aarjaneiro/planetmath huggingface dataset:
Dehn surgery

Dehn surgery

Let

M

be a smooth

3-manifold

, and

K

⊂

M

a smooth knot. Since

K

is an

embedded submanifold

, by the tubular neighborhood theorem there is a closed

neighborhood

U

of

K

diffeomorphic

to the solid torus

D

2

×

S

1

. We let

U

′

denote the

interior

of

U

. Now, let

φ

:

∂

⁡

U

→

∂

⁡

U

be an

automorphism

of the torus, and consider the

manifold

M

′

=

M

\

U

′

⁢

∐

φ

U

, which is the

disjoint union

of

M

\

U

′

and

U

, with points in the

boundary

of

U

identified with their images in the boundary of

M

\

U

′

under

φ

.

It’s a bit hard to visualize how this actually results in a different manifold, but it generally does. For example, if

M

=

S

3

, the 3-sphere,

K

is the

trivial knot

, and

φ

is the automorphism exchanging

meridians

and

parallels

(i.e., since

U

≅

D

2

×

S

1

, get an

isomorphism

∂

⁡

U

≅

S

1

×

S

1

, and

φ

is the map interchanging to the two copies of

S

1

), then one can check that

M

′

≅

S

1

×

S

2

(

S

3

\

U

is also a solid torus, and after our automorphism, we glue the two solid tori, meridians to meridians, parallels to parallels, so the two copies of

D

2

paste along the edges to make

S

2

).

Every

compact

3-manifold can obtained from the

S

3

by

surgery

around finitely many knots.

Title

Dehn surgery

Canonical name

DehnSurgery

Date of creation

2013-03-22 13:56:07

Last modified on

2013-03-22 13:56:07

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

4

Author

bwebste (988)

Entry type

Definition

Classification

msc 57M99

Generated on Sat Feb 10 11:47:57 2018 by

LaTeXML
topics on low dimensional topology

topics on low dimensional topology

List of

topic on low dimensional topology

that should be here at PM

1

Manifolds in general

•

manifold

•

homotopy

,

homotopy groups

,

•

isotopy

,

mapping class group

, homeotopy

•

homology

and

cohomology

•

fiber bundle

,

characteristic classes

•

piecewise linear topology

•

topological methods in

group theory

•

CW complex

2

1-manifolds

•

real line

vs

projective line

•

circle from a

Euclidean

loop to a

Lie group

•

curve

•

knot, link and

Jones polynomial

.

Prime knots

•

torus knots

•

braid

•

loop and string, Witten M-Theory

•

fundamental group

•

graph and

graph theory

3

2-manifolds

•

Mobius strip

•

surface

•

two sphere

•

torus

•

projective plane

•

Klein bottle

•

genus (

genus of topological surface

)

•

Euler characteristic

(Euler’s

polyhedron

formula)

•

Gaussian curvature

•

Nielsen-Thurston classification of surface

homeomorphisms

4

3-manifolds

•

ℝ

3

Euclidean three-space

•

S

3

three sphere,

S

⁢

O

⁢

(

3

)

,

S

⁢

U

⁢

(

2

)

•

surface bundle

•

circle bundle

,

lens space

, Seifert fiber space (

http://planetmath.org/SeifertFiberSpace

)

•

homology sphere

•

Poincare conjecture

•

Dehn’s lemma,

loop theorem

,

sphere theorem

•

annulus

and torus theorems

•

Hopf bundle

5

4-manifolds

•

Donalson’s theory

•

quaternions

•

relativity, space-time

•

surface bundles over surfaces

•

spin structures

•

there are homeomorphic four-manifolds with differentiable structures each, which make them non-diffeomorphic.

