aarjaneiro/planetmath huggingface dataset:
round complexity

round complexity

Mimicking the

Lusternik-Schnirelmann category

invariant

for a

smooth manifold

M

we can ask about the

minimal number

of critical loops of smooth scalar maps

M

→

ℝ

which are

round functions

, that is

functions

whose

critical points

are aligned in a

disjoint union

of closed curves (a link).

This number is called the

round complexity

of

M

and it is symbolized as

roc

⁢

(

M

)

Then

roc

(

M

)

=

min

#

{

critical loops of

f

|

f

:

M

→

ℝ

is round function

}

This

concept

is related to the invariant called t-cat.

Theorem

1:

The round complexity for the 2-torus and the Klein bottle is two; all the other closed

surfaces

have a round complexity of three.

Theorem 2:

For each

closed manifold

,

t

-

c

⁢

a

⁢

t

≤

r

⁢

o

⁢

c

Bibliography

D. Siersma, G. Khimshiasvili, On

minimal

round functions, Preprint 1118, Department of Mathematics, Utrecht University, 1999, pp. 18.

Title

round complexity

Canonical name

RoundComplexity

Date of creation

2013-03-22 15:54:52

Last modified on

2013-03-22 15:54:52

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

11

Author

juanman (12619)

Entry type

Definition

Classification

msc 55M30

Related topic

LusternikSchnirelmannCategory

Related topic

TCat

Generated on Sat Feb 10 11:41:15 2018 by

LaTeXML
t-cat

t-cat

The

t-cat

of a

topological space

X

is the

minimal number

of

open sets

that cover

X

such that each open set in the cover has the

homotopy type

of the

unit circle

S

1

. This means that for each open set

U

, the

inclusion

U

↪

i

X

is

homotopic

to some factorization

U

→

a

S

1

→

b

X

, i.e.

i

≃

b

∘

a

.

When

X

is

manifold

, this is related to the

round complexity

of

X

.

References

1

D. Siersma, G. Khimshiasvili,

On

minimal

round functions

, Preprint 1118, Department of Mathematics, Utrecht University, 1999, pp. 18.

Title

t-cat

Canonical name

Tcat

Date of creation

2013-03-22 15:54:54

Last modified on

2013-03-22 15:54:54

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

7

Author

juanman (12619)

Entry type

Definition

Classification

msc 55M30

Related topic

RoundComplexity

Generated on Sat Feb 10 11:41:16 2018 by

LaTeXML
genus of topological surface

genus of topological surface

The

genus

is a

topological invariant

of

surfaces

. It is one of the oldest known topological invariants and, in fact, much of

topology

has been created in to generalize this notion to more general situations than the topology of surfaces. Also, it is a

complete

invariant

in the sense that, if two

orientable

closed surfaces have the same genus, then they must be

topologically equivalent

. This important topological invariant may be defined in several

equivalent

ways as given in the result below:

Theorem

.

Let

Σ

be a

compact

, orientable

connected

2

–dimensional

manifold

(a.k.a. surface) without

boundary

. Then the following two numbers are equal (in particular the first number is an integer)

(i)

half the first

Betti number

of

Σ

1

2

⁢

dim

⁡

H

1

⁢

(

Σ

;

ℚ

)

,

(ii)

the

cardinality

of a set

C

of mutually non-intersecting

simple
closed curves

with the

property

that

Σ

∖

C

is a

connected surface

.

Definition

.

The integer of the above

theorem

is called the genus of the
surface.

Theorem

.

Any compact

orientable surface

without boundary is a

connected sum

of

g

tori, where

g

is its genus.

Remark

.

The previous theorem is the reason why genus is sometimes referred to as “the number of handles”.

Theorem

.

The genus is a homeomorphism , i.e. two
compact orientable surfaces without boundary are homeomorphic if and only if they have the same genus.

