aarjaneiro/planetmath huggingface dataset:
fundamental groupoid

fundamental groupoid

Definition 1

.

Given a

topological space

X

the

fundamental groupoid

Π

1

⁢

(

X

)

of

X

is defined as
follows:

•

The objects of

Π

1

⁢

(

X

)

are the points of

X

Obj

⁢

(

Π

1

⁢

(

X

)

)

=

X

,

•

morphisms

are

homotopy classes

of paths “rel

endpoints

” that is

Hom

Π

1

⁢

(

X

)

(

x

,

y

)

=

Paths

(

x

,

y

)

/

∼

,

where,

∼

denotes

homotopy

rel endpoints, and,

•

composition

of morphisms is defined via

concatenation

of paths.

It is easily checked that the above defined

category

is indeed a

groupoid

with the

inverse

of (a morphism represented by) a path being (the homotopy
class of) the “reverse” path.
Notice that for

x

∈

X

, the group of

automorphisms

of

x

is the

fundamental group

of

X

with basepoint

x

,

Hom

Π

1

⁢

(

X

)

⁢

(

x

,

x

)

=

π

1

⁢

(

X

,

x

)

.

Definition 2

.

Let

f

:

X

→

Y

be a

continuous function

between two topological spaces.
Then there is an

induced

functor

Π

1

⁢

(

f

)

:

Π

1

⁢

(

X

)

→

Π

1

⁢

(

Y

)

defined as follows

•

on objects

Π

1

⁢

(

f

)

is just

f

,

•

on morphisms

Π

1

⁢

(

f

)

is given by “composing with

f

”, that is
if

α

:

I

→

X

is a path representing the morphism

[

α

]

:

x

→

y

then a representative of

Π

1

⁢

(

f

)

⁢

(

[

α

]

)

:

f

⁢

(

x

)

→

f

⁢

(

y

)

is determined by the following

commutative diagram

Generated on Sat Feb 10 11:43:08 2018 by

LaTeXML
generalized Fourier transform

generalized Fourier transform

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, that is a finite set which contains only a small number of values. 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

.

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

generalized Fourier transform

Canonical

name

GeneralizedFourierTransform

Date of creation

2013-03-22 18:16:07

Last modified on

2013-03-22 18:16:07

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

14

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55P99

Classification

msc 55R10

Classification

msc 55R65

Classification

msc 55R37

Classification

msc 42B10

Classification

msc 42A38

Synonym

Stieltjes-Fourier transform

Related topic

FourierStieltjesAlgebraOfAGroupoid

Related topic

TwoDimensionalFourierTransforms

Related topic

DiscreteFourierTransform

Defines

positive definite- measurable function

Generated on Sat Feb 10 11:43:12 2018 by

LaTeXML
homotopy with a contractible domain

homotopy with a contractible domain

Theorem

.

Assume that

Y

is an arbitrary

topological space

and

X

is a

contractible

topological space. Then all maps

f

:

X

→

Y

are

homotopic

if and only if

Y

is

path connected

.

Proof:

Assume that all maps are homotopic. In particular

constant maps

are homotopic, so if

y

1

,

y

2

∈

Y

, then there exists a continous map

H

:

I

×

Y

→

Y

such that

H

⁢

(

0

,

y

)

=

y

1

and

H

⁢

(

1

,

y

)

=

y

2

for all

y

∈

Y

. Thus the map

α

:

I

→

Y

defined by the

formula

α

⁢

(

t

)

=

H

⁢

(

t

,

y

0

)

for a fixed

y

0

∈

Y

is the wanted path.

On the other hand assume that

Y

is path connected. Since

X

is contractible, then for any

c

∈

X

there exists a continous

homotopy

H

:

I

×

X

→

X

connecting the

identity map

and a constant map

c

. Let

f

:

X

→

Y

be an arbitrary map. Define a map

F

:

I

×

X

→

Y

by the formula:

F

⁢

(

t

,

x

)

=

f

⁢

(

H

⁢

(

t

,

x

)

)

. This map is a homotopy from

f

to a constant map

f

⁢

(

c

)

. Thus every map is homotopic to some constant map.

