1550251515-Classical_Complex_Analysis__Gonzalez_

162 Chapter^3

Fig. 3.21

In our later work the following two special types of restricted homotopies

will be of particular interest:

1. Homotopy of arcs with fixed endpoints. In addition to the
   conditions (3.15-1), the following two are imposed:

ry(a, >.) = zo and ry(b, >.) = z* for all >. E [O, l], (3.15-2)

so that all arcs rJ>.. have the same initial point z 0 and the same terminal

point z* (Fig. 3.21 ).

2. Homotopy of curves. In addition to (3.15-1) the following condition
   is imposed:

ry(a, >.) = ry(b, >.) for each A E [O, 1] (3.15-3)

so that each of the arcs 'f/>.. have the same initial and terminal point (Fig.
3.22). In this case we say that the curves /l and 12 are freely homotopic
in E.
The relation ~ in E, for arcs or curves, is an equivalence relation. In
fact, we have ')'1 ~ 11 by the identity deformation ry(t, >.) = z 1 (t) for all>..
If 11 ~ /2 by ry(t,>.), then /2 ~ ')' 1 by ry(t, 1->.). Finally, if ')' 1 ~ / 2 by ry 1
and 12 ~ 13 by ry 2 , then 11 ~ 13 by ry 3 , where ry 3 is defined as follows:

for 0 ~ >. ~^1 / 2

for^1 / 2 ~ >. ~ 1

Therefore, the set of all arcs can be divided into homotopy classes.

Theorem 3.14 Let Ebe a convex set, and let /l and 12 be any two arcs
(or curves) with graphs contained in E. Then 11 ~ 12 in E.

Fig. 3.22