aarjaneiro/planetmath huggingface dataset:
variable category

variable category

Definition 0.1

.

A

variable category

is defined as an indexed family or class of either

categories

and, or

functor categories

[

𝒢

i

]

with

i

∈

I

in the functor category

ℱ

c

⁢

t

of categories and

functors

with additional axioms, rules, or properties of the underlying categories, that specify for example how such categories can be ’

freely generated

’(

[

1

]

).

References

1

Baianu, I.C., R. Brown and J.F. Glazebrook. : 2007a,

Categorical Ontology

of Complex Spacetime Structures: The Emergence of Life and Human Consciousness, Axiomathes, 17: 35-168.

2

Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006b, Complex Nonlinear Biodynamics in Categories,

Higher Dimensional Algebra

and Łukasiewicz–Moisil Topos:

Transformations

of Neuronal, Genetic and Neoplastic Networks.,

Axiomathes

,

16

Nos. 1-2: 65-122.

Title

variable category

Canonical name

VariableCategory

Date of creation

2013-03-22 18:15:48

Last modified on

2013-03-22 18:15:48

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

13

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55U40

Classification

msc 18G55

Classification

msc 18B40

Synonym

variable topos

Related topic

Supercategories3

Related topic

VariableTopology3

Defines

sequence of different categories

Defines

dynamic supercategory

Generated on Sat Feb 10 11:46:18 2018 by

LaTeXML
table of generalized Fourier and measured groupoid transforms

table of generalized Fourier and measured groupoid transforms

0.1

Generalized Fourier transforms

Fourier-Stieltjes

transforms

and

measured groupoid

transforms are useful generalizations of the (much simpler)

Fourier transform

, as concisely shown in the following table-
with the same format as C. Woo’s Feature on Fourier transforms (

http://planetmath.org/TableOfFourierTransforms

)
- for the purpose of direct comparison with the latter transform. Unlike the more general

Fourier-Stieltjes
transform

, the Fourier transform exists if and only if the

function

to be transformed is

Lebesgue integrable

over the whole real axis for

t

∈

ℝ

, or over the entire

ℂ

domain when

m

ˇ

⁢

(

t

)

is a complex function.

Definition 0.1

.

Fourier-Stieltjes transform

.

Given a

positive definite

,

measurable function

f

⁢

(

x

)

on the

interval

(

-

∞

,

∞

)

there exists a

monotone increasing

, real-valued bounded
function

α

⁢

(

t

)

such that:

f

(

x

)

=

∫

ℝ

e

i

⁢

t

⁢

x

d

(

α

(

t

)

,

(0.1)

for all

x

∈

ℝ

except a small set. When

f

⁢

(

x

)

is defined as above and if

α

⁢

(

t

)

is nondecreasing and bounded then the measurable function defined by the above integral is called

the Fourier-Stieltjes transform of

α

⁢

(

t

)

, and it is

continuous

in addition to being positive definite.

FT Generalizations

Title

table of generalized Fourier and measured groupoid transforms

Canonical name

TableOfGeneralizedFourierAndMeasuredGroupoidTransforms

Date of creation

2013-03-22 18:10:27

Last modified on

2013-03-22 18:10:27

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

46

Author

bci1 (20947)

Entry type

Topic

Classification

msc 55U99

Synonym

Fourier-Stieltjes transforms

Related topic

FourierTransform

Related topic

TwoDimensionalFourierTransforms

Defines

Fourier-Stieltjes and measured groupoid transforms

\@unrecurse

Generated on Sat Feb 10 11:46:41 2018 by

LaTeXML
connected sum

connected sum

Let

M

and

N

be two

n

-

manifolds

. Choose points

m

∈

M

and

n

∈

N

, and let

U

,

V

be

neighborhoods

of these points, respectively. Since

M

and

N

are manifolds, we may assume
that

U

and

V

are balls, with

boundaries

homeomorphic

to

(

n

-

1

)

-spheres, since this is possible
in

ℝ

n

. Then let

φ

:

∂

⁡

U

→

∂

⁡

V

be a

homeomorphism

. If

M

and

N

are

oriented

,
this should be

orientation

preserving with respect to the

induced

orientation (that is, degree 1).
Then the

connected sum

M

⁢

♯

⁢

N

is

M

-

U

and

N

-

V

glued along the boundaries by

φ

.

