surface bundle over the circle

surface bundle over the circle

A

bundle over

S

1

is a closed

3-manifold

which is constructed as a

fiber bundle

over the circle with fiber a closed

surface

.

F

⊂

E

→

S

1

.

This construction it is also a particular case of a more general concept called

mapping torus

.

The precise construction is as follows: Take any surface

F

and multiply by the unit interval

I

to get

F

×

I

. Choose any

ϕ

autohomeomorphism

of

F

.
Then the

quotient space

E

ϕ

=

F

×

I

(

x

,

0

)

∼

(

ϕ

⁢

(

x

)

,

1

)

defines a 3-manifold, characterized by the

isotopy

class of

ϕ

, that is, any other representative of the same class is going to produce a bundle homeomorphic to the original one. The isotopy class is called the

monodromy

for the bundle. It is also used for

E

ϕ

the symbol:

F

×

ϕ

S

1

This construction is an important source of examples in low dimensional

topology

as well in geometric group theory, because the geometry associated to the monodromy’s action and because the bundle’s

fundamental group

can be viewed as a particular kind of

HNN extension

: the fundamenal group of

F

extended by the integers. More precisely, if

π

*

=

g

1

⁢

g

2

⁢

g

1

-

1

⁢

g

2

-

1

⁢

⋯

⁢

g

k

-

1

⁢

g

k

⁢

g

k

-

1

-

1

⁢

g

k

-

1

or

π

*

=

g

1

2

⁢

⋯

⁢

g

k

2

then

π

1

⁢

(

E

ϕ

)

=

⟨

g

1

,

…

,

g

k

,

x

|

π

*

=

1

,

x

⁢

g

k

⁢

x

-

1

=

ϕ

*

⁢

(

g

k

)

⟩

,

depending on

F

is

orientable

or non-orientable.

When one considers

periodic

monodromies it is an amusing situation since, in this case, the bundles can be seen as

Seifert fiber space

i.e. bundles of the form

S

1

⊂

E

→

G

where

G

may be, perhaps, an

orbifold

.

For example, it is known that the extended

mapping class group

of the torus is

G

⁢

L

2

⁢

(

ℤ

)

, so there are only seven periodic elements, corresponding to seven Seifert fiber space already studied by J.Hempel.

Seven torus bundles

T

⊂

M

→

S

1

.

It is known that the following matrices generate

ℳ

*

⁢

(

T

)

t

a

=

(

1

-

1

0

1

)

,

t

b

=

(

1

0

1

1

)

and

y

=

(

1

0

0

-

1

)

which obey

⟨

t

a

,

t

b

,

y

:

(

t

a

⁢

t

b

⁢

t

a

)

4

=

1

,

t

a

⁢

t

b

⁢

t

a

=

t

b

⁢

t

a

⁢

t

b

,

y

2

=

1

,

y

⁢

t

a

⁢

y

-

1

=

t

a

-

1

,

y

⁢

t

b

⁢

y

-

1

=

t

b

-

1

⟩

The first two are left twists from

a

a simple

meridian curve

and

b

a simple longitude curve.
The matrix for

y

represents a autohomeomorphism which is not a twist and inverts

orientation

.
It is obtained by inverting curve

b

’s direction and extending in a

regular

neighborhood

N

⁢

(

b

)

, then extending to

N

⁢

(

a

∪

b

)

and finally in a disk to the whole torus.

Now we can represent the periodic monodromies of example 12.4 in [Hempel, pp.122-123] in terms of those

generators

as

1

=

(

1

0

0

1

)

,

(

t

b

⁢

t

a

)

3

=

(

-

1

0

0

-

1

)

,

y

=

(

1

0

0

-

1

)

,

t

a

⁢

y

=

(

1

1

0

-

1

)

(

t

b

⁢

t

a

)

2

=

(

0

-

1

1

-

1

)

,

(

t

a

⁢

t

b

⁢

t

a

)

3

=

(

0

1

-

1

0

)

,

t

a

⁢

t

b

=

(

0

-

1

1

1

)

of periods 1, 2, 2, 2, 3, 4, 6 respectively [Hempel, pp.123].

And in turn give the Seifert fiber spaces

(

O

o

,

1

|

0

)

=

T

×

S

1

(

O

n

,

2

|

0

)

=

(

O

o

,

0

|

(

1

,

-

2

)

,

(

2

,

1

)

,

(

2

,

1

)

,

(

2

,

1

)

,

(

2

,

1

)

)

(

N

o

,

1

|

0

)

=

(

N

n

I

,

2

|

0

)

=

K

×

S

1

(

N

o

,

1

|

1

)

=

(

N

n

I

,

2

|

1

)

)

=

K

×

τ

S

1

(

O

o

,

0

|

(

3

,

2

)

,

(

3

,

-

1

)

,

(

3

,

-

1

)

)

(

O

o

,

0

|

(

2

,

1

)

,

(

4

,

-

1

)

,

(

4

,

-

1

)

)

(

O

o

,

0

|

(

2

,

1

)

,

(

3

,

-

1

)

,

(

6

,

-

1

)

)

:)

References

:

1.

J. Hempel,

3-manifolds

, Annals of Math. Studies, 86, Princeton Univ. Press 1976.

2.

P. Orlik,

Seifert Manifolds

, Lecture Notes in Math. 291, 1972 Springer-Verlag.

3.

P. Orlik, F. Raymond,

On 3-manifolds with local

SO

2

action

, Quart. J. Math. Oxford Ser.(2) 20 (1969), 143-160.

4.

H. Seifert,

Topologie dreidimensionaler gefaserter Räume

, 60(1933), 147-238.

Title

surface bundle over the circle

Canonical name

SurfaceBundleOverTheCircle

Date of creation

2013-03-22 15:42:37

Last modified on

2013-03-22 15:42:37

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

24

Author

juanman (12619)

Entry type

Definition

Classification

msc 55R10

Classification

msc 57M50

Classification

msc 57N10

Related topic

FiberBundle

Related topic

FourSurfaceBundles

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