Title

genus of topological surface

Canonical

name

GenusOfTopologicalSurface

Date of creation

2013-03-22 12:56:21

Last modified on

2013-03-22 12:56:21

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

29

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 55M99

Synonym

genus

Generated on Sat Feb 10 11:41:18 2018 by

LaTeXML
standard n-ball

standard n-ball

The

standard n-ball

is

B

n

=

{

x

∈

ℝ

n

|

|

|

x

|

|

≤

1

}

,
where

||

x

||

2

=

∑

i

=

1

n

x

i

2

.
The standard n-ball is also called the

unit n-ball

or the

n-disk

.

The notation in the literature for the standard n-ball is not uniform.
Sometimes it is denoted by

V

n

,

D

n

,

E

n

or

E

¯

n

.

The

boundary

of the standard n-ball is

S

n

-

1

, the n-1-sphere.

A space that is

homeomorphic

to the standard n-ball is called
an

n-cell

.

A 1-cell is called an arc and a 2-cell is called a disk.

Title

standard n-ball

Canonical name

StandardNball

Date of creation

2013-03-22 16:52:49

Last modified on

2013-03-22 16:52:49

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 55M99

Synonym

unit n-ball

Synonym

unit ball

Synonym

standard

n

-ball

Defines

n-cell

Generated on Sat Feb 10 11:41:19 2018 by

LaTeXML
de Rham cohomology

de Rham cohomology

Let

X

be a

paracompact

𝒞

∞

differential manifold

. Let

Ω

⁢

X

=

⊕

i

=

0

∞

Ω

i

⁢

X

denote the

graded-commutative

ℝ

-

algebra

of

differential forms

on

X

. Together with the

exterior derivative

d

i

:

Ω

i

X

→

Ω

i

+

1

X

(

i

=

0

,

1

,

…

)

,

Ω

⁢

X

forms a

chain complex

(

Ω

⁢

X

,

d

)

of

ℝ

-

vector spaces

. The

H

dR

i

⁢

X

of

X

are defined as the

homology groups

of this complex, that is to say

H

dR

i

X

:=

(

ker

d

i

)

/

(

im

d

i

-

1

)

(

i

=

0

,

1

,

…

)

,

where

Ω

-

1

⁢

X

is taken to be 0, so

d

-

1

:

0

→

Ω

0

⁢

X

is the

zero map

. The

wedge product

in

Ω

⁢

X

induces the structure of a graded-commutative

ℝ

-algebra on

H

dR

⁢

X

:=

⊕

i

=

0

∞

H

dR

i

⁢

X

.

If

X

and

Y

are both paracompact

𝒞

∞

manifolds

and

f

:

X

→

Y

is a

differentiable map

, there is an

induced map

f

*

:

H

dR

⁢

Y

→

H

dR

⁢

X

,

defined by

f

*

⁢

[

ω

]

:=

[

f

*

⁢

ω

]

for

ω

∈

ker

⁡

d

.

Here

[

ω

]

denotes the class of

ω

modulo

im

d

, and the second

f

*

is the map

Ω

⁢

Y

→

Ω

⁢

X

induced by the

functor

Ω

. This action on differentiable maps makes the

de Rham cohomology

into a

contravariant functor

from the

category

of paracompact

𝒞

∞

manifolds to the category of graded-commutative

ℝ

-algebras. It turns out to be

homotopy invariant

; this implies that

homotopy equivalent

manifolds have

isomorphic

de Rham cohomology.

Title

de Rham cohomology

Canonical name

DeRhamCohomology

Date of creation

2013-03-22 14:24:40

Last modified on

2013-03-22 14:24:40

Owner

pbruin (1001)

Last modified by

pbruin (1001)

Numerical id

9

Author

pbruin (1001)

Entry type

Definition

Classification

msc 55N05

Classification

msc 58A12

Defines

de Rham cohomology group

Generated on Sat Feb 10 11:41:21 2018 by

LaTeXML
example of de Rham cohomology

example of de Rham cohomology

If

ω

is a

differential form

on a

smooth manifold

X

, then it is always true that if

ω

is

exact

(

ω

=

d

⁢

η

for some other differential form

η

), then

ω

is

closed

(

d

⁢

ω

=

0

). On some

manifolds

, the opposite is also the case: all

closed forms

of degree at least 1 are exact. However, in general this is not true. The idea of

de Rham cohomology

is to measure the extent to which closed differential forms are not exact in terms of real vector spaces.