•

gauge theory: monopoles

•

classification

of

simply connected

4-manifolds

•

intersection form

•

geometrization

6

Mathematical physics

Title

topics on low dimensional topology

Canonical name

TopicsOnLowDimensionalTopology

Date of creation

2013-03-22 16:01:46

Last modified on

2013-03-22 16:01:46

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

14

Author

juanman (12619)

Entry type

Topic

Classification

msc 57M99

Classification

msc 57-00

Related topic

Topology

Generated on Sat Feb 10 11:47:59 2018 by

LaTeXML
3-manifold

3-manifold

In this brief note we define and give

instances

of the notion of a

3-manifold

.

A

3-manifold

is a

Hausdorff topological space

which is

locally homeomorphic

to the

Euclidean space

ℝ

3

.

One can see from simple constructions the great

variety

of objects that indicate how they are worth to study.

First examples without

boundary

:

1.

For example, with the

Cartesian product

we can get:

–

S

2

×

S

1

–

ℝ

⁢

P

2

×

S

1

–

T

×

S

1

–

K

×

S

1

where

S

1

and

S

2

are the 1- and 2-dimensional spheres respectively,

T

is a torus,

K

a

Klein bottle

, and

ℝ

⁢

P

2

is the 2-dimensional real projective space.

2.

Also by the

generalization

of the Cartesian product:

fiber bundles

, one can build bundles

E

of the type

F

⊂

E

→

S

1

where

F

is any closed

surface

.

3.

Or interchanging the roles, bundles as:

S

1

⊂

E

→

F

4.

knots and links complements

For the second type it is known that for each

isotopy

class

[

ϕ

]

of maps

F

→

F

correspond to an unique bundle

E

ϕ

. Any

homeomorphism

f

:

F

→

F

representing the isotopy class

[

ϕ

]

is called a

monodromy

for

E

ϕ

.

From the previous paragraph we infer that the

mapping class group

play a important role in the understanding at least for this

subclass

of objets.

For the third class above one can use an

orbifold

instead of a simple surface to get a class of 3-manifolds called

Seifert fiber spaces

which are a large class of spaces needed to understand the modern classifications for 3-manifolds.

References

•

J.C. Gómez-Larrañaga.

3-manifolds which are unions of three solid tori

,
Manuscripta Math. 59 (1987), 325-330.

•

J.C. Gómez-Larrañaga, F.J. González-Acuña, J. Hoste.

Minimal

Atlases on 3-manifolds

,
Math. Proc. Camb. Phil. Soc. 109 (1991), 105-115.

•

J. Hempel.

3-manifolds

, Princeton University Press 1976.

•

P. Orlik.

Seifert Manifolds

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

•

P. Scott.

The

geometry

of 3-manifolds

, Bull. London Math. Soc. 15 (1983), 401-487.

Title

3-manifold

Canonical name

3manifold

Date of creation

2013-03-22 15:40:55

Last modified on

2013-03-22 15:40:55

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

17

Author

juanman (12619)

Entry type

Definition

Classification

msc 57N10

Related topic

manifold

Related topic

DehnsLemma

Related topic

SphereTheorem

Related topic

LoopTheorem

Related topic

SeifertFiberSpace

Related topic

Manifold

Generated on Sat Feb 10 11:48:01 2018 by

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irreducible n-manifold

irreducible n-manifold

An

n

-

manifold

(

http://planetmath.org/TopologicalManifold

)

M

is called

irreducible

if for each

embedding

of a standard

(

n

-

1

)

-sphere

S

n

-

1

in

M

, there is an embedding of a standard

n

-ball (

http://planetmath.org/StandardNBall

)

D

n

in

M

such that the
image of the boundary

∂

⁡

D

n

coincides with the image of

S

n

-

1

.

In case of

dimension

three it can be proved that each irreducible

3-manifold

is also a prime (

http://planetmath.org/Prime3Manifold

) 3-manifold.