That is,

M

⁢

♯

⁢

N

is the

disjoint union

of

M

-

U

and

N

-

V

modulo the

equivalence relation

x

∼

y

if

x

∈

∂

⁡

U

,

y

∈

∂

⁡

V

and

φ

⁢

(

x

)

=

y

.

Title

connected sum

Canonical name

ConnectedSum1

Date of creation

2013-03-22 13:17:59

Last modified on

2013-03-22 13:17:59

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Definition

Classification

msc 57-00

Generated on Sat Feb 10 11:46:58 2018 by

LaTeXML
fundamental group

fundamental group

Let

(

X

,

x

0

)

be a

pointed topological space

(that is, a

topological space

with a chosen basepoint

x

0

).
Denote by

[

(

S

1

,

1

)

,

(

X

,

x

0

)

]

the set of

homotopy classes

of maps

σ

:

S

1

→

X

such that

σ

⁢

(

1

)

=

x

0

.
Here,

1

denotes the basepoint

(

1

,

0

)

∈

S

1

.
Define a

product

[

(

S

1

,

1

)

,

(

X

,

x

0

)

]

×

[

(

S

1

,

1

)

,

(

X

,

x

0

)

]

→

[

(

S

1

,

1

)

,

(

X

,

x

0

)

]

by

[

σ

]

⁢

[

τ

]

=

[

σ

⁢

τ

]

,
where

σ

⁢

τ

means “travel along

σ

and then

τ

”.
This gives

[

(

S

1

,

1

)

,

(

X

,

x

0

)

]

a group structure
and we define the

fundamental group

of

(

X

,

x

0

)

to be

π

1

⁢

(

X

,

x

0

)

=

[

(

S

1

,

1

)

,

(

X

,

x

0

)

]

.

In general, the fundamental group of a topological space
depends upon the choice of basepoint.
However, basepoints in the same path-component of the space
will give

isomorphic groups

.
In particular, this means that the fundamental group of a (non-empty) path-connected space is well-defined, up to

isomorphism

,
without the need to specify a basepoint.

Here are some examples of fundamental groups of familiar spaces:

•

π

1

⁢

(

ℝ

n

)

≅

{

0

}

for each

n

∈

ℕ

.

•

π

1

⁢

(

S

1

)

≅

ℤ

.

•

π

1

⁢

(

T

)

≅

ℤ

⊕

ℤ

, where

T

is the torus.

It can be shown that

π

1

is a

functor

from the category of pointed topological spaces to the category of groups.
In particular, the fundamental group is a

topological invariant

,
in the sense that
if

(

X

,

x

0

)

is

homeomorphic

to

(

Y

,

y

0

)

via a basepoint-preserving map,
then

π

1

⁢

(

X

,

x

0

)

is isomorphic to

π

1

⁢

(

Y

,

y

0

)

.

It can also be shown that two

homotopically equivalent

path-connected spaces
have isomorphic fundamental groups.

Homotopy groups

generalize the

concept

of the fundamental group to higher

dimensions

.
The fundamental group is the first homotopy group,
which is why the notation

π

1

is used.

Title

fundamental group

Canonical name

FundamentalGroup

Date of creation

2013-03-22 11:58:44

Last modified on

2013-03-22 11:58:44

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

16

Author

yark (2760)

Entry type

Definition

Classification

msc 57M05

Classification

msc 55Q05

Classification

msc 20F34

Synonym

first homotopy group

Related topic

Group

Related topic

Curve

Related topic

EtaleFundamentalGroup

Generated on Sat Feb 10 11:47:03 2018 by

LaTeXML
orbifold

orbifold

Roughly, an

orbifold

is the

quotient

of a manifold by a

finite group

. For example, take a sheet of paper and add a small crease perpendicular to one side at the halfway point. Then, line up the two halves of the side. This may be thought of as the plane

ℝ

2

modulo the group

ℤ

2

. Now, let us give the

definition

.

