Colloquium publications.

36

THE CAMBRIDGE COLLOQUIUM.

al2 a22 a32 a42
al1 0 1 0
a2l1 0 1 0
a31 1 1 0 0
a41 1 0 0 1
a510 1 0 1
a610 0 1 1
5. Since each column of H2 contains ai elements it may be
regarded as a symbol (x1, x2, * *, Xa) in the sense of ~ 15, Chap. I
for a set of 1-cells. The jth column of H2 is, in fact the symbol
for the 1-cells on the boundary of the 2-cell aj2. It is therefore
the symbol for a 1-circuit. Hence the columns of H2 are solutions
of the equations (Hi). That is to say
al
'ijjk2 = 0 (i = 1,,, k = 1,..., a2)
j=1
or, in terms of the multiplication of matrices,
(1) H1iH2 = 0,
where 0 stands for the matrix all of whose elements are zero.
It should be recalled here that we have already proved in
~ 28, Chap. 1 that
Ho -H1 = 0.
The ranks of the matrices Ho, Hi, H2, computed modulo 2,
will be denoted by po, pi, P2 respectively.
6. From the point of view of Analysis Situs a two-dimensional
complex is fully described by the three matrices Ho, H1, H2 for
there is no difficulty in proving that if two two-dimensional
complexes have the same matrices there is a (1 - 1) continuous
correspondence between them. Our definitions are such that the
boundary of every 1-cell is a pair of distinct points and the
boundary of every 2-cell a non-singular curve. Hence a figure
composed of a 1-cell incident with a 0-cell or a 2-cell is in (1 - 1)
continuous correspondence with any other such figure.
If two complexes C2 and C2 have the same matrices their
0-cells, 1-cells and 2-cells may be denoted by ai0, a/l, ak2 and