aarjaneiro/planetmath huggingface dataset:
Grothendieck’s publications

Grothendieck’s publications

0.1

Bibliography- Link to a list of Grothendieck’s Publications

This list of 64 published papers can be downloaded online

http://www.math.jussieu.fr/ leila/grothendieckcircle/GrothBiblio.pdf

at:

Alexander Grothendieck

’s Publications, (in French)

Title

Grothendieck’s publications

Canonical name

GrothendiecksPublications

Date of creation

2013-03-22 18:28:12

Last modified on

2013-03-22 18:28:12

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

13

Author

bci1 (20947)

Entry type

Bibliography

Classification

msc 55-00

Classification

msc 18-00

Generated on Sat Feb 10 11:40:29 2018 by

LaTeXML
invariance of dimension

invariance of dimension

The following non-trivial result was proven by Brouwer

[

1

]

around 1910

[

2

]

.

Theorem  (

Invariance of dimension

)

Suppose

U

and

V

are

open subsets

of

ℝ

n

respectively

ℝ

m

.
If

U

and

V

are non-empty and

homeomorphic

, then

n

=

m

.

References

1

The MacTutor History of Mathematics archive,

http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Brouwer.html

entry on Luitzen Egbertus Jan Brouwer

2

A. Hatcher,

Algebraic Topology

, Cambridge University Press, 2002. Also available

http://www.math.cornell.edu/ hatcher/AT/ATpage.html

online.

Title

invariance of dimension

Canonical name

InvarianceOfDimension

Date of creation

2013-03-22 13:42:38

Last modified on

2013-03-22 13:42:38

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Theorem

Classification

msc 55-00

Generated on Sat Feb 10 11:40:31 2018 by

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non-commutative structure

non-commutative structure

Definition 0.1

.

Let

(

C

,

∘

)

be a

structure

consisting of a

class

,

C

, together with a

binary operation

∘

defined for pairs of objects in

C

(or elements of

C

when the latter is a small class, i.e., a set). The structure– and the

operation

∘

– are said to be

noncommutative

if

a

∘

b

≠

b

∘

a

(0.1)

for either at least some or all of the

a

,

b

pairs in

C

for which the operation is defined.

A structure that is noncommutative is also called sometimes a

non-Abelian structure

, although the latter term is, in general, more often used to specify

non-Abelian theories

(

http://planetmath.org/NonAbelianTheories

).

A binary operation that is not

commutative

(

http://planetmath.org/Commutative

) is said to be

non-commutative

(or

noncommutative

). Thus, a

noncommutative structure

can be alternatively defined as any structure whose binary operation is not commutative (

http://planetmath.org/Commutative

) (that is, in the commutative (

http://planetmath.org/Commutative

) case one has

a

∘

b

=

b

∘

a

(0.2)

for all

a

,

b

pairs in

C

, and also that the operation

∘

is defined for all pairs in

C

).

An example of a commutative structure is the field of real numbers– with two commutative operations in this case–
which are the

addition

and

multiplication

over the reals.

Remark 0.1

.

A commutative group is also called

Abelian

, whereas a

category

with structure that has

commutative diagrams

is not necessarily Abelian –unless it does

satisfy

the Ab1 to Ab6 axioms that define an

Abelian category

(or equivalently, if it has the properties specified in Mitchell’s alternative definition of an Abelian category (

http://planetmath.org/AlternativeDefinitionOfAnAbelianCategory

) .)

An example of a non-commutative operation is the multiplication over

n

×

n

matrices.
Another example of a

noncommutative algebra

is a general

Clifford algebra

(

http://planetmath.org/CCliffordAlgebra

), which is of fundamental importance in the algebraic theory of observable quantum

operators

and also in quantum algebraic topology.

Title

non-commutative structure

Canonical name

NoncommutativeStructure

Date of creation

2013-03-22 18:18:06

Last modified on

2013-03-22 18:18:06

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

24

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55-00

Classification

msc 18-00

Synonym

noncommutative

Synonym

nonabelian

Synonym

non-Abelian

Related topic

Commutative

Related topic

QuantumTopos

Defines

non-commutative operation

Generated on Sat Feb 10 11:40:33 2018 by

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Sorgenfrey line

Sorgenfrey line

The

Sorgenfrey line

is a nonstandard

topology

on the

real line

ℝ

.
Its topology is defined by the following base of half

open intervals

ℬ

=

{

[

a

,

b

)

∣

a

,

b

∈

ℝ

,

a

<

b

}

.

