Overview article for algebraic topology


1 An Overview of Algebraic Topology topics

1.1 Introduction

Algebraic topology (AT) utilizes algebraicMathworldPlanetmath approaches to solve topological problems, such as the classification of surfacesMathworldPlanetmath, proving duality theorems for manifoldsMathworldPlanetmath and approximation theorems for topological spaces. A central problem in algebraic topology is to find algebraic invariants of topological spacesMathworldPlanetmath, which is usually carried out by means of homotopyMathworldPlanetmathPlanetmath, homologyMathworldPlanetmathPlanetmath and cohomology groupsPlanetmathPlanetmath. There are close connections between algebraic topology, Algebraic GeometryMathworldPlanetmathPlanetmath (AG) (http://planetmath.org/AlgebraicGeometry), and Non-commutative GeometryPlanetmathPlanetmathPlanetmath/NAAT. On the other hand, there are also close ties between algebraic geometry and number theoryMathworldPlanetmath.

1.2 Outline

  1. 1.
  2. 2.

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

  3. 3.

    Homology and cohomologyMathworldPlanetmathPlanetmath theories

  4. 4.

    Duality

  5. 5.

    Category theoryMathworldPlanetmathPlanetmathPlanetmath 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-AbelianMathworldPlanetmathPlanetmath 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.

    Whitehead groups, torsionPlanetmathPlanetmathPlanetmath and towers

  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 groupoidsPlanetmathPlanetmathPlanetmath

  6. 6.

    Groupoid pushout theorem

  7. 7.

    Double groupoidsPlanetmathPlanetmath and crossed modules

  8. 8.

    new4

1.5 Homology theory

  1. 1.
  2. 2.

    Homology sequenceMathworldPlanetmath

  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.

    new4

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 GroupoidPlanetmathPlanetmath 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 categoryMathworldPlanetmath
Related topic GroupoidCategory
Related topic topology
Related topic HomotopyDoubleGroupoidOfAHausdorffSpace
Related topic QuantumGeometry
Related topic TopologicalSpace
Related topic HigherDimensionalAlgebra