Colloquium publications.

ANALYSIS SITUS.

55

2-cell on a2. Let c be the given curve and 0 a point of a2 not
on c. Let OX be the straight 1-cell joining 0 to a variable
point X of c. Let O' be a point interior to a triangle t of a
Euclidean plane and let X' be a variable point of the boundary
of this triangle. Let F be a continuous (1 - 1) correspondence
between the set of points [X'] and the set of points CX~. If we
let each point of O'X' correspond to the point of OX which
divides it in the same ratio, a continuous correspondence F' is
defined in which each point of the interior and boundary of the
triangle t corresponds to one point of a2. By ~ 1 there is thus
defined a 2-cell (in general, singular) which is bounded by c.
It is not essential that 0 shall not coincide with a point of c,
for in case X coincides with 0 the interval OX may be taken
to be a singular one coinciding with 0. Hence we have without
restrictions the theorem that any closed curve on a 2-cell a is
the boundary of a 2-cell on a.
The theorem may be generalized slightly as follows: Any
curve c on a triangle star (~ 14) is the boundary of a 2-cell on the
triangle star. The 2-cell is constructed as above, taking the
center of the triangle star as 0.
Congruences and Homologies, Modulo 2
37. Before going on to the solution of the problem stated in
~ 35, let us introduce a notation which is adapted from that of
Poincare. We shall say that a complex Cn (n = 1, 2) is congruent
(modulo 2) to a set of (n - 1)-circuits Cn-_ if and only if Cnis the boundary of Cn. This is represented by the notation
(1) Cn _ Cn_- (mod. 2).
In case Cn_ fails to exist, so that Cn is a set of n-circuits, Cn is
said to be congruent to zero (mod. 2) and (1) is replaced by
(2) Cn =0 (mod. 2).
Expressions of the form (1) and (2) are called congruences (mod. 2).
They have been defined thus far only for n = 1 and n = 2, but
these definitions will apply for all values of n as soon as the
terms complex, n-circuit, and boundary of an n-dimensional
complex have been defined for all values of tn.