The space

Y

is path connected, so for all

y

1

,

y

2

∈

Y

there exists a path

α

:

I

→

Y

from

y

1

to

y

2

. Therefore constant maps are homotopic via the homotopy

H

⁢

(

t

,

x

)

=

α

⁢

(

t

)

.

Finaly for any continous maps

f

,

g

:

X

→

Y

and any point

c

∈

X

we get:

f

≃

f

⁢

(

c

)

≃

g

⁢

(

c

)

≃

g

,

which

completes

the proof.

□

Corollary.

If

X

is a contractible space, then for any topological space

Y

there exists a

bijection

between the set

[

X

,

Y

]

of

homotopy classes

of maps from

X

to

Y

and the set

π

0

⁢

(

Y

)

of

path components

of

Y

.

Proof:

Assume that

Y

=

⋃

Y

i

, where

Y

i

are path components of

Y

. It is well known that contractible spaces are path connected, thus the image of any continous map

f

:

X

→

Y

is

contained

in

Y

i

for some

i

. It follows from the theorem that two maps from

X

to

Y

are homotopic if and only if their images are contained in the same

Y

i

. Thus we have a well defined, injective map

ψ

:

[

X

,

Y

]

→

π

0

⁢

(

Y

)

ψ

⁢

(

[

f

]

)

=

Y

i

,

where

i

is such that

f

⁢

(

X

)

⊆

Y

i

. This map is also

surjective

, since for any

i

there exists

y

∈

Y

i

, so the class of the constant map

f

⁢

(

x

)

=

y

is mapped into

Y

i

.

□

Title

homotopy with a contractible domain

Canonical name

HomotopyWithAContractibleDomain

Date of creation

2013-03-22 18:02:12

Last modified on

2013-03-22 18:02:12

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

26

Author

joking (16130)

Entry type

Theorem

Classification

msc 55P99

Related topic

homotopy

Related topic

contractible

Generated on Sat Feb 10 11:43:17 2018 by

LaTeXML
homotopy invariance

homotopy invariance

Let

ℱ

be a

functor

from the category of topological spaces to some

category

𝒞

. Then

ℱ

is called

homotopy invariant

if for any two

homotopic maps

f

,

g

:

X

→

Y

between

topological spaces

X

and

Y

the

morphisms

ℱ

⁢

f

and

ℱ

⁢

g

in

𝒞

induced by

ℱ

are identical.

Suppose

ℱ

is a homotopy invariant functor, and

X

and

Y

are

homotopy equivalent

topological spaces. Then there are

continuous maps

f

:

X

→

Y

and

g

:

Y

→

X

such that

g

∘

f

≃

id

X

and

f

∘

g

≃

id

Y

(i.e.

g

∘

f

and

f

∘

g

are

homotopic

to the

identity maps

on

X

and

Y

, respectively). Assume that

ℱ

is a

covariant functor

. Then the homotopy invariance of

ℱ

implies

ℱ

⁢

g

∘

ℱ

⁢

f

=

ℱ

⁢

(

g

∘

f

)

=

id

ℱ

⁢

X

and

ℱ

⁢

f

∘

ℱ

⁢

g

=

ℱ

⁢

(

f

∘

g

)

=

id

ℱ

⁢

Y

.

From this we see that

ℱ

⁢

X

and

ℱ

⁢

Y

are

isomorphic

in

𝒞

. (The same

argument

clearly holds if

ℱ

is contravariant instead of covariant.)

An important example of a homotopy invariant functor is the

fundamental group

π

1

; here

𝒞

is the category of groups.

Title

homotopy invariance

Canonical name

HomotopyInvariance

Date of creation

2013-03-22 14:24:51

Last modified on

2013-03-22 14:24:51

Owner

pbruin (1001)

Last modified by

pbruin (1001)

Numerical id

4

Author

pbruin (1001)

Entry type

Definition

Classification

msc 55Pxx

Related topic

HomotopyEquivalence

Defines

homotopy invariant

Generated on Sat Feb 10 11:43:19 2018 by

LaTeXML
deformation retract

deformation retract

Let

X

and

Y

be

topological spaces

such that

Y

⊂

X

.
A

deformation retract

of

X

onto

Y

is a

collection

of

mappings

f

t

:

X

→

X

,

t

∈

[

0

,

1

]

such that

1.

f

0

=

i

⁢

d

X

, the

identity mapping

on

X

,

2.

f

1

⁢

(

X

)

⊆

Y

,

3.