The simplest example of a differential manifold (apart from the empty manifold) is the zero-dimensional manifold consisting of a single point. Here the only differential forms are those of degree 0; actually,

Ω

⁢

X

=

Ω

0

⁢

X

≅

ℝ

if

X

is a single point. Applying the definition of the de Rham cohomology gives

H

dR

⁢

X

=

H

dR

0

⁢

X

≅

ℝ

.

Next, we use the fact that the de Rham cohomology is a

homotopy invariant

functor to show that for any

n

≥

0

the de Rham cohomology groups of

ℝ

n

are

H

dR

0

⁢

(

ℝ

n

)

≅

ℝ

and

H

dR

i

⁢

(

ℝ

n

)

=

0

for

i

>

0

.

The reason for this is that

ℝ

n

is

contractible

(

homotopy equivalent

to a point), and so has the same de Rham cohomology. More generally, any contractible manifold has the de Rham cohomology of a point; this is essentially the statement of the Poincaré lemma.

The first example of a non-trivial

H

dR

i

for

i

>

0

is the circle

S

1

. In fact, we have

H

dR

0

⁢

(

S

1

)

≅

ℝ

and

H

dR

1

⁢

(

S

1

)

≅

ℝ

⋅

[

ω

]

,

where

ω

is any 1-form on

S

1

with

∫

S

1

ω

≠

0

. The standard

volume form

d

⁢

ϕ

on

S

1

, which it inherits from

ℝ

2

if we view

S

1

as the

unit circle

, is such a form. The notation

d

⁢

ϕ

is somewhat misleading since it is not the

differential

of a global

function

ϕ

; this is exactly the reason it appears in

H

dR

1

⁢

(

S

1

)

. (However, by the Poincaré lemma, it can locally be viewed as the differential of a function.)

For arbitrary

n

>

0

, the dimensions of the de Rham cohomology groups of

S

n

are given by

dim

⁡

H

dR

i

⁢

(

S

n

)

=

1

for

i

=

0

or

i

=

n

, and

dim

⁡

H

dR

i

⁢

(

S

n

)

=

0

otherwise. A couple of methods exist for calculating the de Rham cohomology groups for

S

n

and other, more complicated, manifolds. The

Mayer-Vietoris sequence

is an example of such a tool.

Title

example of de Rham cohomology

Canonical name

ExampleOfDeRhamCohomology

Date of creation

2013-03-22 14:25:01

Last modified on

2013-03-22 14:25:01

Owner

pbruin (1001)

Last modified by

pbruin (1001)

Numerical id

5

Author

pbruin (1001)

Entry type

Example

Classification

msc 55N05

Classification

msc 58A12

Generated on Sat Feb 10 11:41:23 2018 by

LaTeXML
Betti number

Betti number

Let

X

denote a

topological space

, and let

H

k

⁢

(

X

,

ℤ

)

denote the

k

-th

homology group

of

X

. If

H

k

⁢

(

X

,

ℤ

)

is

finitely generated

, then its rank is called the

k

-th

Betti number

of

X

.

