aarjaneiro/planetmath huggingface dataset:
cubically thin homotopy

cubically thin homotopy

0.1

Cubically thin homotopy

Let

u

,

u

′

be squares in

X

with common vertices.

1.

A

cubically thin homotopy

U

:

u

≡

T

□

u

′

between

u

and

u

′

is a cube (

http://planetmath.org/Polyhedron

)

U

∈

R

3

□

⁢

(

X

)

such that

–

U

is a

homotopy

between

u

and

u

′

,

i.e.

∂

1

-

⁡

(

U

)

=

u

,

∂

1

+

⁡

(

U

)

=

u

′

,

–

U

is rel. vertices of

I

2

,

i.e.

∂

2

-

⁡

∂

2

-

⁡

(

U

)

,

∂

2

-

⁡

∂

2

+

⁡

(

U

)

,

∂

2

+

⁡

∂

2

-

⁡

(

U

)

,

∂

2

+

⁡

∂

2

+

⁡

(

U

)

are

constant

,

–

the faces

∂

i

α

⁡

(

U

)

are thin for

α

=

±

1

,

i

=

1

,

2

.

2.

The square

u

is

cubically

T

-

equivalent

to

u

′

,

denoted

u

≡

T

□

u

′

if there is a cubically
thin homotopy between

u

and

u

′

.

This definition enables one to

construct

the

homotopy double groupoid

scheme

𝝆

2

□

⁢

(

X

)

, by defining a

relation

of cubically thin homotopy on the set

R

2

□

⁢

(

X

)

of squares.

References

1

K.A. Hardie, K.H. Kamps and R.W. Kieboom, A homotopy

2-groupoid

of a

Hausdorff space

,

Applied Cat.

Structures

,

8

(2000): 209-234.

2

R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoid of a Hausdorff
space,

Theory and Applications of Categories

10

,(2002): 71-93.

Title

cubically thin homotopy

Canonical name

CubicallyThinHomotopy

Date of creation

2013-03-22 18:15:06

Last modified on

2013-03-22 18:15:06

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

17

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55N33

Classification

msc 55N20

Classification

msc 55U40

Classification

msc 18D05

Synonym

higher dimensional thin homotopy

Related topic

HomotopyDoubleGroupoidOfAHausdorffSpace

Related topic

HomotopyAdditionLemma

Related topic

WeakHomotopyAdditionLemma

Related topic

Polyhedron

Defines

higher dimensional thin Homotopy

Generated on Sat Feb 10 11:41:52 2018 by

LaTeXML
example of cohomology and Mayer-Vietoris sequence

example of cohomology and Mayer-Vietoris sequence

Consider n-dimensional sphere

S

n

=

{

v

∈

R

n

+

1

|

|

v

|

=

1

}

.

Let

A

=

{

(

x

0

,

…

,

x

n

)

∈

S

n

|

x

0

>

-

1

/

2

}

and

B

=

{

(

x

0

,

…

,

x

n

)

∈

S

n

|

x

0

<

1

/

2

}

.

Of course both

A

and

B

are open (in

S

n

) and their union is

S

n

. Furthermore, it can be easily seen, that their

intersection

can be contracted into ”big circle”, i.e.

A

∩

B

has

homotopy type

of

S

n

-

1

. Also both

A

and

B

are

contractible

(they are

homeomorphic

to

R

n

via

stereographic projection

). So, write part of a Meyer-Vietoris

sequence

(for the

cohomology

H

m

⁢

(

X

)

=

H

m

⁢

(

X

,

G

)

, where

G

is a fixed

Abelian group

):

⋯

→

H

m

⁢

(

A

)

⊕

H

m

⁢

(

B

)

→

H

m

⁢

(

A

∩

B

)

→

H

m

+

1

⁢

(

S

n

)

→

H

m

+

1

⁢

(

A

)

⊕

H

m

+

1

⁢

(

B

)

→

⋯

Since both

A

and

B

are contractible and

A

∩

B

is

homotopic

to

S

n

-

1

, we have the following

short exact sequence

:

0

→

H

m

⁢

(

S

n

-

1

)

→

H

m

+

1

⁢

(

S

n

)

→

0

which shows that

H

m

⁢

(

S

n

-

1

)

is

isomorphic

to

H

m

⁢

(

S

n

)

for every

n

>

0

and

m

>

0

. So, in order to calculate

cohomology groups

of spheres, we only need to know the cohomology groups of

S

1

. And those can be also calculated, if we once again apply previous schema. Note, that in the case of

S

1

we have that

A

∩

B

has the homotopy type of a

discrete space

with two points. Therefore all their cohomology groups are trivial, except for

H

0

(which can be easily calculated to be equal to

H

0

⁢

(

*

)

⊕

H

0

⁢

(

*

)

, where * is a one-pointed space).