Y

is a

retract

of

X

via

f

1

(that is,

f

1

restricted to

Y

is the

identity

on

Y

)

4.

the mapping

X

×

I

→

X

,

(

x

,

t

)

↦

f

t

⁢

(

x

)

is

continuous

, where the

topology

on

X

×

I

is the

product topology

.

Of course, by condition 3, condition 2 can be improved:

f

1

⁢

(

X

)

=

Y

.

A deformation retract is called a

strong deformation retract

if condition 3 above is replaced by a

stronger

form:

Y

is a retract of

X

via

f

t

for every

t

∈

[

0

,

1

]

.

Properties

•

Let

X

and

Y

be as in the above definition. Then a collection of
mappings

f

t

:

X

→

X

,

t

∈

[

0

,

1

]

is
a deformation retract (of

X

onto

Y

) if and only if it is a

homotopy

(

http://planetmath.org/HomotopyOfMaps

) rel Y

id

X

and some

retraction

r

of

X

onto

Y

.

Examples

•

If

x

0

∈

ℝ

n

, then

f

t

⁢

(

x

)

=

(

1

-

t

)

⁢

x

+

t

⁢

x

0

,

x

∈

ℝ

n

shows that

ℝ

n

deformation retracts onto

{

x

0

}

.
Since

{

x

0

}

⊂

ℝ

n

,
it follows that deformation retract is not an

equivalence relation

.

•

The same map as in the previous example can be used to deformation retract any star-shaped set in

ℝ

n

onto a point.

•

we obtain a
deformation retraction of

ℝ

n

\

{

0

}

onto the

http://planetmath.org/node/186

(

n

-

1

)

-sphere

S

n

-

1

by setting

f

t

⁢

(

x

)

=

(

1

-

t

)

⁢

x

+

t

⁢

x

||

x

||

,

where

x

∈

ℝ

n

\

{

0

}

,

n

>

0

,

•

The

http://planetmath.org/node/3278

Möbius strip deformation retracts onto the circle

S

1

.

•

The

2

-torus with one point removed deformation retracts onto
two copies of

S

1

joined at one point. (The circles can be
chosen to be longitudinal and latitudinal circles of the torus.)

•

The

characters

E,F,H,K,L,M,N, and T all deformation retract onto the charachter I, while the letter Q deformation retracts onto the letter O.

Title

deformation retract

Canonical name

DeformationRetract

Date of creation

2013-03-22 13:31:44

Last modified on

2013-03-22 13:31:44

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Definition

Classification

msc 55Q05

Related topic

Retract

Defines

strong deformation retract

Generated on Sat Feb 10 11:43:24 2018 by

LaTeXML
deformation retract is transitive

deformation retract is transitive

Proposition

.

Let

Z

⊂

Y

⊂

X

be nested

topological spaces

. If there exist a
deformation retraction (

http://planetmath.org/DeformationRetraction

) of

X

onto

Y

and a deformation retraction of

Y

onto

Z

,
then there also exists a deformation retraction of

X

onto

Z

. In other words,
“being a

deformation retract

of” is a

transitive relation

.

Proof.

Since

Y

is a deformation retract of

X

, there is a

homotopy

F

:

I

×

X

→

X

between

id

X

and a

retract

r

:

X

→

Y

of

X

onto

Y

. Similarly, there is a homotopy

G

:

I

×

Y

→

Y

between

id

Y

and a retract

s

:

Y

→

Z

of

Y

onto

Z

.

First notice that since both

r

and

s

fix

Z

, the map

s

⁢

r

:

X

→

Z

is a

retraction

.

Now define a map

G

~

:

I

×

X

→

X

by

G

~

=

i

⁢

G

⁢

(

id

I

×

r

)

, where

i

:

Y

↪

X

is

inclusion

. Observe that

•

G

~

⁢

(

0

,

x

)

=

r

⁢

(

x

)

for any

x

∈

X

;

•

G

~

⁢

(

1

,

x

)

=

s

⁢

r

⁢

(

x

)

for any

x

∈

X

; and

•

G

~

⁢

(

t

,

a

)

=

a

for any

a

∈

Z

.

Hence

G

~

is a homotopy between the retractions

r

and

s

⁢

r

.

