Colloquium publications.

ANALYSIS SITUS.

131

Amer. Math. Soc., Vol. 26 (1920), p. 370) that any orientable
manifold can be regarded as a Riemann manifold.
Theorems on Homotopy
13. By a slight modification of the argument in ~~ 35 to 45,
Chap. III, it can be proved that any k-dimensional complex Ck
on a regular n-dimensional complex C, is homotopic with a complex Ck' consisting of cells each of which covers (~ 8, Chap. IV) a
cell of Cn. The nature of the modification needed will be
sufficiently indicated by a consideration of the case of a 1-circuit
K1 on a regular complex Cn. Let the definition of the 1-cell bi1
in ~ 37, Chap. III, be modified so that bil stands in each case for
a 1-cell joining a vertex of K1 not to a vertex of Cn but to a point
coincident with a vertex of Cn. Likewise let the boundary of each
bi2 be the same as in Chap. III except that if it contains a cell
of Cn this cell is replaced by one coincident with it. Thus when
the boundaries of the 2-cells bi2 are added (mod. 2) the only
1-cells cancelled are the 1-cells bil. Hence the boundary of B2
is the 1-circuit K1 and a 1-circuit or set of 1-circuit K1' composed
of 0-cells and 1-cells each covering a cell of Cn.
It is obvious that K1 is homotopic with K1'. For a definition
of straightness and distance on B2 can be made in such a way
that each cell of B2 is a square with one side on K1 and one on
K1' or a triangle with one side on K1 and the opposite vertex
on K1'. Each point X of K1 may then be joined to a point of
K1' by a straight 1-cell x in such a way that every interior point
of B2 is on one and only one of these 1-cells. A transformation
Ft may be defined as that transformation which carries each
point X of K1 to the point P of the 1-cell x whose distance
along x from X is to the length of the 1-cell x in the ratio t.
The transformations Ft evidently give a one-parameter continuous family which define a, deformation of K1 into K1'.
14. A fundamental theorem of homotopy is the following:
If Kn is a non-singular n-circuit on an n-circuit Cn, then K,
cannot be deformed into a single point on Cn. For if such a
deformation of Kn were possible Kn would bound a singular