Title

Betti number

Canonical name

BettiNumber

Date of creation

2013-03-22 13:47:14

Last modified on

2013-03-22 13:47:14

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 55N10

Related topic

homology

Related topic

HomologyTopologicalSpace

Generated on Sat Feb 10 11:41:25 2018 by

LaTeXML
cellular homology

cellular homology

If

X

is a cell space, then let

(

𝒞

*

⁢

(

X

)

,

𝔡

)

be the

cell complex

where the

n

-th group

𝒞

n

⁢

(

X

)

is the

free abelian group

on the cells of

dimension

n

, and the

boundary map

is as follows: If

e

n

is an

n

-cell, then we can define a map

φ

f

:

∂

⁡

e

n

→

f

n

-

1

, where

f

n

-

1

is any cell of dimension

n

-

1

by the following rule: let

φ

:

e

n

→

sk

n

-

1

⁢

X

be the attaching map
for

e

n

, where

sk

n

-

1

⁢

X

is the

(

n

-

1

)

-

skeleton

of

X

. Then let

π

f

be the

natural projection

π

f

:

sk

n

-

1

⁢

X

→

sk

n

-

1

⁢

X

/

(

sk

n

-

1

⁢

X

-

f

)

≅

f

/

∂

⁡

f

.

Let

φ

f

=

π

f

∘

φ

. Now,

f

/

∂

⁡

f

is a (n-1)-sphere, so the map

φ

f

has a degree

deg

⁡

f

which we use to define the

boundary operator

:

𝔡

⁢

(

[

e

n

]

)

=

∑

dim

⁡

f

=

n

-

1

(

deg

⁡

φ

f

)

⁢

[

f

n

-

1

]

.

The resulting

chain complex

is called the cellular chain complex.

Theorem 1

The

homology

of the cellular complex is the same as the

singular homology

of the space. That is

H

*

⁢

(

𝒞

,

𝔡

)

=

H

*

⁢

(

C

,

∂

)

.

Cellular homology

is tremendously useful for computations because the groups involved are

finitely
generated

.

Title

cellular homology

Canonical

name

CellularHomology

Date of creation

2013-03-22 13:14:55

Last modified on

2013-03-22 13:14:55

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Definition

Classification

msc 55N10

Generated on Sat Feb 10 11:41:27 2018 by

LaTeXML
homology of ℝ⁢ℙ^3.

homology of

ℝ

⁢

ℙ

3

.

We need for this problem knowledge of the

homology groups

of

S

2

and

ℝ

⁢

ℙ

2

. We will simply assume the former:

H

k

⁢

(

S

2

;

ℤ

)

=

{

ℤ

k

=

0

,

2

0

e

⁢

l

⁢

s

⁢

e

Now, for

ℝ

⁢

ℙ

2

, we can argue without Mayer-Vietoris.

X

=

ℝ

⁢

ℙ

2

is connected, so

H

0

⁢

(

X

;

ℤ

)

=

ℤ

.

X

is non-orientable, so

H

2

⁢

(

X

;

ℤ

)

is 0. Last,

H

1

⁢

(

X

;

ℤ

)

is the

abelianization

of the already

abelian

fundamental group

π

1

⁢

(

X

)

=

ℤ

/

2

⁢

ℤ

, so we have:

H

k

⁢

(

ℝ

⁢

ℙ

2

;

ℤ

)

=

{

ℤ

k

=

0

ℤ

/

2

⁢

ℤ

k

=

1

0

k

≥

2

Now that we have the

homology

of

ℝ

⁢

ℙ

2

, we can compute the
homology of

ℝ

⁢

ℙ

3

from Mayer-Vietoris. Let

X

=

ℝ

⁢

ℙ

3

,

V

=

ℝ

⁢

ℙ

3

\

{

p

⁢

t

}

∼

ℝ

⁢

ℙ

2

(by vieweing

ℝ

⁢

ℙ

3

as a

CW-complex

),

U

∼

D

3

∼

{

p

⁢

t

}

, and

U

∩

V

∼

S

2

, where

∼

denotes

equivalence

through a

deformation retract

. Then the

Mayer-Vietoris sequence

gives

Generated on Sat Feb 10 11:41:36 2018 by

LaTeXML
Kunneth theorem

Kunneth theorem

Let

X

,

Y

be

topological spaces

. One can ask a question: how

homology

of

X

×

Y

are related to homologies of

X

and

Y

. The answer to this question depends on the homology theory we’re talking about and also the

coefficients ring

. On the other hand, it is well known, that all homology theories are

isomorphic

on

CW-complexes

. Thus we may restrict to CW-complexes. Nevertheless the following theorem is more general:

Theorem.