Finally we must
glue together the homotopies (

http://planetmath.org/GluingTogentherContinuousFunctions

)

F

and

G

~

to get a
homotopy between

id

X

and

s

⁢

r

. To do this, define a

function

H

:

I

×

X

→

X

by

H

⁢

(

t

,

x

)

=

{

F

⁢

(

2

⁢

t

,

x

)

,

0

≤

t

≤

1

2

G

~

⁢

(

2

⁢

t

-

1

,

x

)

,

1

2

≤

t

≤

1

.

Since

F

⁢

(

1

,

x

)

=

G

~

⁢

(

0

,

x

)

=

r

⁢

(

x

)

, the gluing yieds a

continuous map

.
By construction,

•

H

⁢

(

0

,

x

)

=

x

for all

x

∈

X

;

•

H

⁢

(

1

,

x

)

=

s

⁢

r

⁢

(

x

)

for all

x

∈

X

; and

•

H

⁢

(

t

,

a

)

=

a

for any

a

∈

Z

.

Hence

H

is a homotopy between the

identity map

on

X

and a retraction of

X

onto

Z

. We conclude that

H

is a deformation retraction of

X

onto

Z

.
∎

Title

deformation retract is transitive

Canonical name

DeformationRetractIsTransitive

Date of creation

2013-03-22 15:43:59

Last modified on

2013-03-22 15:43:59

Owner

mps (409)

Last modified by

mps (409)

Numerical id

4

Author

mps (409)

Entry type

Result

Classification

msc 55Q05

Generated on Sat Feb 10 11:43:27 2018 by

LaTeXML
homotopy of maps

homotopy of maps

Let

X

,

Y

be

topological spaces

,

A

a closed subspace of

X

and

f

,

g

:

X

→

Y

continuous maps

. A

homotopy of maps

is a

continuous function

F

:

X

×

[

0

,

1

]

→

Y

satisfying

1.

F

⁢

(

x

,

0

)

=

f

⁢

(

x

)

for all

x

∈

X

2.

F

⁢

(

x

,

1

)

=

g

⁢

(

x

)

for all

x

∈

X

3.

F

⁢

(

x

,

t

)

=

f

⁢

(

x

)

=

g

⁢

(

x

)

for all

x

∈

A

,

t

∈

[

0

,

1

]

.

We say that

f

is

homotopic

to

g

relative to

A

and denote this by

f

≃

g

r

⁢

e

⁢

l

⁢

A

. If

A

=

∅

, this can be written

f

≃

g

. If

g

is the

constant map

(i.e.

g

⁢

(

x

)

=

y

for all

x

∈

X

), then we say that

f

is

nullhomotopic

.

Title

homotopy of maps

Canonical

name

HomotopyOfMaps

Date of creation

2013-03-22 12:13:19

Last modified on

2013-03-22 12:13:19

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Definition

Classification

msc 55Q05

Synonym

homotopic maps

Related topic

HomotopyOfPaths

Related topic

HomotopyEquivalence

Related topic

ConstantFunction

Related topic

Contractible

Defines

homotopic

Defines

nullhomotopic

Generated on Sat Feb 10 11:43:29 2018 by

LaTeXML
homotopy of paths

homotopy of paths

Let

X

be a

topological space

and

p

,

q

paths in

X

with the same

initial point

x

0

and terminal point

x

1

. If there exists a

continuous function

F

:

I

×

I

→

X

such that

1.

F

⁢

(

s

,

0

)

=

p

⁢

(

s

)

for all

s

∈

I

2.

F

⁢

(

s

,

1

)

=

q

⁢

(

s

)

for all

s

∈

I

3.

F

⁢

(

0

,

t

)

=

x

0

for all

t

∈

I

4.

F

⁢

(

1

,

t

)

=

x

1

for all

t

∈

I

we call

F

a

homotopy of paths

in

X

and say

p

,

q

are

homotopic paths

in

X

.

F

is also called a

continuous deformation

.