This schema can be used for other spaces like the torus (which can be also calculated from Kunneth’s formula).

Title

example of cohomology and

Mayer-Vietoris sequence

Canonical name

ExampleOfCohomologyAndMayerVietorisSequence

Date of creation

2013-03-22 19:13:27

Last modified on

2013-03-22 19:13:27

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

7

Author

joking (16130)

Entry type

Example

Classification

msc 55N33

Generated on Sat Feb 10 11:41:54 2018 by

LaTeXML
Fourier-Stieltjes algebra of a groupoid

Fourier-Stieltjes algebra of a groupoid

Definition 0.1

.

The

Fourier-Stieltjes algebra

of a

groupoid

,

G

l

.
In ref.

[

3

]

), A.L.T. Paterson defined the

Fourier-Stieltjes algebra of a
groupoid

,

G

l

, as the space of coefficients

ϕ

=

(

ξ

,

η

)

, where

ξ

,

η

are

L

∞

-

sections

for some measurable

G

l

-

Hilbert bundle

(

μ

,

ℜ

,

L

)

. Thus, for

x

∈

G

l

,

ϕ

⁢

(

x

)

=

L

⁢

(

x

)

⁢

ξ

⁢

(

s

⁢

(

x

)

,

η

⁢

(

r

⁢

(

x

)

)

)

.

(0.1)

Therefore,

ϕ

belongs to

L

∞

⁢

G

l

=

L

∞

⁢

(

G

l

,

ν

)

.

References

1

A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of

locally compact groupoids

,

J.

Functional

Anal

.

148

: 314-367 (1997).

2

A. L. T. Paterson, The Fourier algebra for locally compact groupoids., Preprint, (2001).

3

A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locally
compact groupoids, (2003).

Title

Fourier-Stieltjes algebra of a groupoid

Canonical name

FourierStieltjesAlgebraOfAGroupoid

Date of creation

2013-03-22 18:16:11

Last modified on

2013-03-22 18:16:11

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

11

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55N33

Classification

msc 55N20

Classification

msc 55P10

Classification

msc 55U40

Classification

msc 42B10

Classification

msc 42A38

Classification

msc 43A25

Classification

msc 43A30

Synonym

Fourier-Stieltjes algebra of a groupoid

Related topic

FourierStieltjesTransform

Related topic

Distribution4

Defines

Fourier-Stieltjes algebra

Generated on Sat Feb 10 11:41:57 2018 by

LaTeXML
frame groupoid

frame groupoid

Definition 0.1

.

Let

𝒢

be a

groupoid

, defined as usual by a

category

in which all

morphisms

are

invertible

, with the

structure maps

s

,

t

:

G

1

⟶

G

0

, and

u

:

G

0

⟶

G

1

. Given a

vector bundle

q

:

E

⟶

G

0

, the

frame groupoid

is defined as

Φ

⁢

(

E

)

=

s

,

t

:

ϕ

⁢

(

E

)

⟶

G

0

, with

ϕ

⁢

(

E

)

being the set of all

vector space

isomorphisms

η

:

E

x

⟶

E

y

over all pairs

(

x

,

y

)

∈

G

0

2

, also with the usual conditions for the structure maps of the groupoid.

Definition 0.2

.

Let

G

be a group and

V

a vector space. A

group representation

is then defined as a

homomorphism

h

:

G

⟶

E

⁢

n

⁢

d

⁢

(

V

)

,

with

E

⁢

n

⁢

d

⁢

(

V

)

being the group of

endomorphisms

e

:

V

⟶

V

of the vector space

V

.

Note:

With the notation used above, let us consider

q

:

E

⟶

G

0

to be a vector bundle. Then, consider a

group representation

– which was here defined as the representation

R

G

of a group

G

via

the

group action

on the vector space

V

, or as the homomorphism

h

:

G

⟶

E

⁢

n

⁢

d

⁢

(

V

)

, with

E

⁢

n

⁢

d

⁢

(

V

)

being the group of endomorphisms of the vector space

V

. The

generalization

of group representations to the representations of groupoids then occurs naturally by considering the

groupoid action

on a vector bundle

q

:

E

⟶

G

0

. Therefore, the frame groupoid enters into the definition of

groupoid representations

(

http://planetmath.org/GroupoidRepresentation4

).