(Kunneth)

Assume, that

X

,

Y

are topological spaces and

R

is a

principal ideal domain

. Denote by

H

*

⁢

(

X

,

R

)

the

singular homology

with coefficients in

R

. Then, for any

k

>

0

there exists following

short exact sequence

in the

category

of

R

-modules:

0

→

⊕

i

+

j

=

k

H

i

⁢

(

X

,

R

)

⊗

R

H

j

⁢

(

Y

,

R

)

→

H

k

⁢

(

X

×

Y

,

R

)

→

⊕

i

+

j

=

k

-

1

Tor

1

R

⁢

(

H

i

⁢

(

X

,

R

)

,

H

j

⁢

(

Y

,

R

)

)

→

0

,

where

Tor

denotes the Tor functor. Furthermore this sequence splits, i.e. the middle term is a

direct sum

(up to an

isomorphism

) of left and right term.

It should be mentioned, that if

R

=

𝔽

is a field, then the Tor functor is always trivial (i.e.

Tor

1

𝔽

⁢

(

M

,

N

)

=

0

for all

vector spaces

M

,

N

over

𝔽

) and in this case

Kunneth formula

can be stated as

H

k

⁢

(

X

×

Y

,

𝔽

)

≃

⊕

i

+

j

=

k

H

i

⁢

(

X

,

𝔽

)

⊗

𝔽

H

j

⁢

(

Y

,

𝔽

)

for any

k

>

0

.

Title

Kunneth theorem

Canonical name

KunnethTheorem

Date of creation

2013-03-22 19:13:57

Last modified on

2013-03-22 19:13:57

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

7

Author

joking (16130)

Entry type

Theorem

Classification

msc 55N10

Synonym

Kunneth formula

Synonym

Kunneth’s formula

Synonym

Kunneth’s theorem

Synonym

Kunneth theorem

Generated on Sat Feb 10 11:41:38 2018 by

LaTeXML
long exact sequence (of homology groups)

long exact sequence (of homology groups)

If

X

is a

topological space

, and

A

and

B

are

subspaces

with

X

⊃

A

⊃

B

,
then there is a

long exact sequence

:

⋯

→

H

n

⁢

(

A

,

B

)

→

i

*

H

n

⁢

(

X

,

B

)

→

j

*

H

n

⁢

(

X

,

A

)

→

∂

*

H

n

-

1

⁢

(

A

,

B

)

→

where

i

*

is induced by the

inclusion

i

:

(

A

,

B

)

↪

(

X

,

B

)

,

j

*

by the inclusion

j

:

(

X

,

B

)

↪

(

X

,

A

)

, and

∂

is the following map: given

a

∈

H

n

⁢

(

X

,

A

)

,
choose a chain representing it.

∂

⁡

a

is an

(

n

-

1

)

-chain of

A

, so it

represents

an element
of

H

n

-

1

⁢

(

A

,

B

)

. This is

∂

*

⁡

a

.

When

B

is the

empty set

, we get the long exact sequence of the pair

(

X

,

A

)

:

⋯

→

H

n

⁢

(

A

)

→

i

*

H

n

⁢

(

X

)

→

j

*

H

n

⁢

(

X

,

A

)

→

∂

*

H

n

-

1

⁢

(

A

)

→

The

existence

of this long exact sequence follows from the

short exact sequence

0

→

C

*

⁢

(

A

,

B

)

→

i

♯

C

*

⁢

(

X

,

B

)

→

j

♯

C

*

⁢

(

X

,

A

)

→

0

where

i

♯

and

j

♯

are the maps on chains induced by

i

and

j

,
by the

Snake Lemma

.

