was true of the columns of H2, contrary
to ~ 30. Hence there remain two possibilities
r2 = C2 - 1
and
r2 - a2
for any C2 which is a 2-circuit. The examples in the last section
show that both possibilities can be realized.
56. A 2-circuit C2 such that r2 = a2- 1 has the property
that the boundaries of its 2-cells can be converted into oriented
1-circuits in such a way that their sum is zero. For the columns
of E2 represent a set of oriented 1-circuits, one bounding each
2-cell, and since r2 = c2 - 1 they are subject to one linear
relation,
(1) bic ++ b22 + ** + ba2 ca =0
in which the c's represent the columns of E2 and the b's are
positive or negative integers or zero. When reduced modulo 2
this relation must state that the sum of the columns of H2 is zero.
Hence the relation must involve all columns of E2. In case C2
defines a manifold each 1-cell is incident with two and only two
2-cells. Hence if an oriented 1-cell oil is to cancel out, the two
oriented 1-circuits formed from the boundaries of the 2-cells
incident with ail must appear in (1) with numerically equal
coefficients. It follows that the coefficients of (1) are numerically
equal and therefore that by removing a common factor (1) can
be reduced to a form in which bi = ~- 1.
Hence by multiplying some of the columns by - 1, E2 can
be reduced to a form in which the sum of the columns is zero.
The columns of E2 then represent a set of oriented 1-circuits
such that if al is any oriented 1-cell formed from a 1-cell of C2,
one of these 1-circuits contains al and another one contains - 1.
It is obvious in view of ~ 34 that this result applies to all
2-circuits and not merely to those defining manifolds.