PlanetPhysics/Overview of Algebraic Topology

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)


 * 1) homotopy theory and fundamental groups #Topology and groupoids; van Kampen theorem
 * 2) Homology and cohomology theories
 * 3) Duality
 * 4) category theory applications in algebraic topology
 * 5) indexes of category, functors and natural transformations
 * 6) Grothendieck's Descent theory
 * 7) `Anabelian Geometry' #Categorical Galois theory
 * 8) higher dimensional algebra (HDA)
 * 9) Quantum algebraic topology (QAT)
 * 10) Non-Abelian Quantum Algebraic Topology

($$http://aux.planetphysics.org/files/lec/61/ANAQAT20c.pdf$$)
 * 1) Quantum Geometry
 * 2) Non-Abelian algebraic topology (NAAT)

Homotopy theory and fundamental groups

 * 1) Homotopy
 * 2) fundamental group of a space
 * 3) Fundamental theorems
 * 4) Van Kampen theorem #Whitehead groups, torsion and towers
 * 5) Postnikov towers

Topology and Groupoids

 * 1) Topology definition, axioms and basic concepts #fundamental groupoid #topological groupoid #van Kampen theorem for groupoids
 * 2) Groupoid pushout theorem
 * 3) double groupoids and crossed modules
 * 4) new4

Homology theory

 * 1) Homology group
 * 2) Homology sequence
 * 3) Homology complex
 * 4) new4

Cohomology theory

 * 1) cohomology group #Cohomology sequence
 * 2) DeRham cohomology
 * 3) new4

Non-Abelian Algebraic Topology

 * 1) crossed complexes #modules #omega-groupoids #double groupoids
 * 2) Higher Homotopy van Kampen Theorems