Chapter Five
Cauchy’s Theorem
5.1. Homotopy.
Suppose
D
is a connected subset of the plane such that every point of
D
is
an interior point—we call such a set a
region
—and let
C
1
and
C
2
be oriented closed curves
in
D
. We say
C
1
is
homotopic
to
C
2
in
D
if there is a continuous function
H
:
S
D
,
where
S
is the square
S
t
,
s
: 0
s
,
t
1
, such that
H
t
,0
describes
C
1
and
H
t
,1
describes
C
2
, and for each fixed
s
, the function
H
t
,
s
describes a closed curve
C
s
in
D
.
The function
H
is called a
homotopy
between
C
1
and
C
2
. Note that if
C
1
is homotopic to
C
2
in
D
,
then
C
2
is
homotopic
to
C
1
in
D
.
Just
observe
that
the
function
K
t
,
s
H
t
,1
s
is a homotopy.
It is convenient to consider a point to be a closed curve. The point
c
is a described by a
constant function
t
c
. We thus speak of a closed curve
C
being homotopic to a
constant—we sometimes say
C
is
contractible
to a point.
Emotionally, the fact that two closed curves are homotopic in
D
means that one can be
continuously deformed into the other in
D
.
Example
Let
D
be the annular region
D
z
: 1

z

5
. Suppose
C
1
is the circle described by
1
t
2
e
i
2
t
, 0
t
1; and
C
2
is the circle described by
2
t
4
e
i
2
t
, 0
t
1. Then
H
t
,
s
2
2
s
e
i
2
t
is a homotopy in
D
between
C
1
and
C
2
.
Suppose
C
3
is the same
circle
as
C
2
but with the
opposite
orientation; that is, a
description is given
by
3
t
4
e
i
2
t
, 0
t
1. A homotopy between
C
1
and
C
3
is not too easy to construct—in
fact, it is not possible! The moral: orientation counts. From now on, the term ”closed
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 Fall '09
 Peter
 Topology, Derivative, Manifold, dt t s, s

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