Overview article for algebraic topology

Overview article for algebraic topology

1

An Overview of Algebraic Topology topics

1.1

Introduction

Algebraic topology

(AT) utilizes

algebraic

approaches to solve topological problems,
such as the classification of

surfaces

, proving duality theorems for

manifolds

and

approximation theorems for topological spaces

. A central problem in algebraic topology
is to find algebraic invariants of

topological spaces

, which is usually carried out by means
of

homotopy

,

homology

and

cohomology groups

. There are close

connections

between algebraic topology,

Algebraic Geometry

(AG) (

http://planetmath.org/AlgebraicGeometry

), and

Non-commutative Geometry

/NAAT. On the other hand, there are also close ties between algebraic geometry and

number theory

.

1.2

Outline

1.

Homotopy theory

and

fundamental groups

2.

Topology and groupoids;

van Kampen theorem

(

http://planetmath.org/VanKampensTheorem

)

3.

Homology and

cohomology

theories

4.

Duality

5.

Category theory

applications in algebraic topology

6.

Index of categories

,

functors

and

natural transformations

7.

http://www.uclouvain.be/17501.html

Grothendieck’s Descent theory

8.

‘Anabelian geometry’

9.

Categorical Galois theory

10.

Higher dimensional algebra

(HDA)

11.

Quantum algebraic topology (QAT)

12.

Quantum Geometry

13.

Non-Abelian

algebraic topology (NAAT)

1.3

Homotopy theory and fundamental groups

1.

Homotopy

2.

Fundamental group of a space

3.

Fundamental theorems

4.

van Kampen theorem

5.

Whitehead groups,

torsion

and towers

6.

Postnikov towers

1.4

Topology and Groupoids

1.

Topology

definition, axioms and basic concepts

2.

Fundamental groupoid

3.

Topological groupoid

4.

Classifying space

5.

van Kampen theorem for

groupoids

6.

Groupoid

pushout

theorem

7.

Double groupoids

and crossed modules

8.

new4

1.5

Homology theory

1.

Homology group

2.

Homology

sequence

3.

Homology complex

4.

Homological Algebra

1.6

Cohomology theory

1.

Cohomology group

2.

Cohomology sequence

3.

DeRham cohomology

4.

new4

1.7

Non-Abelian Algebraic Topology

1.

Crossed Complexes

2.

Modules

3.

Cross-modules

4.

Omega-Groupoids

5.

Double Groupoids:

Homotopy Double Groupoid

of a

Hausdorff Space

6.

Double Category

7.

Groupoid Category

8.

Algebroids

9.

Higher Homotopy van Kampen Theorem

…more to come

Title

Overview article for algebraic topology

Canonical name

OverviewArticleForAlgebraicTopology

Date of creation

2013-03-22 19:15:48

Last modified on

2013-03-22 19:15:48

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

17

Author

bci1 (20947)

Entry type

Topic

Classification

msc 57R19

Classification

msc 57N65

Classification

msc 11F23

Classification

msc 11E72

Classification

msc 18-00

Classification

msc 55N30

Classification

msc 55N15

Classification

msc 55N99

Classification

msc 55N40

Classification

msc 55N20

Classification

msc 55-01

Related topic

groupoid

Related topic

category

Related topic

GroupoidCategory

Related topic

topology

Related topic

HomotopyDoubleGroupoidOfAHausdorffSpace

Related topic

QuantumGeometry

Related topic

TopologicalSpace

Related topic

HigherDimensionalAlgebra

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