Title

frame groupoid

Canonical name

FrameGroupoid

Date of creation

2013-03-22 19:19:14

Last modified on

2013-03-22 19:19:14

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

29

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55N33

Classification

msc 55N20

Classification

msc 55P10

Classification

msc 22A22

Classification

msc 20L05

Classification

msc 18B40

Classification

msc 55U40

Related topic

GroupAction

Related topic

VectorBundle

Related topic

GroupoidRepresentation4

Related topic

RepresentationsOfLocallyCompactGroupoids

Related topic

Functor

Related topic

FunctionalBiology

Defines

group representation

Defines

End(V)

Defines

group endomorphism

Defines

Lie groupoid representation

Defines

structure maps

Generated on Sat Feb 10 11:42:02 2018 by

LaTeXML
groupoid C*-dynamical system

groupoid C*-dynamical system

Definition 0.1

.

A

C*-

groupoid

system

or

groupoid C*-

dynamical system

is a

triple

(

A

,

𝖦

l

⁢

c

,

ρ

)

, where:

A

is a C*-

algebra

, and

𝖦

l

⁢

c

is a locally compact (topological) groupoid
with a countable basis for which there exists an associated

continuous

Haar system

and a continuous
groupoid (homo)

morphism

ρ

:

𝖦

l

⁢

c

⟶

A

⁢

u

⁢

t

⁢

(

A

)

defined
by the assignment

x

↦

ρ

x

⁢

(

a

)

(from

𝖦

l

⁢

c

to

A

)
which is continuous for any

a

∈

A

; moreover, one considers the norm topology
on

A

in defining

𝖦

l

⁢

c

. (Definition introduced in ref.

[

1

]

.)

Remark 0.1

.

A

groupoid C*-dynamical system

can be regarded as an

extension

of the ordinary

concept

of dynamical system. Thus, it can also be utilized to

represent

a quantum dynamical system
upon further

specification

of the C*-algebra as a

von Neumann algebra

(

http://planetmath.org/VonNeumannAlgebra

), and also of

𝖦

l

⁢

c

as a

quantum groupoid

(

http://planetmath.org/QuantumGroupoids2

); in the latter case, with additional conditions it can also simulate either

quantum automata

(

http://planetmath.org/QuantumAutomataAndQuantumComputation2

), or variable classical

automata

, depending on the added

restrictions

(ergodicity, etc.).

References

1

T. Matsuda, Groupoid dynamical systems and crossed product, II-case of C*-systems.,

Publ. RIMS

, Kyoto Univ.,

20

: 959-976 (1984).

Title

groupoid C*-dynamical system

Canonical name

GroupoidCdynamicalSystem

Date of creation

2013-03-22 18:16:33

Last modified on

2013-03-22 18:16:33

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

23

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55N33

Classification

msc 55N20

Classification

msc 55P10

Classification

msc 55U40

Classification

msc 18B30

Classification

msc 46L85

Classification

msc 18D05

Classification

msc 37-00

Classification

msc 37B45

Classification

msc 46L55

Classification

msc 22D25

Classification

msc 28C10

Classification

msc 22A22

Synonym

C*-groupoid system

Synonym

locally compact dynamical system with Haar measure

Related topic

CAlgebra

Related topic

CAlgebra3

Related topic

VonNeumannAlgebra

Related topic

DynamicalSystem

Related topic

NuclearCAlgebra

Related topic

SystemDefinitions

Related topic

SimilarityAndAnalogousSystemsDynamicAdjointness2

Related topic

QuantumAutomataAndQuantumComputation2

Related topic

VariableTopology3

Related topic

QuantumGroupoids2

Related topic

OrganismicSupercategoriesAndComplexS

Defines

C*-groupoid system

Defines

locally compact dynamical system

Defines

continuous groupoid automorphism

Defines

locally compact dynamical system with Haar measure

Defines

continuous groupoid homomorphism

Defines

dynamical system

Generated on Sat Feb 10 11:42:07 2018 by

LaTeXML
groupoid representation

groupoid representation

Let

q

:

E

⟶

M

be a

vector bundle

, with

E

the

total space

, and

M

a smooth manifold. Then, consider the

representation

R

G

of a group

G

as an action on a

vector space

V

, that is, as a

homomorphism

h

:

G

⟶

E

⁢

n

⁢

d

⁢

(

V

)

, with

E

⁢

n

⁢

d

⁢

(

V

)

being the group of endomorphisms of the vector space

V

. The

generalization

of the

group representation

to general representations of

groupoids

then occurs somewhat naturally by considering the

groupoid action

(

http://planetmath.org/GroupoidAction

) on a vector bundle

E

⟶

M

.

