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 Noncommutative Geometry^{}/NAAT. On the other hand, there are also close ties between algebraic geometry and number theory^{}.
1.2 Outline
 1.

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.

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.

13.
NonAbelian^{} 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.
 3.
 4.

5.
van Kampen theorem for groupoids^{}

6.
Groupoid pushout theorem

7.
Double groupoids^{} and crossed modules

8.
new4
1.5 Homology theory
 1.

2.
Homology sequence^{}

3.
Homology complex

4.
Homological Algebra
1.6 Cohomology theory

1.
Cohomology group

2.
Cohomology sequence

3.
DeRham cohomology

4.
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1.7 NonAbelian Algebraic Topology

1.
Crossed Complexes

2.
Modules

3.
Crossmodules

4.
OmegaGroupoids

5.
Double Groupoids: Homotopy Double Groupoid^{} of a Hausdorff Space
 6.

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  20130322 19:15:48 
Last modified on  20130322 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 1800 
Classification  msc 55N30 
Classification  msc 55N15 
Classification  msc 55N99 
Classification  msc 55N40 
Classification  msc 55N20 
Classification  msc 5501 
Related topic  groupoid 
Related topic  category^{} 
Related topic  GroupoidCategory 
Related topic  topology 
Related topic  HomotopyDoubleGroupoidOfAHausdorffSpace 
Related topic  QuantumGeometry 
Related topic  TopologicalSpace 
Related topic  HigherDimensionalAlgebra 