Title

long exact sequence (of

homology groups

)

Canonical name

LongExactSequenceofHomologyGroups

Date of creation

2013-03-22 13:14:50

Last modified on

2013-03-22 13:14:50

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 55N10

Related topic

NChain

Related topic

ProofOfSnakeLemma

Generated on Sat Feb 10 11:41:40 2018 by

LaTeXML
Mayer-Vietoris sequence

Mayer-Vietoris sequence

Let

X

is a

topological space

, and

A

,

B

⊂

X

are such that

X

=

int

⁢

(

A

)

∪

int

⁢

(

B

)

, and

C

=

A

∩

B

. Then there is an

exact sequence

of

homology groups

:

⋯

→

H

n

⁢

(

C

)

→

i

*

⊕

-

j

*

H

n

⁢

(

A

)

⊕

H

n

⁢

(

B

)

→

j

*

+

i

*

H

n

⁢

(

X

)

→

∂

*

H

n

-

1

⁢

(

C

)

→

⋯

Here,

i

*

is

induced

by the inclusions

i

:

B

↪

X

and

j

*

by

j

:

A

↪

X

, and

∂

*

is the following map:
if

x

is in

H

n

⁢

(

X

)

, then it can be written as the sum of a chain in

A

and one in

B

,

x

=

a

+

b

.

∂

⁡

a

=

-

∂

⁡

b

, since

∂

⁡

x

=

0

. Thus,

∂

⁡

a

is a chain in

C

, and so

represents

a class in

H

n

-

1

⁢

(

C

)

. This is

∂

*

⁡

x

. One can easily check (by standard

diagram chasing

) that this map is well defined on the level of

homology

.

Title

Mayer-Vietoris sequence

Canonical name

MayerVietorisSequence

Date of creation

2013-03-22 13:14:52

Last modified on

2013-03-22 13:14:52

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Definition

Classification

msc 55N10

Generated on Sat Feb 10 11:41:42 2018 by

LaTeXML
relative homology groups

relative homology groups

If

X

is a

topological space

, and

A

a

subspace

, then the

inclusion map

A

↪

X

makes

C

n

⁢

(

A

)

into a subgroup of

C

n

⁢

(

X

)

. Since the

boundary map

on

C

*

⁢

(

X

)

restricts to
the boundary map on

C

*

⁢

(

A

)

, we can take the

quotient

complex

C

*

⁢

(

X

,

A

)

,

←

∂

C

n

⁢

(

X

)

/

C

n

⁢

(

A

)

←

∂

C

n

+

1

⁢

(

X

)

/

C

n

+

1

⁢

(

A

)

←

∂

The

homology groups

of this complex

H

n

⁢

(

X

,

A

)

, are called the

relative homology groups

of the pair

(

X

,

A

)

. Under relatively mild hypotheses,

H

n

⁢

(

X

,

A

)

=

H

n

⁢

(

X

/

A

)

where

X

/

A

is
the set of

equivalence classes

of the

relation

x

∼

y

if

x

=

y

or if

x

,

y

∈

A

, given the

quotient
topology

(this is essentially

X

, with

A

reduced

to a single point). Relative homology groups are
important for a number of reasons, principally for computational ones, since they fit into

long
exact sequences

, which are powerful computational tools in

homology

.

Title

relative homology groups

Canonical

name

RelativeHomologyGroups

Date of creation

2013-03-22 13:14:47

Last modified on

2013-03-22 13:14:47

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 55N10

Generated on Sat Feb 10 11:41:43 2018 by

LaTeXML
homological complex of topological vector spaces

homological complex of topological vector spaces

Definition 0.1

.

A

homological complex of topological vector spaces

is a pair

(

E

∙

,

d

)

,
where

E

∙

=

(

E

q

)

q

∈

Z

is a

sequence of topological vector spaces

and

d

=

(

d

q

)

q

∈

Z

is a sequence of continuous linear maps

d

q

from

E

q

+

1

into

E

q

which satisfy

d

q

∘

d

q

+

1

=

0

.