Define a category

𝒳

:
The objects are pairs

(

G

,

X

)

, where

G

is a finite group acting
effectively on a

connected

smooth manifold

X

. A

morphism

Φ

between
two objects

(

G

′

,

X

′

)

and

(

G

,

X

)

is a family of open

embeddings

ϕ

:

X

′

→

X

which

satisfy

•

for each embedding

ϕ

∈

Φ

, there is an injective

homomorphism

λ

ϕ

:

G

′

→

G

such that

ϕ

is

λ

ϕ

equivariant

•

For

g

∈

G

, we have

g

⁢

ϕ

:

X

′

→

X

g

⁢

ϕ

:

x

↦

g

⁢

ϕ

⁢

(

x

)

and if

(

g

⁢

ϕ

)

⁢

(

X

)

∩

ϕ

⁢

(

X

)

≠

∅

, then

g

∈

λ

ϕ

⁢

(

G

′

)

.

•

Φ

=

{

g

ϕ

,

g

∈

G

}, for any

ϕ

∈

Φ

Now, we define orbifolds. Given a

paracompact

Hausdorff space

X

and a
nice open covering

𝒰

which forms a basis for the

topology

on

X

, an orbifold structure

𝒱

on

X

consists of

1.

For

U

∈

𝒰

,

𝒱

⁢

(

U

)

=

(

G

U

,

U

~

)

→

τ

U

is a ramified cover

U

~

→

U

which
identifies

U

~

/

G

U

≅

U

2.

For

U

⊂

V

∈

𝒰

, there exists a morphism

ϕ

V

⁢

U

⁢

(

G

U

,

U

~

)

→

(

G

V

,

V

~

)

covering the

inclusion

3.

If

U

⊂

V

⊂

W

∈

𝒰

,

ϕ

W

⁢

U

=

ϕ

W

⁢

V

∘

ϕ

V

⁢

U

References

:

[1] Kawasaki T., The

Signature

theorem

for V-manifolds. Topology 17 (1978), 75-83. MR0474432
(57:14072)

Title

orbifold

Canonical

name

Orbifold

Date of creation

2013-03-22 15:40:06

Last modified on

2013-03-22 15:40:06

Owner

guffin (12505)

Last modified by

guffin (12505)

Numerical id

8

Author

guffin (12505)

Entry type

Definition

Classification

msc 57M07

Synonym

orbifold structure

Defines

orbifold structure

Generated on Sat Feb 10 11:47:07 2018 by

LaTeXML
example of a space that is not semilocally simply connected

example of a space that is not semilocally simply connected

An example of a space that is

not

semilocally simply connected

is
the following: Let

H

⁢

R

=

⋃

n

∈

ℕ

{

(

x

,

y

)

∈

ℝ

2

|

(

x

-

1

2

n

)

2

+

y

2

=

(

1

2

n

)

2

}

endowed with the

subspace topology

. Then

(

0

,

0

)

has no

simply connected

neighborhood

. Indeed every neighborhood of

(

0

,

0

)

contains (ever diminshing)
homotopically non-trivial loops. Furthermore these loops are homotopically non-trivial even when considered as loops in

H

⁢

R

.

\begin{pspicture}(0.0pt,-56.905502pt)(113.81102pt,85.358261pt)\leavevmode%
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\special{pst:
\pst@dict\tx@STP\pst@newpath\psk@origin\psk@swapaxes\pst@code end
}\ignorespaces\ignorespaces\leavevmode\ignorespaces\ignorespaces\ignorespaces%
\ignorespaces\special{pst:
\pst@dict\tx@STP\pst@newpath\psk@origin\psk@swapaxes\pst@code end
}\ignorespaces\ignorespaces\leavevmode\ignorespaces\ignorespaces\ignorespaces%
\ignorespaces\special{pst:
\pst@dict\tx@STP\pst@newpath\psk@origin\psk@swapaxes\pst@code end
}\ignorespaces\ignorespaces\leavevmode\ignorespaces\ignorespaces\ignorespaces%
\ignorespaces\special{pst:
\pst@dict\tx@STP\pst@newpath\psk@origin\psk@swapaxes\pst@code end
}\ignorespaces\ignorespaces\leavevmode\ignorespaces\ignorespaces\ignorespaces%
\ignorespaces\special{pst:
\pst@dict\tx@STP\pst@newpath\psk@origin\psk@swapaxes\pst@code end
}\ignorespaces\ignorespaces\leavevmode\ignorespaces\ignorespaces\ignorespaces%
\ignorespaces\special{pst:
\pst@dict\tx@STP\pst@newpath\psk@origin\psk@swapaxes\pst@code end
}\ignorespaces\ignorespaces\leavevmode\ignorespaces\ignorespaces\ignorespaces%
\ignorespaces\special{pst:
\pst@dict\tx@STP\pst@newpath\psk@origin\psk@swapaxes\pst@code end
}\ignorespaces\ignorespaces\end{pspicture}

