1550251515-Classical_Complex_Analysis__Gonzalez_

Functions. Limits and Continuity. Arcs and Curves 161

3.15 Deformation of Arcs and Curves. Homotopy

Definition 3.28 Let 11: [a,b]-+ C and 12 : [a,b]-+ C be two oriented

arcs defined on the same interval by z = z 1 (t) and z = z 2 (t), a S t S b,

respectively. Let Q be the rectangle defined in the plane of the two variables

(t,>.) by Q = {(t,>.): a St Sb, 0 S ,\ S 1}.

A homotopy of /'1 into /'2 is a continuous mapping from Q into C such

that the lower side of Q maps onto /'l and the upper side onto 12. More

precisely, a homotopy 'r/ of /'l onto 1'2 is a continuous mapping ry: Q -+ C

such that

ry(t,0) = z1(t)

for all t E [a, b].

and ry(t, 1) = z2(t) (3.15-1)

If we picture the rectangle Q as made up of horizontal line segments

S>.., one for each ,\ in [O, 1], the restriction of 'r/ to any one segment s 7

determines an arc ry: [a, b] -+ C defined by 'r/>..(t) = ry(t, >.), a S t S b, ,\

fixed. Thus we obtain a family of arcs, one for each value of,\ in [O, 1], the
arc corresponding to ,\ = 0 being 11 , that corresponding to ,\ = 1 being
12 (Fig. 3.20).

If ,\ is interpreted as time, we may think of the family of arcs as the

various positions of a single moving arc, or the whole picture as a continuous
deformation of /'l into 12. For this reason a homotopy is also called a
deformation.

If we suppose that t E [a, b] remains fixed while ,\ is allowed to vary in

[O, 1), the function ry( t, A) maps each vertical segment d of Q into the path

or trajectory p described by a point of the moving arc.

If /'1 is homotopic to /'2 in the sense described above, we write 1'1 '.::::'. /'2.

If the family 'r/>.. is contained in a subset E of C, then, by definition /'l '.::::'. (^12)

in E. If 12 is a point arc, we say that /'l is deformable or homotopic to a

point (or, null homotopic ), and we write /'1 '.::::'. 0.

A

1----

S1

Q

d

0 a so b t

Fig. 3.20