Another name is

lower limit topology

, since a sequence

x

α

converges

only if it converges in the

standard topology

and its limit is
a limit from above (which, in this case, means that at most finitely many
points of the sequence lie below the limit). For example, the sequence

(

1

/

n

)

converges to

0

, while

(

-

1

/

n

)

does not.

This topology is finer than the standard topology on

ℝ

.
The Sorgenfrey line is

first countable

and

separable

, but is not

second countable

.
It is therefore not

metrizable

.

References

1

R. H. Sorgenfrey,

On the topological product of

paracompact spaces

,
Bulletin of the

American Mathematical Society

53 (1947) 631–632.
(This paper is

http://projecteuclid.org/euclid.bams/1183510809

available on-line
from

Project

Euclid.)

Title

Sorgenfrey line

Canonical name

SorgenfreyLine

Date of creation

2013-03-22 13:03:45

Last modified on

2013-03-22 13:03:45

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Example

Classification

msc 55-00

Classification

msc 54-00

Classification

msc 22-00

Synonym

Sorgenfrey topology

Defines

lower limit topology

Generated on Sat Feb 10 11:40:37 2018 by

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Thom class

Thom class

Let

h

*

be a generalized cohomology theory (for example, let

h

*

=

H

*

, singular cohomology with integer coefficients). Let

ξ

→

X

be a

vector bundle

of

dimension

d

over a

topological space

X

. Assume for convenience that

ξ

has a Riemannian metric, so that we may speak of its associated sphere and

disk bundles

,

S

⁢

(

ξ

)

and

D

⁢

(

ξ

)

respectively.

Let

x

∈

X

, and consider the fibers

S

⁢

(

ξ

x

)

and

D

⁢

(

ξ

x

)

. Since

D

⁢

(

ξ

x

)

/

S

⁢

(

ξ

x

)

is

homeomorphic

to the

d

-sphere, the

Eilenberg-Steenrod axioms

for

h

*

imply that

h

*

+

d

⁢

(

D

⁢

(

ξ

x

)

,

S

⁢

(

ξ

x

)

)

is

isomorphic

to the coefficient group

h

*

⁢

(

pt

)

of

h

*

. In fact,

h

*

⁢

(

D

⁢

(

ξ

x

)

,

S

⁢

(

ξ

x

)

)

is a

free module

of rank one over the ring

h

*

⁢

(

pt

)

.

Definition 1

An

element

τ

∈

h

*

⁢

(

D

⁢

(

ξ

)

,

S

⁢

(

ξ

)

)

is said to be a

Thom class

for

ξ

if, for every

x

∈

X

, the

restriction

of

τ

to

h

*

⁢

(

D

⁢

(

ξ

x

)

,

S

⁢

(

ξ

x

)

)

is an

h

*

⁢

(

pt

)

-module

generator

.

Note that

τ

lies necessarily in

h

d

⁢

(

D

⁢

(

ξ

)

,

S

⁢

(

ξ

)

)

.

Definition 2

If a Thom class for

ξ

exists,

ξ

is said to be

orientable

with respect to the

cohomology

theory

h

*

.

Remark 1

Notice that we may consider

τ

as an element of the

reduced

h

*

-

cohomology group

h

~

*

⁢

(

X

ξ

)

, where

X

ξ

is the Thom space

D

⁢

(

ξ

)

/

S

⁢

(

ξ

)

of

ξ

. As is the case in the definition of the Thom space, the Thom class may be defined without reference to associated disk and sphere bundles, and hence to a Riemannian metric on

ξ

. For example, the pair

(

ξ

,

ξ

-

X

)

(where

X

is included in

ξ

as the

zero section

) is

homotopy equivalent

to

(

D

⁢

(

ξ

)

,

S

⁢

(

ξ

)

)

.

Remark 2

If

h

*

is singular cohomology with integer coefficients, then

ξ

has a Thom class if and only if it is an orientable vector bundle in the ordinary sense, and the choices of Thom class are in

one-to-one correspondence

with the orientations.

