monodromy

monodromy

Let

(

X

,

*

)

be a

connected

and

locally connected

based space and

p

:

E

→

X

a covering map. We will denote

p

-

1

⁢

(

*

)

, the fiber over the

basepoint

, by

F

, and the

fundamental
group

π

1

⁢

(

X

,

*

)

by

π

. Given a loop

γ

:

I

→

X

with

γ

⁢

(

0

)

=

γ

⁢

(

1

)

=

*

and a point

e

∈

F

there exists a
unique

γ

~

:

I

→

E

,

with

γ

~

⁢

(

0

)

=

e

such that

p

∘

γ

~

=

γ

,
that is, a

lifting

of

γ

starting at

e

. Clearly, the

endpoint

γ

~

⁢

(

1

)

is also a point of the fiber, which we will denote by

e

⋅

γ

.

Theorem 1

.

With notation as above we have:

1.

If

γ

1

and

γ

2

are

homotopic

relative

∂

⁡

I

then

∀

e

∈

F

e

⋅

γ

1

=

e

⋅

γ

2

.

2.

The map

F

×

π

→

F

,

(

e

,

γ

)

↦

e

⋅

γ

defines a

right action

of

π

on

F

.

3.

The

stabilizer

of a point

e

is the image of the fundamental group

π

1

⁢

(

E

,

e

)

under the map induced by

p

:

Stab

⁡

(

x

)

=

p

*

⁢

(

π

1

⁢

(

E

,

e

)

)

.

Proof.

1.

Let

e

∈

F

,

γ

1

,

γ

2

:

I

→

X

two loops homotopic relative

∂

⁡

I

and

γ

~

1

,

γ

~

2

:

I

→

E

their liftings
starting at

e

. Then there is a

homotopy

H

:

I

×

I

→

X

with the
following properties:

–

H

⁢

(

∙

,

0

)

=

γ

1

,

–

H

⁢

(

∙

,

1

)

=

γ

2

,

–

H

⁢

(

0

,

t

)

=

H

⁢

(

1

,

t

)

=

*

,

∀

t

∈

I

.

According to the

lifting theorem

H

lifts to a homotopy

H

~

:

I

×

I

→

E

with

H

⁢

(

0

,

0

)

=

e

. Notice that

H

~

⁢

(

∙

,

0

)

=

γ

~

1

(respectively

H

~

⁢

(

∙

,

1

)

=

γ

~

2

)
since they both are liftings of

γ

1

(respectively

γ

2

) starting at

e

. Also notice that that

H

~

⁢

(

1

,

∙

)

is a path that lies entirely in
the fiber (since it lifts the

constant

path

*

). Since the fiber is

discrete

this means that

H

~

⁢

(

1

,

∙

)

is a constant path. In particular

H

~

⁢

(

1

,

0

)

=

H

~

⁢

(

1

,

1

)

or equivalently

γ

~

1

⁢

(

1

)

=

γ

~

2

⁢

(

1

)

.

2.

By (1) the map is well defined. To prove that it is an action notice
that firstly the constant path

*

lifts to constant paths and therefore

∀

e

∈

F

,

e

⋅

1

=

e

.

Secondly the

concatenation

of two paths lifts to the concatenation of their
liftings (as is easily verified by projecting). In other words, the lifting
of

γ

1

⁢

γ

2

that starts at

e

is the concatenation of

γ

~

1

,
the lifting of

γ

1

that starts at

e

, and

γ

~

2

the lifting of

γ

2

that
starts in

γ

1

⁢

(

1

)

. Therefore

e

⋅

(

γ

1

⁢

γ

2

)

=

(

e

⋅

γ

1

)

⋅

γ

2

.

3.

This is a

tautology

:

γ

fixes

e

if and only if its lifting
starting at

e

is a loop.

∎

Definition 2

.

The action described in the above

theorem

is called the

monodromy
action

and the corresponding

homomorphism

ρ

:

π

→

Sym

⁢

(

F

)

is called the

monodromy

of

p

.

Title

monodromy

Canonical name

Monodromy

Date of creation

2013-03-22 13:26:20

Last modified on

2013-03-22 13:26:20

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 55R05

Related topic

MonodromyGroup

Defines

monodromy

Defines

monodromy action

Defines

monodromy homomorphism

Generated on Sat Feb 10 11:44:38 2018 by

LaTeXML