Title

irreducible n-manifold

Canonical

name

IrreducibleNmanifold

Date of creation

2013-03-22 16:05:43

Last modified on

2013-03-22 16:05:43

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

12

Author

juanman (12619)

Entry type

Definition

Classification

msc 57N10

Generated on Sat Feb 10 11:48:03 2018 by

LaTeXML
prime n-manifold

prime n-manifold

A n-manifold (

http://planetmath.org/TopologicalManifold

)

M

is called if for all the factorizations of

M

, as a

connected sum

(

http://planetmath.org/ConnectedSum2

)

M

=

M

1

⁢

#

⁢

M

2

, one finds that one of the factors

M

1

or

M

2

is the n-sphere

S

n

.

The importance for

3-manifold

is the existence and uniqueness prime decomposition theorem which says: Each 3-manifold can be decomposed as a connected sum of

prime manifolds

.

Title

prime n-manifold

Canonical name

PrimeNmanifold

Date of creation

2013-03-22 16:05:37

Last modified on

2013-03-22 16:05:37

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

14

Author

juanman (12619)

Entry type

Feature

Classification

msc 57N10

Generated on Sat Feb 10 11:48:04 2018 by

LaTeXML
self-intersections of a curve

self-intersections of a curve

self-intersections of a curve

Let

X

be a

topological manifold

and

γ

:

[

0

,

1

]

→

X

a segment of a curve in

X

.

Then the curve is said to have a self-intersection in a point

p

∈

X

if

γ

fails to be injective, i.e. if there exists

a

,

b

∈

(

0

,

1

)

, with

a

≠

b

such that

γ

⁢

(

a

)

=

γ

⁢

(

b

)

.
Usually, the case when the curve is closed i.e.

γ

⁢

(

0

)

=

γ

⁢

(

1

)

, is not considered as a self-intersecting curve.

Title

self-intersections of a curve

Canonical name

SelfintersectionsOfACurve

Date of creation

2013-03-22 14:01:11

Last modified on

2013-03-22 14:01:11

Owner

mike (2826)

Last modified by

mike (2826)

Numerical id

9

Author

mike (2826)

Entry type

Definition

Classification

msc 57N16

Classification

msc 57R42

Generated on Sat Feb 10 11:48:07 2018 by

LaTeXML
h-cobordism

h

-cobordism

A cobordism

(

N

,

M

,

M

′

)

is called an

h

-cobordism if the

inclusion maps

i

:

M

→

N

and

i

′

:

M

′

→

N

are

homotopy equivalences

.

Title

h

-cobordism

Canonical name

Hcobordism

Date of creation

2013-03-22 13:57:19

Last modified on

2013-03-22 13:57:19

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

4

Author

bwebste (988)

Entry type

Definition

Classification

msc 57N70

Generated on Sat Feb 10 11:48:14 2018 by

LaTeXML
oriented cobordism

oriented cobordism

Two

oriented

n

-

manifolds

M

and

M

′

are called

cobordant

if there is an oriented

n

+

1

manifold with boundary

N

such that

∂

⁡

N

=

M

⁢

∐

M

′

⁣

o

⁢

p

⁢

p

where

M

′

⁣

o

⁢

p

⁢

p

is

M

′

with orientation reversed. The triple

(

N

,

M

,

M

′

)

is called a

oriented cobordism

. Cobordism is an

equivalence relation

, and a very coarse

invariant

of manifolds. For example, all

surfaces

are cobordant to the

empty set

(and hence to each other).

There is a cobordism category, where the objects are manifolds, and the

morphisms

are cobordisms between them. This category is important in topological .

Title

oriented cobordism

Canonical

name

OrientedCobordism

Date of creation

2013-03-22 13:56:05

Last modified on

2013-03-22 13:56:05

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 57N70

Classification

msc 57Q20

Synonym

cobordant

Synonym

bordism

Generated on Sat Feb 10 11:48:16 2018 by

LaTeXML
Smale’s h-cobordism theorem

Smale’s

h

-cobordism theorem

Let

(

N

,

M

,

M

′

)

be an

h

-cobordism (

http://planetmath.org/HCobordism

) of

smooth manifolds

with

N

(and hence

M

and

M

′

)

simply connected

, and

dim

⁡

(

M

)

≥

5

. Then

N

≅

M

×

[

0

,

1

]

and

M

and

M

′

are

diffeomorphic

.

