# 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. 1.
2. 2.

Topology and groupoids; van Kampen theorem (http://planetmath.org/VanKampensTheorem)

3. 3.

Homology and cohomology theories

4. 4.

Duality

5. 5.

Category theory applications in algebraic topology

6. 6.
7. 7.

http://www.uclouvain.be/17501.htmlGrothendieck’s Descent theory

8. 8.

‘Anabelian geometry’

9. 9.

Categorical Galois theory

10. 10.
11. 11.

Quantum algebraic topology (QAT)

12. 12.
13. 13.

Non-Abelian algebraic topology (NAAT)

### 1.3 Homotopy theory and fundamental groups

1. 1.

Homotopy

2. 2.

Fundamental group of a space

3. 3.

Fundamental theorems

4. 4.

van Kampen theorem

5. 5.

6. 6.

Postnikov towers

### 1.4 Topology and Groupoids

1. 1.

Topology definition, axioms and basic concepts

2. 2.
3. 3.
4. 4.
5. 5.

van Kampen theorem for groupoids

6. 6.

Groupoid pushout theorem

7. 7.

Double groupoids and crossed modules

8. 8.

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### 1.5 Homology theory

1. 1.
2. 2.

Homology sequence

3. 3.

Homology complex

4. 4.

Homological Algebra

### 1.6 Cohomology theory

1. 1.

Cohomology group

2. 2.

Cohomology sequence

3. 3.

DeRham cohomology

4. 4.

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### 1.7 Non-Abelian Algebraic Topology

1. 1.

Crossed Complexes

2. 2.

Modules

3. 3.

Cross-modules

4. 4.

Omega-Groupoids

5. 5.

Double Groupoids: Homotopy Double Groupoid of a Hausdorff Space

6. 6.
7. 7.

Groupoid Category

8. 8.

Algebroids

9. 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