The

Hawaiian rings

It is essential in this example that

H

⁢

R

is endowed with the

topology

induced

by its inclusion in the plane. In contrast, the same set endowed with
the CW topology is just a bouquet of countably many circles and (as any

CW
complex

) it is semilocaly simply connected.

Title

example of a space that is not semilocally simply connected

Canonical name

ExampleOfASpaceThatIsNotSemilocallySimplyConnected

Date of creation

2013-03-22 13:25:10

Last modified on

2013-03-22 13:25:10

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

16

Author

mathcam (2727)

Entry type

Example

Classification

msc 57M10

Classification

msc 54D05

Defines

Hawaiian rings

Defines

Hawaiian earrings

Generated on Sat Feb 10 11:47:15 2018 by

LaTeXML
surface

surface

A

surface

is a two-dimensional

topological manifold

.
A closed surface is a surface without

boundary

.

A result called the “classification theorem” gives us a symbolic semantics, matching the geometrical view point, in terms of genera, orientability and number of boundary components. Together with the

connected sum

operation, they make available a powerful language to be explored and exploited.

As an example of a surface take

T

=

S

1

×

S

1

the two torus, the boundary of a solid sugar donut shaped cake

D

2

×

S

1

, where

S

1

is the familiar modulus one

complex numbers

.

Title

surface

Canonical name

Surface

Date of creation

2013-03-22 16:01:43

Last modified on

2013-03-22 16:01:43

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

13

Author

juanman (12619)

Entry type

Definition

Classification

msc 57M20

Related topic

Manifold

Related topic

NonOrientableSurface

Generated on Sat Feb 10 11:47:17 2018 by

LaTeXML
alternating knot

alternating knot

An

alternating knot

is a knot with a diagram that has crossings that
alternate between over and under as one travels around the knot in a fixed
direction.

Examples of alternating knots include the

trefoil knot

(

http://planetmath.org/ConnectedSum

) and the figure-eight knot.

Title

alternating knot

Canonical

name

AlternatingKnot

Date of creation

2013-03-22 16:29:18

Last modified on

2013-03-22 16:29:18

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

9

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 57M25

Defines

alternating

Generated on Sat Feb 10 11:47:21 2018 by

LaTeXML
amphicheiral knot

amphicheiral knot

A knot is which is

equivalent

to its

mirror image

is called an

amphicheiral knot

. Chemists use the term

achiral

.

At least twenty amphicheiral knots are known which have nine or less crossings.

Title

amphicheiral knot

Canonical name

AmphicheiralKnot

Date of creation

2013-03-22 16:29:11

Last modified on

2013-03-22 16:29:11

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

5

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 57M25

Defines

achiral

Generated on Sat Feb 10 11:47:23 2018 by

LaTeXML
connected sum

connected sum

The

connected sum

of knots

K

and

J

is a knot, denoted by

K

⁢

#

⁢

J

,
constructed by removing a short segment from each of

K

and

J

and joining each free end of

K

to a different free end of

J

to form a new knot. The connected sum of two knots always exists but is not necessarily unique.

The

connected sum of

oriented knots

K

and

J

is a connected sum of knots which has a consistent

orientation

inherited from that of

K

and

J

. This sum always exists and is unique.

Example

.

Suppose

K

and

J

are both the

trefoil knot

.

Figure 1:

The trefoil knot

By one choice of segment deletion and reattachment,

K

⁢

#

⁢

J

is the quatrefoil knot.

Figure 2:

K

⁢

#

⁢

J

is the quatrefoil knot

Title

connected sum

Canonical

name

ConnectedSum

Date of creation

2013-03-22 13:17:33

Last modified on

2013-03-22 13:17:33

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

11

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 57M25

Synonym

knot sum

Related topic

KnotTheory

Generated on Sat Feb 10 11:47:28 2018 by

LaTeXML
crossing change

crossing change

In a knot, a

crossing change

occurs when a positive crossing (or crossing over) is followed by a negative crossing (or crossing under). An

alternating knot

, for example, has exactly as many crossing changes as it has crossings.

