overview of algebraic topology(Topic)
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),
Non-commutative Geometry and NAAT. On the other hand, there are also close ties between
algebraic geometry and number theory.
1.2 Outline
- Homotopy theory and fundamental groups
- Topology and groupoids; van Kampen theorem
- Homology and cohomology theories
- Duality
- category theory applications in algebraic topology
- indexes of category, functors and natural transformations
- Grothendieck’s Descent theory
- ‘Anabelian geometry’
- Categorical Galois theory
- higher dimensional algebra (HDA)
- Quantum algebraic topology (QAT)
- Non-Abelian Quantum Algebraic Topology
(http : ∕∕aux.planetphysics.org∕files∕lec∕61∕ANAQAT20c.pdf)
- Quantum Geometry
- Non-Abelian algebraic topology (NAAT)
1.3 Homotopy theory and fundamental groups
- Homotopy
- Fundamental group of a space
- Fundamental theorems
- van Kampen theorem
- Whitehead groups, torsion and towers
- Postnikov towers
1.4 Topology and Groupoids
- Topology definition, axioms and basic concepts
- Fundamental groupoid
- topological groupoid
- van Kampen theorem for groupoids
- Groupoid pushout theorem
- double groupoids and crossed modules
- new4
1.5 Homology theory
- homology group
- Homology sequence
- Homology complex
- new4
1.6 Cohomology theory
- Cohomology group
- Cohomology sequence
- DeRham cohomology
- new4
1.7 Non-Abelian Algebraic Topology
- crossed complexes
- Modules
- omega-groupoids
- double groupoids
- Higher Homotopy van Kampen Theorems
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See Also: non-Abelian algebraic topology
| Also defines: |
omega-groupoid, higher homotopy, algebraic topology, nonabelian algebraic topology, manifold, homology group, cohomology theory, nonabelian theory, quantum geometry, quantum algebra, quantum algebraic topology, category thyeory, noncommutative geometry |
| Keywords: |
overview, algebraic topology, nonabelian, algebraic topology invarinats, groupoid, topological space, higher homotopy theory, higher homotopy van Kampen theorem, crossed complex, module, omega-groupoid, double groupoid |
Cross-references: omega-groupoids, crossed complexes, homology group, modules, double groupoids, pushout, topological groupoid, concepts, QAT, HDA, higher dimensional algebra, natural transformations, functors, indexes of category, category theory applications, groupoids, fundamental groups, cohomology groups, homotopy, theorems, duality, topological, algebraic
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