Title

Thom class

Canonical name

ThomClass

Date of creation

2013-03-22 15:40:48

Last modified on

2013-03-22 15:40:48

Owner

antonio (1116)

Last modified by

antonio (1116)

Numerical id

5

Author

antonio (1116)

Entry type

Definition

Classification

msc 55-00

Related topic

Orientation2

Defines

orientability with respect to a generalized homology theory

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

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Thom isomorphism theorem

Thom isomorphism theorem

Let

ξ

→

X

be a

d

-dimensional

vector bundle

over a

topological space

X

, and let

h

*

be a multiplicative generalized cohomology theory, such as ordinary cohomology. Let

τ

∈

h

d

⁢

(

D

⁢

(

ξ

)

,

S

⁢

(

ξ

)

)

be a

Thom class

for

ξ

, where

D

⁢

(

ξ

)

and

S

⁢

(

ξ

)

are the associated disk and

sphere bundles

of

ξ

.

Since

h

*

is a multiplicative theory, there is a generalized

cup product

map

h

*

⁢

(

D

⁢

(

ξ

)

)

⊗

h

*

h

*

⁢

(

D

⁢

(

ξ

)

,

S

⁢

(

ξ

)

)

→

h

*

⁢

(

D

⁢

(

ξ

)

,

S

⁢

(

ξ

)

)

,

where the

tensor product

is over the

coefficient ring

h

*

⁢

(

pt

)

of the theory. Using the

isomorphism

p

*

:

h

*

⁢

(

X

)

≅

h

*

⁢

(

D

⁢

(

ξ

)

)

induced by the

homotopy equivalence

p

:

D

⁢

(

ξ

)

→

X

, we obtain a

homomorphism

T

:

h

n

⁢

(

X

)

→

h

n

+

d

⁢

(

D

⁢

(

ξ

)

,

S

⁢

(

ξ

)

)

≅

h

~

n

+

d

⁢

(

X

ξ

)

taking

α

to

p

*

⁢

(

α

)

⋅

τ

. Here

X

ξ

stands for the Thom space

D

⁢

(

ξ

)

/

S

⁢

(

ξ

)

of

ξ

.

Thom isomorphism theorem

T

is an isomorphism

h

*

⁢

(

X

)

≅

h

~

*

+

d

⁢

(

X

ξ

)

of graded modules over

h

*

⁢

(

pt

)

.

Remark 1

When

ξ

is a

trivial bundle

of

dimension

1

, this generalizes the

suspension isomorphism

. In fact, a typical proof of this theorem for

compact

X

proceeds by induction over the number of

open sets

in a

trivialization

of

ξ

, using the suspension isomorphism as the base case and the

Mayer-Vietoris sequence

to carry out the inductive step.

Remark 2

There is also a

homology

Thom isomorphism

h

~

*

+

d

⁢

(

X

ξ

)

≅

h

*

⁢

(

X

)

, in which the map is given by

cap product

with the Thom class rather than cup product.

Title

Thom isomorphism theorem

Canonical name

ThomIsomorphismTheorem

Date of creation

2013-03-22 15:40:52

Last modified on

2013-03-22 15:40:52

Owner

antonio (1116)

Last modified by

antonio (1116)

Numerical id

6

Author

antonio (1116)

Entry type

Theorem

Classification

msc 55-00

Generated on Sat Feb 10 11:40:42 2018 by

LaTeXML
topic entry on topology

topic entry on topology

Topology

is a branch of mathematics that arose from the study of geometry. Topology can be viewed as the study of spaces, and more specifically as the study of those properties of spaces that are invariant under smooth

deformation

. It has been said that a topologist is a person who cannot tell a doughnut from a coffee cup, as both spaces have a single “hole”, and each can be continuously deformed into the other.

A

topological space

(

http://planetmath.org/TopologicalSpace

) is a set of points together with a definition of which sets are

open sets

. This definition is called a topology (

http://planetmath.org/TopologicalSpace

) on the space. For example, any

metric space

can be seen as a topological space, where the prototypical open set is an open ball of any radius around a center.

There are several major branches of topology:

•

is the most basic study of topology. It investigates concepts such as

connectedness

, compactness, properties of different topologies, and topologies on

subspaces

and product spaces. See the articles

list of common topologies

and

index of properties of topological spaces

for more information.

•

Algebraic topology

(

http://planetmath.org/AlgebraicTopology

)

studies topological spaces by attaching algebraic invariants to them. Key concepts here are those of

homotopy

,

homology

, and

cohomology

.

