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.
-
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.
-
11.
Quantum algebraic topology (QAT)
- 12.
-
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.
- 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.
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.
-
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 |