Definition 0.1

.

Let

𝒢

be a groupoid, and given a vector bundle

q

:

E

⟶

M

consider the

frame groupoid

Φ

⁢

(

E

)

=

s

,

t

:

ϕ

⁢

(

E

)

⟶

M

,

with

ϕ

⁢

(

E

)

being the set of all vector space

isomorphisms

η

:

E

x

⟶

E

y

over all pairs

(

x

,

y

)

∈

M

2

, also with the associated

structure maps

. Then, a general

representation

R

d

of a groupoid

𝒢

is defined as a homomorphism

R

d

:

𝒢

⟶

Φ

⁢

(

E

)

Example 0.1: Lie groupoid representations

Definition 0.2

.

Let

𝒢

L

=

s

,

t

:

G

1

⟶

M

be a

Lie groupoid

. A

representation of a Lie groupoid

𝒢

L

=

s

,

t

:

G

1

⟶

M

on a vector bundle

q

:

E

⟶

M

is defined as a smooth homomorphism (or a

functor

)

ρ

:

𝒢

L

⟶

Φ

⁢

(

E

)

of Lie groupoids over

M

.

Note:

A

Lie groupoid representation

ρ

thus yields a functor,

R

:

𝒢

L

⟶

𝐕𝐞𝐜𝐭

,

with

𝐕𝐞𝐜𝐭

being the category of vector spaces and

R

⁢

(

x

)

=

E

x

being the fiber at each

x

∈

M

, as well as an isomorphism

R

⁢

(

g

)

for each

g

:

x

→

y

.

Example 0.2: Group representations

If one restricts the vector bundle to a single vector space in

Definition 0.1

then one obtains a group representation, in the same manner as a groupoid that reduces to a group when its object space is

reduced

to a single object.

Title

groupoid representation

Canonical name

GroupoidRepresentation

Date of creation

2013-03-22 19:19:17

Last modified on

2013-03-22 19:19:17

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

40

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55N33

Classification

msc 55N20

Classification

msc 55P10

Classification

msc 22A22

Classification

msc 20L05

Classification

msc 55U40

Related topic

GroupoidAction

Related topic

FrameGroupoid

Related topic

RepresentationsOfLocallyCompactGroupoids

Related topic

Functor

Related topic

FrameGroupoid

Related topic

LieGroupoid

Related topic

CategoryOfRepresentations

Related topic

FunctionalBiology

Defines

frame groupoid

Defines

Vect

Defines

End(V)

Defines

group endomorphism

Defines

Lie groupoid representation

Generated on Sat Feb 10 11:42:12 2018 by

LaTeXML
homotopy double groupoid of a Hausdorff space

homotopy double groupoid of a Hausdorff space

Homotopy double groupoid

of a Hausdorff space

Let

X

be a Hausdorff space. Also consider the HDA concept of a

double groupoid

(

http://planetmath.org/HigherDimensionalAlgebraHDA

),
and how it can be completely specified for a Hausdorff space,

X

. Thus, in
ref.

[

BHKP

]

Brown et al. associated to

X

a double groupoid,

𝝆

2

□

⁢

(

X

)

, called the

homotopy double groupoid of X

which is completely defined by the data specified in Definitions 0.1 to 0.3 in this entry and related objects.

Generally, the geometry of squares and their

compositions

leads to a common

representation

of a

double groupoid

in the following form:

𝖣

=

(0.1)

Generated on Sat Feb 10 11:42:16 2018 by

LaTeXML
thin double track

thin double track

Definition 0.1

.

Let us consider a

square

map

u

:

I

2

⟶

X

in a

topological space

X

.
Then the

homotopy class

of

u

relative to the

boundary

∂

⁡

I

2

of

I

is called a

double track

. A double track is

thin

if it has a thin representative.