For the purpose of illustrating crossing changes, a non-alternating knot is more interesting. In the following diagram, if we start at the point indicated by the blue dot and proceed clockwise, the red arrows indicate crossing changes while the green arrows indicate a crossing the same as the previous crossing. One crossing change after that and there are no further arrows in the diagram until the blue dot.

Title

crossing change

Canonical name

CrossingChange

Date of creation

2013-03-22 16:29:14

Last modified on

2013-03-22 16:29:14

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 57M25

Generated on Sat Feb 10 11:47:29 2018 by

LaTeXML
mirror image of a knot

mirror image of a knot

The

mirror image

of a knot

K

is the knot

K

′

whose diagram obtained by changing all the crossings in a diagram of

K

. This notion is well defined since

Reidemeister moves

are preserved under mirror images.

Title

mirror image of a knot

Canonical name

MirrorImageOfAKnot

Date of creation

2013-03-22 16:29:24

Last modified on

2013-03-22 16:29:24

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 57M25

Defines

mirror image

Generated on Sat Feb 10 11:47:37 2018 by

LaTeXML
prime knot

prime knot

A knot which is not the

connected sum

of two nontrivial knots is called
a

prime knot

.

Examples include the

trefoil knot

and the figure-eight knot.

Title

prime knot

Canonical name

PrimeKnot

Date of creation

2013-03-22 16:39:31

Last modified on

2013-03-22 16:39:31

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

5

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 57M25

Generated on Sat Feb 10 11:47:38 2018 by

LaTeXML
unknot

unknot

The

unknot

is the knot with a projection containing no crossings, or equivalently the knot with three vertices. The unknot forms an identity when working with the

connected sum

of knots; that is, if

U

is the unknot, then

K

=

K

⁢

#

⁢

U

for any knot

K

.

Title

unknot

Canonical name

Unknot

Date of creation

2013-03-22 13:17:24

Last modified on

2013-03-22 13:17:24

Owner

basseykay (877)

Last modified by

basseykay (877)

Numerical id

6

Author

basseykay (877)

Entry type

Definition

Classification

msc 57M25

Synonym

trivial knot

Related topic

KnotTheory

Generated on Sat Feb 10 11:47:47 2018 by

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writhe

writhe

The

writhe

of a link, also called its

twist number

, is
the sum of the signs of the link’s crossings. In short, it is the sum of the

positive

crossings minus the sum of the negative crossings. Note that the writhe is not an isotopy invariant and that is actually depends on the regular homotopy class of its projection.

Title

writhe

Canonical

name

Writhe

Date of creation

2013-03-22 16:29:22

Last modified on

2013-03-22 16:29:22

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 57M25

Synonym

twist number

Generated on Sat Feb 10 11:47:48 2018 by

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Dehn’s lemma

Dehn’s lemma

The Dehn’s lemma states that if a

continuous map

from a
2-disk (

http://planetmath.org/StandardNBall

) to a 3-manifold

f

:

D

2

→

M

for which

f

-

1

⁢

f

⁢

∂

⁡

D

2

=

D

2

, then there exists an

embedding

g

:

D

2

→

M

such that the

restrictions

g

|

∂

and

f

|

∂

are
equal.

Title

Dehn’s lemma

Canonical name

DehnsLemma

Date of creation

2013-03-22 15:45:32

Last modified on

2013-03-22 15:45:32

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

11

Author

juanman (12619)

Entry type

Theorem

Classification

msc 57M35

Synonym

Dehn’s lemma

Related topic

3Manifolds

Generated on Sat Feb 10 11:47:50 2018 by

LaTeXML
loop theorem

loop theorem

In the

topology

of 3-manifolds,

the loop theorem

is

generalization

of an ansatz discovered by Max Dehn (namely, Dehn’s lemma),
who saw that

if a

continuous map

from a 2-disk to a 3-manifold whose

restriction

to the

boundary

’s disk has no singularities,
then there exists another

embedding

whose restriction to the boundary’s disk is equal to the boundary’s restriction original map

.

