Colloquium publications.

ANALYSIS SITUS.

69

linear relation involving the oriented 1-circuits of K2 and one
analogous linear relation for each of the spheres S2P. Since each
S2P has just one 2-cell in common with K2, the linear relations
corresponding to the spheres S2P can be multiplied by i= 1 and
added to the linear relation corresponding to K2 in such a way
that all terms involving oriented 1-circuits of K2 cancel out, thus
giving a linear relation, R, among oriented 1-circuits bounding
2-cells of the spheres S2p which does not involve any oriented
1-circuit bounding a 2-cell of K2.
Among the 2-cells of the spheres S2P are the 2-cells bi2 each
determined as explained in ~ 41 by a 1-cell kil of K2. Each such
2-cell is in two and only two spheres S2P and since the two
oriented circuits bounding 2-cells of K2 which are incident with kil
were cancelled out in forming R, the oriented 1-circuit formed
from the boundary of bi2 is also cancelled out. Hence R contains
none of the oriented 1-circuits formed from the boundaries of the
2-cells bi2. Hence R can only contain oriented 1-circuits formed
from the boundaries of 2-cells of C2. It must contain some of
these, for otherwise each 2-cell of C2 would be in an even number
of spheres S2P and hence the sum (mod. 2) of these spheres
S2P and the complex K2 would be zero contrary to ~ 51.
Hence the set of oriented 1-circuits formed from the boundaries
of the 2-cells of C2 is subject to one linear condition. Hence by
~ 55 r2= a2- for C2. Hence by ~ 58 r2 = a2- 1 for C2.
60. The theorem of ~ 53 was that if C2 is a 2-circuit any
complex homeomorphic with C2 is a 2-circuit. The theorem of
the last section adds to this result the theorem that if C2 is
orientable so is also any complex homeomorphic with C2. It
follows that if one of the complexes into which a manifold can
be decomposed is orientable so are all the complexes into which
it can be decomposed. Thus the property of orientability or nonorientability is a property of a manifold and is invariant under
the group of homeomorphisms.
As a corollary of this it follows that any complex defining a
sphere is orientable. The same follows for any sphere with p
handles on observing that the particular complexes used in