Colloquium publications.

ANALYSIS SITUS.

67

Orientable Complexes
57. The theorem of the last section is equivalent to the statement that if r2 = a2 - 1 for a 2-circuit C2 the boundaries of the
2-cells of C2 can be converted into oriented 1-circuits in such a
way that their sum is zero. If r2 = a2 the boundaries of the
2-cells evidently cannot be thus oriented. In the first case
C2 is said to be two-sided or orientable and in the second case to
be one-sided or non-orientable. A manifold is said to be orientable
or non-orientable according as the complex defining it is or is not
orientable. This extension of the term is justified by the theorems of ~~ 58-60 below according to which the complexes
defining a given manifold M2 are all orientable or all non-orientable.
This definition is equivalent to the one given in 1865 by A. F.
Mobius, Ueber die Bestimmung des Inhaltes eines Polyeders,
Werke, Vol. 2, p. 475; see also p. 519. The term "orientable"
was suggested by J. W. Alexander as preferable to "two-sided"
because the latter term connotes the separation of a threedimensional manifold into two parts, the two "sides," by the
two-dimensional manifold, whereas the property which we are
dealing with is an internal property of the two-dimensional
manifold.*
The intuitional significance of orientableness is perhaps best
grasped by experiments with the well-known M6bius paper
strip described in the article referred to above. These experiments can also be used to verify the theorems on deformation
and on the indicatrix in Chap. V.
58. Suppose that a 2-cell a,2 of a complex C2, the cells of which
have been oriented in the manner described above, is separated
into two 2-cells by a 1-cell a1. The two new 2-cells are bounded
* On the relation between orientableness and two-sidedness, see E. Steinitz,
Sitzungsberichte der Berliner Math. Ges., Vol. 7 (1908), p. 35; and D. Konig,
Archiv. der Math. u. Phys., 3d Ser., Vol. 19 (1912), p. 214. The term orientable (orientierbar) has also been used by H. Tietze in an article in the
Jahresbericht der Deutschen Math. Ver., Vol. 29 (1920), p. 95, which came to
my attention while these lectures were in proof-sheets. This article contains a
general discussion of orientability covering a number of the questions referred
to in the beginning of Chap. V below, and also a useful collection of references.