The following statement called the loop theorem is a version from J. Stallings, but written in W. Jaco’s book.

Let

M

be a three-manifold and let

S

be a connected surface in

∂

⁡

M

. Let

N

⊂

π

1

⁢

(

M

)

be a

normal subgroup

.
Let

f

:

D

2

→

M

be a

continuous map

such that

f

⁢

(

∂

⁡

D

2

)

⊂

S

and

[

f

|

∂

D

2

]

∉

N

.

Then there exists an

embedding

g

:

D

2

→

M

such that

g

⁢

(

∂

⁡

D

2

)

⊂

S

and

[

g

|

∂

D

2

]

∉

N

,

The proof is a clever construction due to C. Papakyriakopoulos about a

sequence

(a tower) of covering spaces.
Maybe the best detailed

presentation

is due to A. Hatcher.
But in general, accordingly to Jaco’s opinion,

”… for anyone unfamiliar with the techniques of 3-manifold-topology and are here to gain a working knowledge for the study of problems in this
…, there is no better
to start.”

References

W. Jaco,

Lectures on 3-manifolds topology

, A.M.S. regional conference series in Math 43.

J. Hempel,

3-manifolds

, Princeton University Press 1976.

A. Hatcher,

Notes on 3-manifolds

, available on-line.

Title

loop theorem

Canonical name

LoopTheorem

Date of creation

2013-03-22 15:49:13

Last modified on

2013-03-22 15:49:13

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

11

Author

juanman (12619)

Entry type

Theorem

Classification

msc 57M35

Related topic

3Manifolds

Generated on Sat Feb 10 11:47:52 2018 by

LaTeXML
sphere theorem

sphere theorem

Theorem 1

(sphere theorem).

If

M

is a

differentiable

orientable

3-manifold

such that

π

2

⁢

(

M

)

is not trivial, then there exists an

embedding

S

2

→

M

such that its image homotopy class is not equal to zero.

This theorem was established and proved by C. Papakyriakopoulos in 1957.

Title

sphere theorem

Canonical name

SphereTheorem

Date of creation

2013-03-22 15:45:35

Last modified on

2013-03-22 15:45:35

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

11

Author

juanman (12619)

Entry type

Theorem

Classification

msc 57M35

Related topic

3Manifolds

Generated on Sat Feb 10 11:47:53 2018 by

LaTeXML
Seifert fiber space

Seifert fiber space

In the field of three dimensional

manifolds

there is a kind which can be considered as spaces which are fibered by
circles (

http://planetmath.org/Circle

), that is, for each point in the space there is a unique circle (a homeomorph of

S

1

) which contains it.

A

Seifert fiber space

can be defined as a

circle bundle

over an

orbifold

,
however these kind of

3-manifolds

appeared much earlier than the concept of fiber-bundle.

To construct them one must begin learning how to fiber a solid torus.
Naturally a solid torus

D

2

×

S

1

is foliated by circles

p

×

S

1

where

p

∈

D

2

.

To get different fibered solid tori one must begin with a solid torus

D

2

×

S

1

, cut it along a meridian disk and twist the resulting solid cylinder by a rational angle

2

⁢

π

⁢

(

a

/

b

)

(being

a

,

b

coprime) and reglue to obtain a new fibering by circles of the old torus, where the fibers all are still longitudinal circles but differs by the previous one in the fact that now only the center’s disk is a trivial circle but any other point in the disk is in a circle which is

b

times longer that the

central

one.

It is possible construct an associated 2-manifold identifying each fiber to a point to get the so called

orbit

surface

.

References

H. Seifert,

Topologie drei-dimensionaler gefaserter Räume

, Acta. Math. 60 (1933), 147 - 238.

M.Brin,

Seifert Fibered Spaces, Notes for a course given in the Spring of 1993

, on line at
ftp://ftp.math.binghamton.edu/pub/matt/seifert.pdf

Title

Seifert fiber space

Canonical name

SeifertFiberSpace

Date of creation

2013-03-22 15:50:53

Last modified on

2013-03-22 15:50:53

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

10

Author

juanman (12619)

Entry type

Definition

Classification

msc 57M50

Synonym

circle bundle

Related topic

3Manifolds

Generated on Sat Feb 10 11:47:55 2018 by

LaTeXML