Title

thin double track

Canonical

name

ThinDoubleTrack

Date of creation

2013-03-22 18:14:49

Last modified on

2013-03-22 18:14:49

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

7

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55N33

Classification

msc 55N20

Classification

msc 55U40

Classification

msc 18D05

Synonym

double track

Related topic

GeometricallyAndorAlgebraicallyThinSquares

Related topic

HomotopyDoubleGroupoidOfAHausdorffSpace

Related topic

WeakHomotopyAdditionLemma

Related topic

HomotopyAdditionLemma

Defines

thin representattion

Generated on Sat Feb 10 11:42:20 2018 by

LaTeXML
weak homotopy double groupoid

weak homotopy double groupoid

Definition 0.1

.

a

weak homotopy double groupoid

(WHDG)

of a

compactly–generated space

X

c

⁢

g

, (weak Hausdorff space) is
defined through a construction method

similar

to that developed by R. Brown (ref.

[

1

]

) for the

homotopy double groupoid

of a Hausdorff space

. The key changes here involve replacing the regular homotopy

equivalence relation

from the cited ref. with the

weak homotopy equivalence

relation

in the definition of the

fundamental groupoid

, as well as replacing the Hausdorff space by the compactly-generated space

X

c

⁢

g

. Therefore, the weak

homotopy

data for the

weak homotopy double groupoid

of

X

c

⁢

g

,

𝝆

□

⁢

(

X

c

⁢

g

)

, will now be:

(

𝝆

2

□

(

X

)

,

𝝆

1

□

(

X

)

,

∂

1

-

,

∂

1

+

,

+

1

,

ε

1

)

,

𝝆

2

□

(

X

)

,

𝝆

1

□

(

X

)

,

∂

2

-

,

∂

2

+

,

+

2

,

ε

2

)

(

𝝆

1

□

⁢

(

X

)

,

X

,

∂

-

,

∂

+

,

+

,

ε

)

.

References

1

R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoid of a Hausdorff
space,

Theory and Applications of Categories

10

,(2002): 71-93.

Title

weak homotopy double groupoid

Canonical name

WeakHomotopyDoubleGroupoid

Date of creation

2013-03-22 18:15:12

Last modified on

2013-03-22 18:15:12

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

16

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55N33

Classification

msc 55N20

Classification

msc 55P10

Classification

msc 55U40

Classification

msc 18B40

Classification

msc 18D05

Synonym

homotopy double groupoid

Related topic

WeakHomotopyAdditionLemma

Related topic

OmegaSpectrum

Related topic

FEquivalenceInCategory

Defines

higher dimensional weak homotopy

Generated on Sat Feb 10 11:42:22 2018 by

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cap product

cap product

Let

X

be a

topological space

,

(

C

*

⁢

(

X

)

,

∂

)

the

singular

chain complex

, and

(

C

*

⁢

(

X

;

𝕂

)

,

δ

)

the singular

cochain complex

in any coefficient group

𝕂

. We can define a

bilinear pairing

operation

⌢

:

C

i

(

X

;

𝕂

)

×

C

n

(

X

)

→

C

n

-

i

(

X

)

,

(

n

≥

i

)

in the following way: for each cochain

b

∈

C

i

⁢

(

X

;

𝕂

)

and each chain

σ

∈

C

n

⁢

(

X

)

we define their

cap product

b

⌢

σ

as the unique

(

n

-

i

)

-singular chain such that

a

(

b

⌢

σ

)

=

(

a

⌣

b

)

(

σ

)

,

where

⌣

:

C

j

(

X

;

𝕂

)

×

C

h

(

X

;

𝕂

)

→

C

j

+

h

(

X

;

𝕂

)

denotes the

cup product

.
Combining the definition of cap product with the standard properties of cup product we obtain that

∂

⁡

(

b

⌢

ξ

)

=

(

∂

⁡

b

)

⌢

ξ

+

(

-

1

)

dim

⁢

(

b

)

⁢

b

⌢

∂

⁡

(

ξ

)

,

thus there is a corresponding operation in

cohomology

⌢

:

H

i

(

X

;

𝕂

)

⊗

H

n

(

X

)

→

H

n

-

i

(

X

)

,

(

n

≥

i

)

that we also call

cap product

.

Title

cap product

Canonical name

CapProduct

Date of creation

2013-03-22 16:26:10

Last modified on

2013-03-22 16:26:10

Owner

Mazzu (14365)

Last modified by

Mazzu (14365)

Numerical id

9

Author

Mazzu (14365)

Entry type

Definition

Classification

msc 55N45

Defines

cap product

Generated on Sat Feb 10 11:42:24 2018 by

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cup product

cup product

Let

X

be a

topological space

and

R

be a

commutative ring

. The

diagonal map

Δ

:

X

→

X

×

X

induces a

chain map

between

singular

cochain complexes

:

Δ

*

:

C

*

⁢

(

X

×

X

;

R

)

→

C

*

⁢

(

X

;

R

)

.