Title

Smale’s

h

-cobordism theorem

Canonical name

SmalesHcobordismTheorem

Date of creation

2013-03-22 13:57:22

Last modified on

2013-03-22 13:57:22

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Theorem

Classification

msc 57N70

Generated on Sat Feb 10 11:48:18 2018 by

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Donaldson Freedman exotic R4

Donaldson Freedman exotic R4

Let

K

denote the simply connected closed 4-

manifold

given by

K

=

{

x

:

y

:

z

:

w

∈

ℂ

P

3

|

x

4

+

y

4

+

z

4

+

w

4

=

0

}

Let

E

8

denote the unique rank 8

unimodular

symmetric bilinear form

over

ℤ

, which is

positive definite

and with respect to which, the norm of any vector is even. Let

B

denote the rank 2

bilinear form

over

ℤ

which may be represented by the matrix

(

0

1

1

0

)

Then we may regard

H

2

⁢

(

K

;

ℤ

)

as a

direct sum

M

⊕

N

, where the cup product induces the form

E

8

⊕

E

8

on

M

and

B

⊕

B

⊕

B

on

N

and we have

M

orthogonal

to

N

. (This does not contradict Donaldson’s theorem as

B

has 1 and -1 as

eigenvalues

.)

We may choose a (topological) open ball,

U

, in

#

3

⁢

S

2

×

S

2

which contains a (topological) closed ball,

V

, such that we have a smooth

embedding

,

f

:

#

3

⁢

S

2

×

S

2

-

V

→

K

satisfying the following property:

The map

f

induces an

isomorphism

from

H

2

⁢

(

#

3

⁢

S

2

×

S

2

-

U

;

ℤ

)

into the summand

N

.

If we could smoothly embed

S

3

into

U

-

V

, enclosing

V

, then by replacing the outside of the embedded

S

3

with a copy of

B

4

, and regarding

U

-

V

as
lying in

K

, we obtain a smooth simply connected closed 4- manifold, with
bilinear form

E

8

⊕

E

8

induced by the cup product. This contradicts Donaldson’s theorem.

Therefore,

U

has the property of containing a compact set which is not enclosed by any smoothly embedded

S

3

. Hence

U

is an exotic

ℝ

4

.

By considering the three copies of

B

one at a time, we could have obtained our exotic

ℝ

4

as an open subset of

S

2

×

S

2

.

Title

Donaldson Freedman exotic R4

Canonical name

DonaldsonFreedmanExoticR4

Date of creation

2013-03-22 15:37:36

Last modified on

2013-03-22 15:37:36

Owner

whm22 (2009)

Last modified by

whm22 (2009)

Numerical id

13

Author

whm22 (2009)

Entry type

Application

Classification

msc 57R12

Classification

msc 14J80

Related topic

Donaldsonstheorem

Related topic

exoticR4s

Related topic

ExoticR4s

Related topic

DonaldsonsTheorem

Generated on Sat Feb 10 11:48:22 2018 by

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Donaldson’s theorem

Donaldson’s theorem

Theorem

Let

M

be a smooth

simply connected

, closed 4-

manifold

. If the

bilinear form

induced on

H

2

⁢

(

M

;

𝒁

)

by the

cup product

is

positive definite

, then it may be represented by the

identity matrix

.

One application is the existence of exotic

𝑹

4

’s. See

Donaldson Freedman exotic R4

.

