Colloquium publications.

ANALYSIS SITUS.

101

oriented n-cells formed from ain is denoted by oi" and the other
by - ai. Any set of oriented n-cells of Cn is called an oriented
n-dimensional complex and may be denoted by a symbol (x1, x2,
* *, xa) in which xi = + 1 (i = 1, 2, * *, an) if (an is in the set,
= - 1 if - ai" is in the set and xi = 0 if neither of them is in
the set. The sum of two such symbols is defined as in ~ 45, Chap.
1; and if the sum of the symbols for two oriented complexes,
rn', rn" is the symbol for an oriented complex I,"', the complex
Fn"' is called the sum* of Fr' and Fn" and denoted by rI' + n".
Now suppose that Cn is an n-circuit and let one of the two
oriented (n - 1)-circuits into which the boundary of ani (i = 1,
2,., an) can be converted be denoted by rn-i. Since each
(n - 1)-cell of Cn is incident with an even number of n-cells of
Cn, the number of oriented (n - 1)-circuits i-li which contain
a given oriented (n - 1)-cell jn-1n or its negative is even. If
the oriented (n - 1)-circuits r,-li (i = 1, 2, *., an) can be so
chosen that for each j (j = 1, 2,..., an-i), jn-1 and - ojnappear in equal numbers of them, Cn is said to be orientable.
In other words, Cn is orientable if and only if the oriented (n - 1)circuits nl-i (i = 1, 2, *., an) can be so chosen that their sum
is zero.
A set of oriented n-cells formed from the n-cells of an orientable
n-circuit in such a way that the sum of the oriented (n - 1)circuits associated with the n-cells is zero is called an oriented ncircuit. Thus an oriented n-circuit is an oriented n-dimensional
complex formed from an orientable n-circuit in a particular way.
3. By reference to ~ 22, Chap. III, it is obvious that the
boundaries of the n-cells into which an n-cell ain of Cn is decomposed by a regular subdivision can be converted into a set
of oriented (n - 1)-circuits whose sum is the oriented (n - 1)circuit rn-, formed from the boundary of ai". Hence an
n-circuit Cn is orientable if and only if a regular sub-division Cn of
it is orientable.
It can now be proved by exactly the method used in ~~ 58, 59,
Chap. II that if Gn is any n-circuit homeomorphic with Cn, Gn is
* This term is given a more extensive significance in ~ 9 below.