Colloquium publications.

ANALYSIS SITUS.

13

19. In the symbol (x1, x2, * *, xao) for a bounding 0-circuit
all the x's are 0 except two which correspond to a pair of vertices
belonging to one of the connected complexes into which Ci falls
according to ~ 11. This symbol must therefore satisfy the
following equations,
Xl + X2 + * * + Xmi = 0,
Xmi+l + + Xm2 = O,
(Ho)
XmRO_-+l + + Xao = O.,
in which the variables are reduced modulo 2, as explained in
~ 14. The matrix of these equations is Ho.
Since the symbol for any set of bounding 0-circuits is the sum
(mod. 2) of the symbols for the 0-circuits of the set, it follows
that any such symbol satisfies the equations (Ho). This is also
evident because in the symbol for any set of bounding 0-circuits
an even number of the x's in each of these equations must be 1.
Hence any such symbol satisfies (Ho). On the other hand, the
symbol for a non-bounding 0-circuitwill not satisfy the equations
(Ho) because the two x's which are not zero in this symbol appear
in different equations; and, in general, any symbol for a set of
vertices of which an odd number are in some connected subcomplex of Ci will fail to satisfy these equations. Hence the set
of all solutions of (Ho) is the set of all symbols for sets of bounding
0-circuits.
Since no two of these equations have a variable in common,
they are linearly independent. Hence all solutions of (Ho0) are
linearly dependent (mod. 2) on a set of ao - Ro linearly independent
solutions.
20. Denoting the connected sub-complexes of Ci by C11, Ci2,
*., C1R~ as in ~ 16 let the notation be so assigned that all, * *,
am,1 are the 1-cells in Ci1; am,+41,., am21 the 1-cells in C12;
and so on. The matrix Hi then must take the form