Colloquium publications.

CHAPTER V

THE FUNDAMENTAL GROUP AND CERTAIN UNSOLVED
PROBLEMS
Homotopic and Isotopic Deformations
1. Let Ki be a generalized complex on a generalized complex
Cn. A set of transformations Fx (O _ x _ 1) is called a oneparameter continuous family of transformations if each Fx for each
number x (0 _- x 1) is a transformation of Ki and if for each
point P of Ki the set of points [Fx(P)] for which 0 < x < 1,
constitutes a 1-cell whose ends are Fo(P) and F1(P). A (1-1)continuous transformation F of Ki into a generalized complex
K' on C is called a deformation on Cn if there exists a continuous
family of (1-1) continuous transformations Fx (0 -- x 1) such
that Fo is the identity, F1 = F, and each Fx transforms Ki into
a complex on C,.
For example, C, may be taken to be a 2-cell and Ki to be a
single point. The points Fx(P) then constitute a 1-cell with its
ends, the 1-cell being singular or not according to the properties
of Fx. As another example, Ki may be taken to be a 1-cell
with its ends, the complexes into which Ki is deformed then are
all 1-cells and constitute a 2-cell and its boundary.
Under the conditions described above, the complex Ki is said
to be deformed into the complex K' and the complexes into which
Ki is transformed by the functions Fx (0 < x < 1) are called the
intermediate positions of Ki.
It is an obvious consequence of the definitions made that if
F1 is a deformation on Cn which carries Ki to a generalized
complex K' on C, and F2 a deformation on Cn of K' and if F3 is
the resultant of F1 and F2, then F3 is a deformation on Cn.
2. Following the nomenclature introduced in the DehnHeegaard article on Analysis Situs in the Encyklopadie we shall
distinguish between isotopic and homotopic deformations. A
deformation is called an isotopy or an isotopic deformation if it is
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