Title

Donaldson’s theorem

Canonical name

DonaldsonsTheorem

Date of creation

2013-03-22 15:37:30

Last modified on

2013-03-22 15:37:30

Owner

whm22 (2009)

Last modified by

whm22 (2009)

Numerical id

9

Author

whm22 (2009)

Entry type

Theorem

Classification

msc 57R12

Classification

msc 14J80

Related topic

exoticR4s

Related topic

DonaldsonFreedmanexoticR4

Related topic

DonaldsonFreedmanExoticR4

Related topic

ExoticR4s

Generated on Sat Feb 10 11:48:24 2018 by

LaTeXML
exotic R4’s

exotic R4’s

If

n

≠

4

then the

smooth manifolds

homeomorphic

to a given topological

n

-

manifold

,

M

, are parameterized by some

discrete

algebraic invariant of

M

. In particular there is a unique smooth manifold homeomorphic to

ℝ

n

.

By contrast one may choose uncountably many

open sets

in

ℝ

4

, which are all homeomorphic to

ℝ

4

, but which are pairwise non-diffeomorphic.

A smooth manifold homeomorphic to

ℝ

4

, but not

diffeomorphic

to it is called an

exotic

ℝ

4

.

Given an exotic

ℝ

4

,

E

, we have a

diffeomorphism

E

×

ℝ

→

ℝ

5

. (As there is only one smooth manifold homeomorphic to

ℝ

5

). Hence exotic

ℝ

4

’s may be identified with closed

submanifolds

of

ℝ

5

. In particular this means the cardinality of the set of exotic

ℝ

4

’s is precisely

continuum

.

Historically, Donaldson’s theorem led to the discovery of the Donaldson Freedman exotic

ℝ

4

.

Title

exotic R4

’s

Canonical name

ExoticR4s

Date of creation

2013-03-22 15:37:33

Last modified on

2013-03-22 15:37:33

Owner

whm22 (2009)

Last modified by

whm22 (2009)

Numerical id

21

Author

whm22 (2009)

Entry type

Definition

Classification

msc 57R12

Classification

msc 14J80

Related topic

DonaldsonsTheorem

Related topic

DonaldsonFreedmanExoticR4

Generated on Sat Feb 10 11:48:26 2018 by

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𝒞^r topologies

𝒞

r

topologies

The

𝒞

r

Whitney (or strong)

topology

is a topology
assigned to the space

𝒞

r

⁢

(

M

,

N

)

of

mappings

from
a

𝒞

r

manifold

M

to a

𝒞

r

manifold

N

having

r

continuous

derivatives

. It gives a notion of

proximity

of two

𝒞

r

mappings, and it allows us to speak of “robustness”
of properties of a mapping. For example, the
property of being an

embedding

is robust: if

f

:

M

→

N

is a

𝒞

r

embedding, then there is a strong

𝒞

r

neighborhood

of

f

in which any

𝒞

r

mapping

g

:

M

→

N

is an embedding.

Given a

locally finite

atlas

{

(

U

i

,

ϕ

i

)

:

i

∈

I

}

and

compact sets

K

i

⊂

U

i

such that there are charts

{

(

V

i

,

ψ

i

)

:

i

∈

I

}

of

N

for which

f

⁢

(

K

i

)

⊂

V

i

for all

i

∈

I

, and given a

sequence

{

ϵ

i

>

0

:

i

∈

I

}

, we define the basic neighborhood

𝒰

r

⁢

(

f

,

ϕ

,

ψ

,

{

K

i

:

i

∈

I

}

,

{

ϵ

i

:

i

∈

I

}

)

as the set of

C

r

mappings

g

:

M

→

N

such that for all

i

∈

I

we have

g

⁢

(

K

i

)

⊂

V

i

and

sup

x

∈

ϕ

i

⁢

(

K

i

)

,

0

≤

k

≤

r

⁡

||

D

k

⁢

(

ψ

i

⁢

f

⁢

ϕ

i

-

1

)

⁢

(

x

)

-

D

k

⁢

(

ψ

i

⁢

g

⁢

ϕ

i

-

1

)

⁢

(

x

)

||

<

ϵ

i

.

