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topics in algebraic topology (Topic)

Algebraic Topology topics

Freely contributed, open topics

Introduction

Algebraic Topology (AT) utilizes algebraic approaches to solving 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 and Algebraic Geometry (AG). On the other hand, there are also close ties between algebraic geometry and number theory.

Background

Latin quote: “Non multa sed multum”

Outline

  1. Homotopy theory and fundamental groups
  2. Topology and Groupoids
  3. Homology and cohomology theories
  4. Duality
  5. Category theory applications in algebraic topology
  6. Index of categories, functors and natural transformations
  7. Grothendieck's Descent theory
  8. `Anabelian Geometry'
  9. Categorical Galois theory
  10. Higher Dimensional Algebra (HDA)
  11. Quantum Algebraic Topology (QAT)
  12. 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
  5. Whitehead groups, torsion and towers
  6. Postnikov towers

Topology and Groupoids

  1. Topology definition, axioms and basic concepts
  2. Fundamental groupoid
  3. Topological groupoid
  4. van Kampen theorem for groupoids
  5. Groupoid pushout theorem
  6. Double groupoids and crossed modules
  7. new4

Homology theory

  1. Homology group
  2. Homology sequence
  3. Homology complex
  4. new4

Cohomology theory

  1. Cohomology group
  2. Cohomology sequence
  3. DeRham cohomology
  4. new4

Duality in algebraic topology and category theory

  1. Tanaka-Krein duality
  2. Grothendieck duality
  3. Categorical duality
  4. Tangled duality
  5. DA5
  6. DA6
  7. DA7

Category theory applications

  1. Abelian categories
  2. Topological category
  3. Fundamental Groupoid Functor
  4. Categorical Galois theory
  5. Non-Abelian algebraic topology
  6. Group category
  7. Groupoid category
  8. $ \mathcal{T}op$ category
  9. Topos and topoi axioms
  10. Generalized toposes
  11. Categorical logic and algebraic topology
  12. Metatheorems
  13. Duality between Spaces and Algebras

Index of categories

The following is a listing of categories relevant to algebraic topology:
  1. Algebraic categories
  2. Topological category
  3. Category of sets, Set
  4. Category of topological spaces
  5. Category of Riemannian manifolds
  6. Category of CW-complexes
  7. Category of Hausdorff spaces
  8. Category of Borel spaces
  9. Category of CR-complexes
  10. Category of graphs
  11. Category of spin networks
  12. Category of groups
  13. Galois category
  14. Category of fundamental groups
  15. Category of Polish groups
  16. Groupoid category
  17. Category of groupoids (or groupoid category)
  18. Category of Borel groupoids
  19. Category of fundamental groupoids
  20. Category of functors (or functor category)
  21. Double groupoid category
  22. Double category
  23. Category of Hilbert spaces
  24. Category of quantum automata
  25. R-category
  26. Category of algebroids
  27. Category of double algebroids
  28. Category of dynamical systems

Index of functors

The following is a contributed listing of functors:
  1. Covariant functors
  2. Contravariant functors
  3. Adjoint functors
  4. Preadditive functors
  5. Additive functor
  6. Representable functors
  7. Fundamental groupoid functor
  8. Forgetful functors
  9. Grothendieck group functor
  10. Exact functor
  11. Multi-functor
  12. Section functors
  13. NT2
  14. NT3

Index of natural transformations

The following is a contributed listing of natural transformations:
  1. Natural equivalence
  2. Natural transformations in a 2-category
  3. NT3
  4. NT1
  5. NT2
  6. NT3

Grothendieck proposals

  1. Esquisse d'un Programme
  2. Pursuing Stacks
  3. S2
  4. S3
  5. S4

Descent Theory

  1. D1
  2. D2
  3. D3
  4. D4

Higher Dimensional Algebra (HDA)

  1. Categorical groups
  2. Double groupoids
  3. Double algebroids
  4. Bi-algebroids
  5. $ R$-algebroid
  6. $ 2$-category
  7. $ n$-category
  8. Super-category
  9. weak n-categories
  10. Bi-dimensional Geometry
  11. Noncommutative geometry
  12. Higher-Homotopy theories
  13. Higher-Homotopy Generalized van Kampen Theorem (HGvKT)
  14. H1
  15. H2
  16. H3
  17. H4

Axioms of cohomology theory

  1. A1
  2. A2
  3. A3
  4. A4
  5. A5
  6. A6
  7. A7

Axioms of Homology Theory

  1. A1
  2. A2
  3. A3
  4. A4
  5. A5
  6. A6

Quantum Algebraic Topology (QAT)

(a). Quantum Algebraic Topology (QAT) is described as the mathematical and physical study of general theories of quantum algebraic structures from the standpoint of Algebraic Topology, Category Theory and their non-Abelian extensions in Higher Dimensional Algebra and Supercategories