Title

homotopy of paths

Canonical name

HomotopyOfPaths

Date of creation

2013-03-22 12:13:16

Last modified on

2013-03-22 12:13:16

Owner

RevBobo (4)

Last modified by

RevBobo (4)

Numerical id

8

Author

RevBobo (4)

Entry type

Definition

Classification

msc 55Q05

Synonym

homotopic paths

Synonym

continuous deformation

Synonym

homotopy

Related topic

HomotopyOfMaps

Related topic

HomotopyWithAContractibleDomain

Related topic

PathConnectnessAsAHomotopyInvariant

Generated on Sat Feb 10 11:43:30 2018 by

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long exact sequence (locally trivial bundle)

long exact sequence (locally trivial bundle)

Let

π

:

E

→

B

is a

locally trivial bundle

, with fiber

F

. Then there is a long exact sequence
of homotopy groups

⋯

→

π

n

⁢

(

F

)

→

i

*

π

n

⁢

(

E

)

→

π

*

π

n

⁢

(

B

)

→

∂

*

π

n

-

1

⁢

(

F

)

→

⋯

Here

i

*

is

induced

by the

inclusion

i

:

F

↪

E

as the fiber over the

basepoint

of

B

, and

∂

*

is the following
map: if

[

φ

]

∈

π

n

⁢

(

B

)

, then

φ

lifts to a map of

(

D

n

,

∂

⁡

D

n

)

into

(

E

,

F

)

(that is a map of the

n

-disk into

E

, taking its boundary to

F

), sending the
basepoint on the boundary to the base point of

F

⊂

E

. Thus the map on

∂

⁡

D

n

=

S

n

-

1

, the

n

-

1

-sphere, defines an element of

π

n

-

1

⁢

(

F

)

. This is

∂

*

⁡

[

φ

]

. The covering homotopy property of a locally trivial bundle shows that this is well-defined.

Title

long exact sequence

(locally trivial bundle)

Canonical name

LongExactSequencelocallyTrivialBundle

Date of creation

2013-03-22 13:14:58

Last modified on

2013-03-22 13:14:58

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Definition

Classification

msc 55Q05

Related topic

Fibration

Related topic

Fibration2

Related topic

HomotopyLiftingProperty

Generated on Sat Feb 10 11:43:32 2018 by

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quantum fundamental groupoid

quantum fundamental groupoid

Definition 0.1

.

A

quantum fundamental groupoid

F

𝒬

is defined as a

functor

F

𝒬

:

𝐇

B

→

𝒬

G

, where

𝐇

B

is the

category

of

Hilbert space

bundles, and

𝒬

G

is the category of

locally compact

quantum groupoids

and their

homomorphisms

.

0.1

Fundamental groupoid functors and functor categories

The natural setting for the definition of a quantum fundamental groupoid

F

𝒬

is in one of the

functor categories

– that of

fundamental groupoid functors

(

http://planetmath.org/FundamentalGroupoidFunctor

),

F

𝒢

, and their

natural transformations

(

http://planetmath.org/NaturalTransformation

) defined in the

context

of

quantum categories

of

quantum spaces

𝒬

represented by Hilbert space bundles or rigged Hilbert (also called Frechét) spaces

𝐇

B

.

Other related functor categories are those specified with the general definition of the

fundamental groupoid functor

,

F

𝒢

:

𝐓𝐨𝐩

→

𝒢

2

, where

Top

is the
category of topological spaces and

𝒢

2

is the

groupoid category

(

http://planetmath.org/GroupoidCategory

).

Example 0.1

.

A specific example of a quantum fundamental groupoid can be given for spin foams of spin networks, with a spin foam defined as a functor between spin network categories. Thus, because spin networks or graphs are specialized
one-dimensional

CW-complexes

whose cells are linked quantum spin states, their quantum fundamental groupoid is defined as a

functor representation

of CW-complexes on

rigged Hilbert spaces

(also called Frechét

nuclear spaces

).

Title

quantum fundamental groupoid

Canonical name

QuantumFundamentalGroupoid

Date of creation

2013-03-22 18:15:53

Last modified on

2013-03-22 18:15:53

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

62

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55Q05

Classification

msc 55U40

Classification

msc 20L05

Synonym

quantum spacetimes homotopy theory

Related topic

FundamentalGroupoidFunctor

Related topic

SpinNetworksAndSpinFoams

Related topic

QuantumGroupoids2

Related topic

GroupoidCategory

Related topic

SpinNetworksAndSpinFoams

Related topic

FunctorCategories

Related topic

FundamentalGroupoid

Related topic

HomotopyGroups

Related topic

HigherDimensionalAlgebraHDA

Related topic

GeneralizedVanKampenTheoremsHigherDimensional

Related topic

Quantu

Defines

category of quantum groupoids

Defines

Frechét spaces

Generated on Sat Feb 10 11:43:37 2018 by

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Van Kampen’s theorem

Van Kampen’s theorem

Van Kampen’s theorem for

fundamental groups

may be stated as
follows:

Theorem 1

.