That is, those maps

g

that are close to

f

and have their first

r

derivatives close to the respective first

r

-th
derivatives of

f

, in local coordinates.
It can be checked that the set of all such neighborhoods forms a basis
for a topology, which we call the Whitney or strong

𝒞

r

topology of

𝒞

r

⁢

(

M

,

N

)

.

The weak

𝒞

r

topology, or

𝒞

r

compact-open topology

, is defined
in the same fashion but instead of choosing

{

(

U

i

,

ϕ

i

)

:

i

∈

I

}

to be a locally finite atlas for

M

,
we require it to be an arbitrary

finite

family of charts
(possibly not covering

M

).

The space

𝒞

r

⁢

(

M

,

N

)

with the weak or strong topologies is denoted by

𝒞

W

r

⁢

(

M

,

N

)

and

𝒞

S

r

⁢

(

M

,

N

)

, respectively.

We have that

𝒞

W

r

⁢

(

M

,

N

)

is always

metrizable

(with a

complete metric

)
and

separable

. On the other hand,

𝒞

S

r

⁢

(

M

,

N

)

is not even

first countable

(thus, not metrizable) when

M

is not

compact

; however, it is a

Baire space

. When

M

is compact, the weak and strong topologies coincide.

Title

𝒞

r

topologies

Canonical name

mathcalCrTopologies

Date of creation

2013-03-22 14:08:27

Last modified on

2013-03-22 14:08:27

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Definition

Classification

msc 57R12

Synonym

Whitney topology

Synonym

compact-open

𝒞

r

topology

Synonym

weak

𝒞

r

topology

Synonym

strong

\mathcak

⁢

C

r

topology

Related topic

ConjectureApproximationTheoremHoldsForWhitneyCrMNSpaces

Related topic

ApproximationTheoremAppliedToWhitneyCrMNSpaces

Generated on Sat Feb 10 11:48:28 2018 by

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handle decomposition

handle decomposition

Let

M

n

be a smooth, connected, closed

n

dimensional

manifold

.
A

handle

is

H

λ

n

=

B

λ

×

B

n

-

λ

where

B

λ

is a

λ

-ball.

Any such manifold

M

is

diffeomorphic

to the union of finitely many such handles where each handle

H

λ

n

is in a

one-to-one correspondence

with the

critical points

of

λ

of a

Morse function

on

M

.

Title

handle decomposition

Canonical name

HandleDecomposition

Date of creation

2013-03-22 15:21:28

Last modified on

2013-03-22 15:21:28

Owner

RobKing (9598)

Last modified by

RobKing (9598)

Numerical id

8

Author

RobKing (9598)

Entry type

Definition

Classification

msc 57R19

Generated on Sat Feb 10 11:48:30 2018 by

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hairy ball theorem

hairy ball theorem

Theorem

.

If

X

is a vector field on

S

2

⁢

n

, then

X

has a zero.
Alternatively, there are no

continuous

unit vector field on
the sphere. Moreover, the

tangent bundle

of the sphere is
nontrivial as a bundle, that is, it is not simply a

product

.

There are two proofs for this. The first proof is based
on the fact that the

antipodal map

on

S

2

⁢

n

is not

homotopic

to the
identity map. The second proof gives the
as a corollary of the Poincaré-Hopf index theorem.

Near a zero of a vector field, we can consider a small sphere around the zero, and restrict the vector field to that. By normalizing, we get a map from the sphere to itself. We define the index of the vector field at a zero to be the degree of that map.

Theorem

(Poincaré-Hopf index theorem).

If

X

is a vector field on a

compact manifold

M

with

isolated

zeroes, then

χ

⁢

(

M

)

=

∑

v

∈

Z

⁢

(

X

)

ι

⁢

(

v

)

where

Z

⁢

(

X

)

is the set of zeroes of

X

, and

ι

⁢

(

v

)

is
the index of

x

at

v

, and

χ

⁢

(

M

)

is the

Euler characteristic

of

M

.

