Colloquium publications.

ANALYSIS SITUS.

11

With this choice of notation, the sets of vertices of Cil, C12,
*.., CR~, respectively, are represented by the symbols (xi, x2,
*, x0a) which constitute the rows of the following matrix.
mi m2 - mi ao - mRo —
1 1... 1 0 0 *.. 0. 0 0. * 0
0 0 11. * 1. ** 0 0.** 0
Ho = I. - || ijl ~
00... 0 00... 0 *.. 11 *.. 1
For most purposes it is sufficient to limit attention to connected
complexes. In such cases Ro = 1, and Ho consists of one row all
of whose elements are 1.
17. By the definition in ~ 5 a 0-cell is incident with a 1-cell
if it is one of the ends of the 1-cell, and under the same conditions
the 1-cell is incident with the 0-cell. The incidence relations
between the 0-cells and 1-cells may be represented in a table or
matrix of ao rows and a1 columns as follows: The 0-cells of C1
having been denoted by ag~, (i = 1, 2, *, ao) and the 1-cells by
aj, (j = 1, 2,..., a), let the element of the ith row and the jth
column of the matrix be 1 if ai~ is incident with aj1 and let it be
0 if ai~ is not incident with aj1.
For example, the table for the linear graph of fig. 1 formed by
the vertices and edges of a tetrahedron is as follows:
al1 a21 a31 a41 a51 a61
ai 1 0 0 0 1 1
a2 0 1 0 1 0 1
a3 0 0 1 1 0
a40 1 1 1 0 0 0
In the case of the complex used in ~ 5 to define a simple closed
curve the incidence matrix is
1 1
1 1 '