•

Geometric topology

studies

manifolds

(which are spaces that are locally Euclidean) and their

embeddings

into

Euclidean space

, including

knot theory

.

•

Differential topology

studies properties of manifolds with differentiable structures and

differentiable maps

on those manifolds.

Other general topological topics include

•

Topics on low dimensional topology

•

Graph theory

•

Noncommutative topology

•

Generalizations of topological spaces

See also the attached

bibliography for topology

.

Title

topic entry on topology

Canonical name

TopicEntryOnTopology

Date of creation

2013-03-22 17:59:57

Last modified on

2013-03-22 17:59:57

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

13

Author

rm50 (10146)

Entry type

Topic

Classification

msc 55-00

Classification

msc 54-00

Related topic

OverviewOfTheContentOfPlanetMath

Generated on Sat Feb 10 11:40:44 2018 by

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universal coefficient theorem

universal coefficient theorem

The

Universal Coefficient Theorem

for

homology

expresses the

homology groups

with coefficients in an arbitrary

abelian group

G

in terms of the homology groups with coefficients in

ℤ

.

Theorem

(

Universal

Coefficients for Homology)

Let

K

be a

chain complex

of

free abelian groups

, and let

G

any abelian group. Then there exists a

split short exact sequence

Generated on Sat Feb 10 11:40:49 2018 by

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natural equivalence of C_G and C_M categories

natural equivalence of

C

G

and

C

M

categories

Theorem 0.1

.

(with proof by Verdier

[

1

]

)
The

category

C

G

of

categorical

groups and functorial

homomorphisms

between categorical groups, and the category

C

M

of crossed modules of groups and homomorphisms between them, are

naturally equivalent

.

References

1

Jean-Louis Verdier, Des cat

é

gories d

é

riv

é

es des cat

é

gories ab

é

liennes, Ast

é

risque, vol. 239, Soci

$́

⁢

e

⁢

t

⁢

é

M

⁢

a

⁢

t

⁢

h

⁢

$́

⁢

e

⁢

m

⁢

a

⁢

t

⁢

i

⁢

q

⁢

u

⁢

e

⁢

d

⁢

e

⁢

F

⁢

r

⁢

a

⁢

n

⁢

c

⁢

e

,

1996

⁢

(

i

⁢

n

⁢

F

⁢

r

⁢

e

⁢

n

⁢

c

⁢

h

)

.

Title

natural equivalence of

C

G

and

C

M

categories

Canonical name

NaturalEquivalenceOfCGAndCMCategories

Date of creation

2013-03-22 18:25:47

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2013-03-22 18:25:47

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

13

Author

bci1 (20947)

Entry type

Theorem

Classification

msc 55M05

Classification

msc 18E05

Classification

msc 18-00

Related topic

HomotopyGroupoidsAndCrossComplexesAsNonCommutativeStructuresInHigherDimensionalAlgebraHDA

Related topic

EquivalenceOfCategories2

Related topic

FunctorCategory2

Related topic

GroupCohomology

Related topic

IndexOfCategories

Generated on Sat Feb 10 11:40:54 2018 by

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Poincaré duality

Poincaré duality

If

M

is a

compact

, oriented,

n

-dimensional manifold, then there is a canonical (though non-natural (

http://planetmath.org/NaturalTransformation

))

isomorphism

D

:

H

q

⁢

(

M

,

ℤ

)

→

H

n

-

q

⁢

(

M

,

ℤ

)

(where

H

k

⁢

(

M

,

ℤ

)

is the

k

th

homology group

of

M

with integer coefficients and

H

k

⁢

(

M

,

ℤ

)

the

k

th

cohomology

(

http://planetmath.org/DeRhamCohomology

) group) for all

q

, which is given by

cap product

with a

generator

of

H

n

⁢

(

M

,

ℤ

)

(a choice of a generator here corresponds to an orientation). This isomorphism exists with
coefficients in

ℤ

/

2

⁢

ℤ

regardless of orientation.

This isomorphism gives a nice

interpretation

to

cup product

. If

X

,

Y

are transverse submanifolds of

M

, then

X

∩

Y

is also a submanifold. All of these submanifolds

represent

homology classes

of

M

in the appropriate

dimensions

, and

D

-

1

⁢

(

[

X

]

)

∪

D

-

1

⁢

(

[

Y

]

)

=

D

-

1

⁢

(

[

X

∩

Y

]

)

,

where

∪

is cup product, and

∩

in

intersection

, not cap product.