Remarks

•

The

homological complex of topological vector spaces

is a specifc example of a

chain complex

.

•

A

sequence

of

R

-modules and their

homomorphisms

is said to be a

R

-

complex

.

Title

homological complex of topological vector spaces

Canonical name

HomologicalComplexOfTopologicalVectorSpaces

Date of creation

2013-03-22 18:17:24

Last modified on

2013-03-22 18:17:24

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

15

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55N25

Classification

msc 55N99

Classification

msc 55R65

Classification

msc 55P99

Synonym

ChainComplex

Related topic

ChainComplex

Related topic

CategoricalSequence

Related topic

ExactSequence

Related topic

TangentialCauchyRiemannComplexOfCinftySmoothForms

Related topic

HomolgyOfMathbbRP3

Related topic

CohomologicalComplexOfTopologicalVectorSpaces

Related topic

ACRcomplex

Defines

homological complex of topological vector spaces

Generated on Sat Feb 10 11:41:48 2018 by

LaTeXML
cohomology group theorem

cohomology group theorem

The following theorem involves

Eilenberg-MacLane spaces

in

relation

to

cohomology groups

for
connected

CW-complexes

.

Theorem 0.1

.

Cohomology group theorem

for connected CW-complexes (

[

1

]

):

Let

K

⁢

(

π

,

n

)

be Eilenberg-MacLane spaces for connected
CW complexes (

http://planetmath.org/CWComplexDefinitionRelatedToSpinNetworksAndSpinFoams

)

X

,

Abelian groups

π

and integers

n

≥

0

. Let us also consider the set of non-basepointed

homotopy classes

[

X

,

K

⁢

(

π

,

n

)

]

of non-basepointed maps

η

:

X

→

K

⁢

(

π

,

n

)

and the cohomolgy groups (

http://planetmath.org/GroupCohomology

)

H

¯

n

⁢

(

X

;

π

)

. Then, there exist the following

natural isomorphisms

:

[

X

,

K

⁢

(

π

,

n

)

]

≅

H

¯

n

⁢

(

X

;

π

)

,

(0.1)

0.1

Related remarks:

1.

In order to determine all cohomology operations one needs only to compute the

cohomology

of all
Eilenberg-MacLane spaces

K

⁢

(

π

,

n

)

; (source: ref

[

1

]

);

2.

When

n

=

1

, and

π

is

non-Abelian

, one still has that

[

X

,

K

⁢

(

π

,

1

)

]

≅

H

⁢

o

⁢

m

⁢

(

π

1

⁢

(

X

)

,

π

)

/

π

, that is, the

conjugacy class

or

representation

of

π

1

into

π

;

3.

A

derivation

of this result based on the fundamental cohomology theorem is also attached.

References

1

May, J.P. 1999.

A Concise Course in Algebraic Topology

, The University of Chicago Press: Chicago.,p.173.

Title

cohomology group theorem

Canonical name

CohomologyGroupTheorem

Date of creation

2013-03-22 18:14:43

Last modified on

2013-03-22 18:14:43

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

48

Author

bci1 (20947)

Entry type

Theorem

Classification

msc 55N33

Classification

msc 55N20

Synonym

fundamental cohomology theorem

Related topic

GroupCohomology

Related topic

EilenbergMacLaneSpace

Related topic

HomotopyGroups

Related topic

HomotopyCategory

Related topic

GroupCohomologyTopologicalDefinition

Related topic

TangentialCauchyRiemannComplexOfCinftySmoothForms

Related topic

HomologyTopologicalSpace

Related topic

ProofOfCohomologyGroupTheorem

Related topic

OmegaSpectrum

Related topic

ACRcomplex

Defines

conjugacy class or representation of

π

1

into

π

Defines

set of based homotopy classes of based maps

Generated on Sat Feb 10 11:41:50 2018 by

LaTeXML