Colloquium publications.

ANALYSIS SITUS.

43

process of subdivision shall be the mid-point of the 1-cell, that
each point interior to a triangular 2-cell (the point Pk2 of ~ 13)
shall be the center of gravity (intersection point of the medians)
of the triangle, and that the 1-cells introduced shall be straight.
With these conventions, it is evident that for every number
6 > 0 there exists a number Ns such that if n > Ns every 1-cell
in C2' is of length less than 6.
16. The relationship between C2 and C2 may be stated as
follows: (1) each 2-cell ak2 of C2 is the sum (mod. 2) of all 2-cells
Pi0PjPk2 of C2 incident with Pk2; (2) each 1-cell ajl of C2 is the
sum of the two 1-cells PjiPf of C2 incident with Pjl; and (3) each
0-cell a? of C2 is the vertex PiO of C2.
Hence the complex C2 may be converted into C2 by a series of
steps of two sorts; (1) combine two 2-cells whose boundaries have
one and only one 1-cell in common into a new 2-cell, suppressing
the common 1-cell and (2) combine two 1-cells both incident
with a 0-cell which is not incident with any other 1-cell into a new
1-cell, suppressing the common 0-cell.
The first type of step requires that the matrix H2 of C2 be
modified by adding the column representing one of the two 2-cells
to the one representing the other, removing the column representing the first of the two 2-cells, and also removing the row
corresponding to the 1-cell which is suppressed. The row which
is removed contained only two l's before the two columns were
added, because the 1-cell to which it corresponds is incident with
only two 2-cells. After the one column is added to the other
this row contains only one 1 and this 1 is common to the row and
column removed. Hence the first type of step has the effect of
reducing the rank of H2 by 1.
It also has the effect of removing the column of H1 corresponding to the 1-cell suppressed. This 1-cell is on the boundary of a
2-cell. Hence the 0-circuit represented by the column removed
is linearly dependent on the columns corresponding to the other
1-cells of the boundary of this 2-cell. Hence the removal of this
column leaves the rank of Hi unaltered.
The first type of step thus changes p2 and pi into P2 - 1 and