Title

Poincaré duality

Canonical name

PoincareDuality

Date of creation

2013-03-22 13:11:36

Last modified on

2013-03-22 13:11:36

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 55M05

Synonym

Poincaré isomorphism

Related topic

DualityInMathematics

Generated on Sat Feb 10 11:40:56 2018 by

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Brouwer fixed point theorem

Brouwer fixed point theorem

Theorem

Let

𝐁

=

{

x

∈

ℝ

n

:

∥

x

∥

≤

1

}

be the closed

unit ball

in

ℝ

n

. Any

continuous function

f

:

𝐁

→

𝐁

has a

fixed point

.

Notes

Shape is not important

The theorem also applies to anything

homeomorphic

to a closed disk, of course. In particular, we can replace

B

in the formulation with a square or
a

triangle

.

Compactness

counts (a)

The theorem is not true if we drop a point from the

interior

of

B

. For example, the map

f

⁢

(

x

→

)

=

1

2

⁢

x

→

has the single fixed point at

0

; dropping it from the domain yields a map with no fixed points (

http://planetmath.org/FixedPoint

).

Compactness counts (b)

The theorem is not true for an

open disk

. For

instance

, the map

f

⁢

(

x

→

)

=

1

2

⁢

x

→

+

(

1

2

,

0

,

…

,

0

)

has its single fixed point on the

boundary

of

B

.

Title

Brouwer fixed point theorem

Canonical name

BrouwerFixedPointTheorem

Date of creation

2013-03-22 12:44:34

Last modified on

2013-03-22 12:44:34

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 55M20

Classification

msc 54H25

Classification

msc 47H10

Related topic

FixedPoint

Related topic

SchauderFixedPointTheorem

Related topic

TychonoffFixedPointTheorem

Related topic

KKMlemma

Related topic

KKMLemma

Generated on Sat Feb 10 11:40:58 2018 by

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the topologist’s sine curve has the fixed point property

the topologist’s sine curve has the fixed point property

The typical example of a

connected space

that is not

path connected

(the topologist’s sine curve) has the

fixed point property

.

Let

X

1

=

{

0

}

×

[

-

1

,

1

]

and

X

2

=

{

(

x

,

sin

⁡

(

1

/

x

)

)

:

0

<

x

≤

1

}

, and

X

=

X

1

∪

X

2

.

If

f

:

X

→

X

is a

continuous map

, then since

X

1

and

X

2

are both path connected, the image of each one of them must be entirely contained in another of them.

If

f

⁢

(

X

1

)

⊂

X

1

, then

f

has a

fixed point

because the

interval

has the fixed point property.
If

f

⁢

(

X

2

)

⊂

X

1

, then

f

⁢

(

X

)

=

f

⁢

(

c

⁢

l

⁢

(

X

2

)

)

⊂

c

⁢

l

⁢

(

f

⁢

(

X

2

)

)

⊂

X

1

, and in particular

f

⁢

(

X

1

)

⊂

X

1

and again

f

has a fixed point.

So the only case that remains is that

f

⁢

(

X

)

⊂

X

2

. And since

X

is

compact

, its projection to the first

coordinate

is also compact so that it must be an interval

[

a

,

b

]

with

a

>

0

. Thus

f

⁢

(

X

)

is contained in

S

=

{

(

x

,

sin

⁡

(

1

/

x

)

)

:

x

∈

[

a

,

b

]

}

. But

S

is

homeomorphic

to a

closed interval

, so that it has the fixed point property, and
the

restriction

of

f

to

S

is a continuous map

S

→

S

, so that it has a fixed point.

This proof is due to Koro.

Title

the topologist’s sine curve has the fixed point property

Canonical name

TheTopologistsSineCurveHasTheFixedPointProperty

Date of creation

2013-03-22 16:59:37

Last modified on

2013-03-22 16:59:37

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Proof

Classification

msc 55M20

Classification

msc 54H25

Classification

msc 47H10

Generated on Sat Feb 10 11:41:05 2018 by

LaTeXML
degree (map of spheres)

degree (map of spheres)

Given a non-negative

integer

n

, let

S

n

denote the

n

-dimensional sphere. Suppose

f

:

S

n

→

S

n

is a continuous map. Applying the

n

t

⁢

h

reduced homology functor

H

~

n

⁢

(

_

)

, we obtain a

homomorphism

f

*

:

H

~

n

⁢

(

S

n

)

→

H

~

n

⁢

(

S

n

)

. Since

H

~

n

⁢

(

S

n

)

≈

ℤ

, it follows that

f

*

is a homomorphism

ℤ

→

ℤ

. Such a map must be

multiplication

by an integer

d

. We define the

degree

of the map

f

, to be this

d

.