It is not difficult to show that

S

2

⁢

n

+

1

has non-vanishing vector
fields for all

n

. A much harder result of Adams shows that the
tangent bundle of

S

m

is trivial if and only if

n

=

0

,

1

,

3

,

7

,
corresponding to the unit spheres in the 4 real

division algebras

.

Proof.

First, the low tech proof. Assume that

S

2

⁢

n

has a unit vector field

X

. Then the antipodal map is homotopic to the identity (

http://planetmath.org/AntipodalMapOnSnIsHomotopicToTheIdentityIfAndOnlyIfNIsOdd

).
But this cannot be, since the degree of the antipodal map is

-

1

and
the degree of the identity map is

+

1

. We therefore reject the
assumption that

X

is a unit vector field.

This also implies that the tangent bundle of

S

2

⁢

n

is non-trivial,
since any

trivial bundle

has a non-zero

section

.
∎

Proof.

Now for the sledgehammer proof.
Suppose

X

is a nonvanishing vector field on

S

2

⁢

n

.
Then by the Poincaré-Hopf index theorem, the Euler characteristic
of

S

2

⁢

n

is

χ

⁢

(

X

)

=

∑

v

∈

X

-

1

⁢

(

0

)

ι

⁢

(

v

)

=

0

. But the Euler characteristic of

S

2

⁢

k

is

2

. Hence

X

must have a zero.
∎

Title

hairy ball theorem

Canonical name

HairyBallTheorem

Date of creation

2013-03-22 13:11:33

Last modified on

2013-03-22 13:11:33

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 57R22

Synonym

porcupine theorem

Synonym

PoincarÃÂ©-

Hopf theorem

Defines

Poincaré-Hopf index theorem

Generated on Sat Feb 10 11:48:38 2018 by

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Thom space

Thom space

Let

ξ

→

X

be a

vector bundle

over a

topological space

X

. Assume that

ξ

has a

Riemannian metric

. We can form its associated

disk bundle

D

⁢

(

ξ

)

and its associated sphere bundle

S

⁢

(

ξ

)

, by letting

D

⁢

(

ξ

)

=

{

v

∈

ξ

:

∥

v

∥

≤

1

}

,

S

⁢

(

ξ

)

=

{

v

∈

ξ

:

∥

v

∥

=

1

}

.

The

Thom space

of

ξ

is defined to be the

quotient space

D

⁢

(

ξ

)

/

S

⁢

(

ξ

)

, obtained by taking the disk bundle and collapsing the sphere bundle to a point. Notice that this makes the Thom space naturally into a

based topological space

.

Two common forms of notation for the Thom space are

Th

⁡

(

ξ

)

and

X

ξ

.

Remark 1

If

ξ

=

X

×

ℝ

d

is a trivial vector bundle, then its Thom space is

homeomorphic

to

Σ

d

⁢

X

+

, where

X

+

stands for

X

with an added disjoint basepoint, and

Σ

d

stands for the

based suspension

iterated

d

times. Thus, we may think of

X

ξ

as a “twisted

suspension

” of

X

+

.

Remark 2

If

X

is

compact

, then

X

ξ

is homeomorphic as a based space to the

one-point compactification

of

ξ

. Even if

X

is not compact,

X

ξ

can be obtained by doing a one-point compactification on each fiber and then collapsing the resulting

section

of

points at infinity

to a point.

Remark 3

The choice of Riemannian metric on

ξ

does not change the

homeomorphism

type of

X

ξ

, and, by the previous remark, the Thom space can be described without reference to associated disk and sphere bundles.

Title

Thom space

Canonical name

ThomSpace

Date of creation

2013-03-22 15:40:46

Last modified on

2013-03-22 15:40:46

Owner

antonio (1116)

Last modified by

antonio (1116)

Numerical id

5

Author

antonio (1116)

Entry type

Definition

Classification

msc 57R22

Classification

msc 55R25

Defines

Thom space

Defines

disk bundle

Defines

sphere bundle

Generated on Sat Feb 10 11:48:40 2018 by

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