Let

X

be a

topological space

which is the union of the

interiors

of two

path connected

subspaces

X

1

,

X

2

. Suppose

X

0

:=

X

1

∩

X

2

is path connected. Let
further

*

∈

X

0

and

i

k

:

π

1

⁢

(

X

0

,

*

)

→

π

1

⁢

(

X

k

,

*

)

,

j

k

:

π

1

⁢

(

X

k

,

*

)

→

π

1

⁢

(

X

,

*

)

be induced by the inclusions for

k

=

1

,

2

. Then

X

is path connected and the natural

morphism

π

1

⁢

(

X

1

,

*

)

⁢

★

π

1

⁢

(

X

0

,

*

)

⁢

π

1

⁢

(

X

2

,

*

)

→

π

1

⁢

(

X

,

*

)

,

is an

isomorphism

, that is, the fundamental group of

X

is the

free product

of the
fundamental groups of

X

1

and

X

2

with amalgamation of

π

1

⁢

(

X

0

,

*

)

.

Usually the morphisms induced by inclusion in this theorem are not
themselves injective, and the more precise version of the statement
is in terms of

pushouts of groups

.

The notion of

pushout

in the

category of groupoids

allows for a
version of the theorem for the non path connected case, using the

fundamental groupoid

π

1

⁢

(

X

,

A

)

on a set

A

of base points,

[

rb1

]

. This

groupoid

consists of

homotopy classes

rel end
points of paths in

X

joining points of

A

∩

X

. In particular,
if

X

is a contractible space, and

A

consists of two distinct
points of

X

, then

π

1

⁢

(

X

,

A

)

is easily seen to be isomorphic to
the groupoid often written

ℐ

with two vertices and
exactly one morphism between any two vertices. This groupoid plays a
role in the theory of groupoids analogous to that of the group of
integers in the theory of groups.

Theorem 2

.

Let the topological space

X

be covered by the interiors of two
subspaces

X

1

,

X

2

and let

A

be a set which meets each

path
component

of

X

1

,

X

2

and

X

0

:=

X

1

∩

X

2

. Then

A

meets each
path component of

X

and the following

diagram

of morphisms induced
by inclusion

Generated on Sat Feb 10 11:43:42 2018 by

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Van Kampen’s theorem result

Van Kampen’s theorem result

There is a more general version of the theorem of van Kampen which involves
the

fundamental groupoid

π

1

⁢

(

X

,

A

)

on a

set

A

of base points,
defined as the full subgroupoid of

π

1

⁢

(

X

)

on the set

A

∩

X

. This
allows one to compute the

fundamental group

of the circle

S

1

and many more
cases.

Theorem

If

X

is the union of open sets

U

,

V

with

intersection

W

,
and

A

meets each path component of

U

,

V

,

W

then the following

induced

diagram

\xymatrix

⁢

π

1

⁢

(

W

,

A

)

⁢

\ar

⁢

[

r

]

⁢

\ar

⁢

[

d

]

⁢

&

⁢

π

1

⁢

(

U

,

A

)

⁢

\ar

⁢

[

d

]

⁢

π

1

⁢

(

V

,

A

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

1

⁢

(

X

,

A

)

is a

pushout

in the

category of groupoids

.

This may be found in R. Brown’s book “

Topology

: a geometric account of general topology
and the fundamental groupoid”, Ellis Horwood 1988 (first edition McGraw Hill, 1968).
It has many useful applications, and was the guide for higher dimensional theorems
involving

higher homotopy groupoids

.

Title

Van Kampen’s theorem result

Canonical name

VanKampensTheoremResult

Date of creation

2013-03-22 14:09:23

Last modified on

2013-03-22 14:09:23

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Result

Classification

msc 55Q05

Generated on Sat Feb 10 11:43:44 2018 by

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