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 andapproximation theorems for topological spaces. A central problem in algebraic topologyis to find algebraic invariants of topological spaces, which is usually carried out by meansof 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.htmlGrothendieck’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