Colloquium publications.

ANALYSIS SITUS.

63

ular subdivision C2 as in ~ 40 and construct the spheres S2P
as in ~ 44. By ~ 44 the result of adding the spheres S2P to K2
(mod. 2) is either zero or a 2-circuit composed of cells of C2.
If it were zero the 2-circuit K2 would be the sum (mod. 2) of
the spheres S2P. But this is impossible, as shown by the following
theorem.
51. There is no set of 2-circuits K2i on a 2-circuit C2 such that
(1) for each 2-circuit K2. there is a 2-cell of C2 on which there is no
point of K2i and (2) the sum (mod. 2) of the 2-circuits K2i is C2.
To prove this theorem, we suppose that there is a set of
2-circuits K2i having the property (1). We let these 2-circuits
take the place of K in ~ 40, make the regular subdivision of C2
into C2 and K2i into K2i, construct a correspondence A and
obtain a set of spheres S2p (which, of course, must not be confused with those in ~ 50). When the spheres having 2-cells in
common with one of the 2-circuits K2g are added to this K2i the
result is either zero or a non-singular 2-circuit composed of cells
of C2. But since C2 is a 2-circuit the only 2-circuit composed of
its cells is C2 itself. Since there is one 2-cell of C2 which contains
no point of K2i it follows that the sum of K2 and the spheres
S2P determined by its 2-cells is zero.
Obviously if each of two 2-circuits is such that the sum (mod. 2)
of it and the spheres S2P determined by its 2-cells is zero the same
is true of the sum (mod. 2) of the two 2-circuits. Hence the
sum of all the 2-circuits K2i has this property. On the other
hand the 2-circuit C2 is such that the sum of the spheres S2P
determined by its 2-cells is C2 itself. Hence the 2-circuits K2
do not have the property (2).
52. Letting the 2-circuit K2 and the spheres S2P of ~ 50 take
the place of the 2-circuit C2 and the 2-circuits K2i of ~ 51 it
follows from the theorem of ~ 51 that K2 is not the sum (mod. 2)
of the spheres S2P. Hence the sum (mod._2) of K2 and the
spheres S2P is a 2-circuit composed of cells of C2. If this 2-circuit
is C2 itself the theorem of ~ 50 is verified. If not, let this 2 -circuit be denoted by C2', let cj2 be one of the 2-cells of C2 which
is not on C2', and let K2 be regularly subdivided into a complex
K2' which has at least one 2-cell, which is interior to cj2.