winding number

winding number

Winding numbers

are a basic notion in

algebraic topology

, and play an
important role in

connection

with

analytic functions

of a complex variable.
Intuitively, given a closed curve

t

↦

S

⁢

(

t

)

in an oriented
Euclidean plane (such as the

complex plane

ℂ

), and a point

p

not in the image of

S

, the winding number (or index) of

S

with respect
to

p

is the net number of times

S

surrounds

p

. It is not altogether
easy to make this notion rigorous.

Let us take

ℂ

for the plane. We have
a

continuous mapping

S

:

[

a

,

b

]

→

ℂ

where

a

and

b

are some
reals with

a

<

b

and

S

⁢

(

a

)

=

S

⁢

(

b

)

. Denote by

θ

⁢

(

t

)

the angle from the

positive real axis

to the ray from

z

0

to

S

⁢

(

t

)

. As

t

moves from

a

to

b

, we expect

θ

to increase or decrease by a multiple of

2

⁢

π

,
namely

2

⁢

ω

⁢

π

where

ω

is the winding number. One therefore thinks
of using integration. And indeed, in the theory of

functions

of a complex variable, it is proved that the value

1

2

⁢

π

⁢

i

⁢

∫

S

d

⁢

z

z

-

z

0

is an integer and has the expected properties of a winding number around

z

0

. To define the winding number in this way, we need to assume
that the closed path

S

is rectifiable (so that the path
integral is defined). An

equivalent

condition is that the real and

imaginary
parts

of the function

S

are of

bounded variation

.

But if

S

is any continuous mapping

[

a

,

b

]

→

ℂ

having

S

⁢

(

a

)

=

S

⁢

(

b

)

, the winding number is still definable, without any integration.
We can break up the domain of

S

into a finite number of

intervals

such that
the image of

S

, on any of those intervals, is contained in a disc which
does not contain

z

0

. Then

2

⁢

ω

⁢

π

emerges as a finite sum: the sum
of the angles subtended at

z

0

by the sides of a polygon.

Let

A

,

B

, and

C

be any three distinct rays from

z

0

.
The three sets

S

-

1

⁢

(

A

)

S

-

1

⁢

(

B

)

S

-

1

⁢

(

C

)

are closed in

[

a

,

b

]

, and they

determine

the winding number of

S

around

z

0

. This result can provide an alternative
definition of winding numbers in

ℂ

, and a definition in some other
spaces also, but the details are rather subtle.

For one more variation on the theme, let

S

be any

topological space

homeomorphic

to a circle, and let

f

:

S

→

S

be any continuous mapping. Intuitively we expect that if a point

x

travels once around

S

, the point

f

⁢

(

x

)

will travel around

S

some

integral

number of times, say

n

times. The notion can be made precise.
Moreover, the number

n

is determined by the three

closed sets

f

-

1

⁢

(

a

)

f

-

1

⁢

(

b

)

f

-

1

⁢

(

c

)

where

a

,

b

, and

c

are any three distinct points in

S

.

Title

winding number

Canonical name

WindingNumber

Date of creation

2013-03-22 12:56:06

Last modified on

2013-03-22 12:56:06

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 55M25

Classification

msc 30A99

Synonym

index

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