Let

h

:

C

*

⁢

(

X

;

R

)

⊗

C

*

⁢

(

X

;

R

)

→

C

*

⁢

(

X

×

X

;

R

)

denote the

chain homotopy equivalence

associated with the Kunneth .

Given

α

∈

C

p

⁢

(

X

;

R

)

and

β

∈

C

q

⁢

(

X

;

R

)

we define

α

⌣

β

=

Δ

*

⁢

h

⁢

(

α

⊗

β

)

.

As

Δ

*

and

h

are chain maps,

⌣

induces a well defined

product

on

cohomology groups

, known as the

cup product

. Hence the

direct sum

of the cohomology groups of

X

has the

structure

of a ring. This is called the cohomology ring of

X

.

Title

cup product

Canonical name

CupProduct

Date of creation

2013-03-22 15:37:42

Last modified on

2013-03-22 15:37:42

Owner

whm22 (2009)

Last modified by

whm22 (2009)

Numerical id

7

Author

whm22 (2009)

Entry type

Definition

Classification

msc 55N45

Generated on Sat Feb 10 11:42:26 2018 by

LaTeXML
cohomological complex of topological vector spaces

cohomological complex of topological vector spaces

Definition 0.1

.

A

cohomological 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

into

E

q

+

1

which satisfy

d

q

∘

d

q

+

1

=

0

.

Remarks

•

The

dual

complex

of a cohomological complex

(

E

∙

,

d

)

of

topological vector spaces

is the homological complex

(

E

∙

′

,

d

′

)

(

http://planetmath.org/HomologicalComplexOfTopologicalVectorSpaces

), where

(

E

∙

′

=

(

E

q

′

)

q

∈

Z

with

E

q

′

being the strong dual of

E

q

and

d

′

=

(

d

q

′

)

q

∈

Z

, and also with

d

q

′

being the

transpose map

of

d

q

.

•

A cohomological complex of topological vector spaces (TVS) is a
specific case of a

cochain complex

, which is the dual of the concept of

chain complex

.

Title

cohomological complex of topological vector spaces

Canonical name

CohomologicalComplexOfTopologicalVectorSpaces

Date of creation

2013-03-22 18:17:27

Last modified on

2013-03-22 18:17:27

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

22

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55N99

Classification

msc 81T70

Classification

msc 32S20

Classification

msc 12G10

Classification

msc 55N33

Classification

msc 13D25

Classification

msc 18G35

Synonym

cohomological complex

Related topic

HomologicalComplexOfTopologicalVectorSpaces

Related topic

ChainComplex

Related topic

CategoricalSequence

Related topic

TangentialCauchyRiemannComplexOfCinftySmoothForms

Related topic

ACRcomplex

Defines

dual of chain complex

Defines

cochain complex

Defines

transpose map

Defines

sequence of topological vector spaces

Defines

sequence of continuous linear maps

Generated on Sat Feb 10 11:42:28 2018 by

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Euler characteristic

Euler characteristic

The term

Euler characteristic

is defined for several objects.

If

K

is a finite

simplicial complex

of

dimension

m

, let

α

i

be the number of
simplexes of dimension

i

. The

Euler characteristic

of

K

is defined to be

χ

⁢

(

K

)

=

∑

i

=

0

m

(

-

1

)

i

⁢

α

i

.

Next, if

K

is a finite CW complex, let

α

i

be the number of i-cells
in

K

. The

Euler characteristic

of

K

is defined to be

χ

⁢

(

K

)

=

∑

i

≥

0

(

-

1

)

i

⁢

α

i

.

If

X

is a

finite polyhedron

, with triangulation

K

, a simplicial complex,
then the

Euler characteristic

of

X

is

χ

⁢

(

K

)

. It can be shown
that all triangulations of

X

have the same value for

χ

⁢

(

K

)

so that
this is well-defined.

Finally, if

C

=

{

C

q

}

is a

finitely generated

graded group, then
the

Euler characteristic

of

C

is defined to be

χ

⁢

(

C

)

=

∑

q

≥

0

(

-

1

)

q

⁢

r

⁢

a

⁢

n

⁢

k

⁢

(

C

q

)

.

Title

Euler characteristic

Canonical name

EulerCharacteristic

Date of creation

2013-03-22 16:12:51

Last modified on

2013-03-22 16:12:51

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

13

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 55N99

Generated on Sat Feb 10 11:42:30 2018 by

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