0.1

Basic Properties

1.

If

f

,

g

:

S

n

→

S

n

are

continuous

, then

deg

⁡

(

f

∘

g

)

=

deg

⁡

(

f

)

⋅

deg

⁡

(

g

)

.

2.

If

f

,

g

:

S

n

→

S

n

are

homotopic

, then

deg

⁡

(

f

)

=

deg

⁡

(

g

)

.

3.

The degree of the

identity map

is

+

1

.

4.

The degree of the

constant map

is

0

.

5.

The degree of a

reflection

through an

(

n

+

1

)

-dimensional

hyperplane

through the origin is

-

1

.

6.

The

antipodal map

, sending

x

to

-

x

, has degree

(

-

1

)

n

+

1

. This follows since the map

f

i

sending

(

x

1

,

…

,

x

i

,

…

,

x

n

+

1

)

↦

(

x

1

,

…

,

-

x

i

,

…

,

x

n

+

1

)

has degree

-

1

by (4), and the compositon

f

1

∘

⋯

∘

f

n

+

1

yields the antipodal map.

0.2

Examples

If we identify

S

1

⊂

ℂ

, then the map

f

:

S

1

→

S

1

defined by

f

⁢

(

z

)

=

z

k

has degree

k

. It is also possible, for any

positive integer

n

, and any integer

k

, to construct a map

f

:

S

n

→

S

n

of degree

k

.

Using degree, one can prove several

theorems

, including the so-called ’hairy ball theorem’, which that there exists a continuous non-zero

vector field

on

S

n

if and only if

n

is odd.

Title

degree (map of spheres)

Canonical name

DegreemapOfSpheres

Date of creation

2013-03-22 13:22:12

Last modified on

2013-03-22 13:22:12

Owner

drini (3)

Last modified by

drini (3)

Numerical id

12

Author

drini (3)

Entry type

Definition

Classification

msc 55M25

Defines

degree

Generated on Sat Feb 10 11:41:08 2018 by

LaTeXML
Lusternik-Schnirelmann category

Lusternik-Schnirelmann category

Let

X

be a

topological space

. An important

topological invariant

of

X

called

Lusternik-Schnirelmann category

cat

is defined as follows:

cat

⁢

(

X

)

=

min

⁢

{

#

⁢

(

C

)

:

where

C

are the coverings of

X

by contractible open sets

}

.

If

X

is a

manifold

,

cat

⁢

(

X

)

coincides with the

minimal number

of

critical points

among all smooth scalars maps

X

→

ℝ

.

This is

equivalent

to saying that

X

has a covering

{

U

s

}

such that
it is posible to factor homotopically each

U

s

↪

i

X

through

U

s

→

a

*

→

b

X

i.e

i

≃

b

∘

a

.

This allows us to define another category, e.g.:

We can ask about the minimal number of

open sets

U

s

that cover

X

and
are

homotopically equivalent

to

S

1

, say,
the

inclusion

U

s

↪

i

X

and

U

s

→

a

S

1

→

b

X

are

i

≃

b

∘

a

.

It is becoming standard to speak of the t-cat of

X

.
This is related to the

round complexity

of the space.

References

1

R.H. Fox,

On the Lusternik-Schnirelmann category

, Annals of Math. 42 (1941), 333-370.

2

F. Takens,

The minimal number of critical points of a

function

on

compact manifolds

and the Lusternik-Schnirelmann category

, Invent. math. 6,(1968), 197-244.

Title

Lusternik-Schnirelmann category

Canonical name

LusternikSchnirelmannCategory

Date of creation

2013-03-22 15:53:30

Last modified on

2013-03-22 15:53:30

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

13

Author

juanman (12619)

Entry type

Definition

Classification

msc 55M30

Related topic

Topology

Related topic

RoundComplexity

Generated on Sat Feb 10 11:41:13 2018 by

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