  1. Quantum Operator Algebras (such as: involution, *-algebras, or $ *$-algebras, von Neumann algebras, , JB- and JL- algebras, $ C^*$ - or C*- algebras,
  2. Quantum von Neumann algebra and subfactors; Jone's towers and subfactors
  3. Kac-Moody and K-algebras
  4. categorical groups
  5. Hopf algebras, Quantum Groups and Quantum group algebras
  6. Quantum Groupoids and weak Hopf $ C^*$-algebras
  7. Groupoid C*-Convolution algebras and *-Convolution Algebroids
  8. Quantum Spacetimes and Quantum Fundamental Groupoids
  9. Quantum Double Algebras
  10. Quantum Gravity, supersymmetries, supergravity, superalgebras and graded `Lie' algebras
  11. Quantum Categorical algebra and Higher Dimensional, $ \L{}-M_n$- Toposes
  12. Quantum R-categories, R-supercategories and Symmetry Breaking
    1. new1
    2. new2
    3. new3
    4. new4

12

  1. new1
  2. new2
  3. new3
  4. new4

13

  1. new1
  2. new2
  3. new3
  4. new4

14

References

Bibliography

Textbooks and Expositions:

  1. A Textbook1
  2. A Textbook2
  3. A Textbook3
  4. A Textbook4
  5. A Textbook5
  6. A Textbook6
  7. A Textbook7
  8. A Textbook8
  9. A Textbook9
  10. A Textbook10
  11. A Textbook11
  12. A Textbook12
  13. A Textbook13
  14. new1
  15. new2
  16. new3
  17. new4



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See Also: category, functor, Grothendieck group, topological group, natural transformation, topics in algebraic topology, algebraic geometry, category theory bibliography: algebraic topology, algebraic foundations of quantum algebraic topology, index of categories

Other names:  category theory, algebraic geometry, topology, algebras
Keywords:  homology and cohomology theory, fundamental functor, fundamental groupoid functor, groupoid category, algebroid category, crossed complexes, complex modules, homology groups and groupoids. homotopy theory, groupoids, categorical algebra, topological categories, topological groupoids, Lie groupoids, Lie algebroids, higher-dimensional algebra, higher-dimensional groupoids, Van Kampen theorems, approximation theorem, Hurewivz, theorem, algebraic theories
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Cross-references: symmetry, R-supercategories, categorical algebra, graded Lie algebras, superalgebras, supergravity, supersymmetries, Quantum Gravity, groupoid C*-convolution algebras, quantum groupoids, quantum group, quantum groups, Hopf algebras, von Neumann algebras, *-algebras, involution, quantum operator algebras, supercategories, extensions, algebraic structures, noncommutative geometry, n-categories, 2-category, natural equivalence, section functors, exact functor, Grothendieck group, forgetful functors, representable functors, additive functor, preadditive functors, adjoint functors, dynamical systems, R-category, category of quantum automata, category of Hilbert spaces, double category, functor category, category of functors, category of Borel groupoids, category of groupoids, category of Polish groups, graphs, category of Borel spaces, Hausdorff spaces, CW-complexes, category of Riemannian manifolds, category of sets, logic, topos, fundamental groupoid functor, category, abelian categories, categorical duality, complex, sequence, homology group, modules, double groupoids, pushout, fundamental groupoid, axioms, torsion, groups, non-Abelian, QAT, quantum algebraic topology, HDA, higher dimensional algebra, Galois theory, categorical, geometry, natural transformations, functors, index of categories, applications, cohomology, groupoids, fundamental groups, theory, number theory, connections, cohomology groups, homology, homotopy, invariants, approximation theorems for topological spaces, manifolds, duality, surfaces, algebraic, open
There are 264 references to this entry.

This is version 54 of topics in algebraic topology, born on 2008-09-16, modified 2008-10-21.
Object id is 11041, canonical name is AlgebraicTopology.
Accessed 850 times total.

Classification:
AMS MSC: 55-01 (Algebraic topology :: Instructional exposition )
 55N20 (Algebraic topology :: Homology and cohomology theories :: Generalized homology and cohomology theories)
 55N40 (Algebraic topology :: Homology and cohomology theories :: Axioms for homology theory and uniqueness theorems)
 55N99 (Algebraic topology :: Homology and cohomology theories :: Miscellaneous)
 55N15 (Algebraic topology :: Homology and cohomology theories :: $K$-theory)
 55N30 (Algebraic topology :: Homology and cohomology theories :: Sheaf cohomology)
 18-00 (Category theory; homological algebra :: General reference works )
 11E72 (Number theory :: Forms and linear algebraic groups :: Galois cohomology of linear algebraic groups)
 11F23 (Number theory :: Discontinuous groups and automorphic forms :: Relations with algebraic geometry and topology)
 57N65 (Manifolds and cell complexes :: Topological manifolds :: Algebraic topology of manifolds)
 57R19 (Manifolds and cell complexes :: Differential topology :: Algebraic topology on manifolds)
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