Colloquium publications.

52

THE CAMBRIDGE COLLOQUIUM.

which is the formula for the characteristic of a complex defining
an open manifold of two dimensions. The same formula holds
for any two-dimensional tree.
Singular Complexes
32. The cells ai0, ajl, ak2 which enter into the definition of a
complex are all non-singular and their boundaries are also nonsingular. This restriction was necessary in order to obtain the
theorem of ~ 6 that the matrices Ho, H1, H2 fully determine the
complex. In many applications, however, it is desirable to drop
the restriction that the boundaries of the cells referred to in the
matrices Hi shall be non-singular. The results of the theory of
matrices can in general be applied whenever it is possible to subdivide the cells having singular boundaries by means of a finite
number of 0-cells and 1-cells in such a way as to obtain a complex
of non-singular cells with non-singular boundaries.
For example, in ~ 21 the anchor ring was defined as consisting
of one 0-cell, represented by the four vertices of the rectangle,
two 1-cells represented by its pairs of opposite edges, and one
2-cell. The matrices of incidence relations of these cells are
0
Ho= I1111, Hi= 0 0|, H2= 0.
Thus po = 1, Pi = 0, P2 = 0, ao = 1, a1 = 2, a2 = 1. Hence
R, = 3 - (ao - a1 + a2) = 3
= ai - P - P2 + 1.
If the rectangle is subdivided into triangles so that a non-singular
complex is obtained it will be found that the same value for R1
will be obtained from the non-singular complex as from the
singular one.
33. The notion of a singular complex on a one-dimensional
complex, as defined in ~ 8, Chap. I, can be generalized directly
to two dimensions as follows:
Let C2 be a two-dimensional complex, C' a generalized complex
of zero, one or two dimensions,* and F a correspondence in which
* The definition may be extended so that C' is of any number of dimensions.