topics in algebraic topology


1 Algebraic topology topics

1.1 Introduction

Algebraic topology (AT) utilizes algebraic 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 theory.

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.

    van Kampen theorem for groupoidsPlanetmathPlanetmathPlanetmath

  5. 5.

    Groupoid pushout theorem

  6. 6.

    Double groupoidsPlanetmathPlanetmath and crossed modules

  7. 7.

    new4

1.5 Homology theory

  1. 1.
  2. 2.

    Homology sequencePlanetmathPlanetmath

  3. 3.

    Homology complex

  4. 4.

    new4

1.6 Cohomology theory

  1. 1.

    Cohomology group

  2. 2.

    Cohomology sequence

  3. 3.

    DeRham cohomology

  4. 4.

    new4

1.7 Duality in algebraic topology and category theory

  1. 1.

    Tanaka-Krein duality

  2. 2.

    Grothendieck duality

  3. 3.
  4. 4.

    Tangled duality

  5. 5.

    DA5

  6. 6.

    DA6

  7. 7.

    DA7

1.8 Category theory applications

  1. 1.
  2. 2.

    Topological category

  3. 3.

    Fundamental groupoid functor

  4. 4.

    Categorical Galois theory

  5. 5.

    Non-Abelian algebraic topology

  6. 6.

    Group categoryMathworldPlanetmath

  7. 7.

    Groupoid category

  8. 8.

    𝒯⁢o⁢p category

  9. 9.

    Topos and topoi axioms

  10. 10.

    Generalized toposes

  11. 11.

    Categorical logic and algebraic topology

  12. 12.
  13. 13.

    Duality between spaces and algebrasMathworldPlanetmathPlanetmath

1.9 Index of categories

The following is a listing of categories relevant to algebraic topology:

  1. 1.

    http://www.uclouvain.be/17501.htmlAlgebraic categoriesPlanetmathPlanetmathPlanetmathPlanetmath

  2. 2.

    Topological category

  3. 3.
  4. 4.

    Category of topological spaces

  5. 5.

    Category of Riemannian manifolds

  6. 6.

    Category of CW-complexesMathworldPlanetmath

  7. 7.

    Category of Hausdorff spaces

  8. 8.
  9. 9.

    Category of CR-complexes

  10. 10.

    Category of graphs

  11. 11.

    Category of spin networks

  12. 12.

    Category of groups

  13. 13.

    Galois category

  14. 14.

    Category of fundamental groups

  15. 15.

    Category of Polish groups

  16. 16.

    Groupoid category

  17. 17.

    Category of groupoidsPlanetmathPlanetmath (or groupoid category)

  18. 18.
  19. 19.

    Category of fundamental groupoids

  20. 20.

    Category of functorsPlanetmathPlanetmath (or functor category)

  21. 21.

    Double groupoid category

  22. 22.
  23. 23.

    Category of Hilbert spacesMathworldPlanetmath

  24. 24.

    Category of quantum automata

  25. 25.
  26. 26.

    Category of algebroids

  27. 27.

    Category of double algebroids

  28. 28.

    Category of dynamical systems

1.10 Index of functors

The following is a contributed listing of functors:

1.11 Index of natural transformations

The following is a contributed listing of natural transformations:

  1. 1.
  2. 2.

    Natural transformations in a 2-category

  3. 3.

    NT3

  4. 4.

    NT1

  5. 5.

    NT2

  6. 6.

    NT3

1.12 Grothendieck proposals

  1. 1.

    Esquisse d’un Programme (http://planetmath.org/AlexSMathematicalHeritageEsquisseDunProgramme)

  2. 2.

    http://www.math.jussieu.fr/ leila/grothendieckcircle/stacks.psPursuing Stacks

  3. 3.

    S2

  4. 4.

    S3

  5. 5.

    S4

1.13 Descent theory

  1. 1.

    D1

  2. 2.

    D2

  3. 3.

    D3

  4. 4.

    D4

1.14 Higher dimensional algebra (HDA)

  1. 1.

    Categorical groups

  2. 2.

    Double groupoids

  3. 3.

    Double algebroids

  4. 4.

    Bi-algebroids

  5. 5.

    R-algebroid

  6. 6.

    2-category

  7. 7.

    n-category

  8. 8.
  9. 9.

    weak n-categories

  10. 10.

    Bi-dimensional Geometry

  11. 11.

    Noncommutative geometry (http://planetmath.org/NoncommutativeGeometry)

  12. 12.

    Higher-Homotopy theories

  13. 13.

    Higher-Homotopy Generalized van Kampen Theorem (HGvKT) (http://planetmath.org/GeneralizedVanKampenTheoremsHigherDimensional)

  14. 14.

    H1

  15. 15.

    H2

  16. 16.

    H3

  17. 17.

    H4

1.14.1 Axioms of cohomology theory

  1. 1.

    A1

  2. 2.

    A2

  3. 3.

    A3

  4. 4.

    A4

  5. 5.

    A5

  6. 6.

    A6

  7. 7.

    A7

1.14.2 Axioms of homology theory

  1. 1.

    A1

  2. 2.

    A2

  3. 3.

    A3

  4. 4.

    A4

  5. 5.

    A5

  6. 6.

    A6

1.15 Quantum algebraic topology (QAT)

(a). Quantum algebraic topology is described as the mathematical and physical study of general theories of quantum algebraic structuresPlanetmathPlanetmath from the standpoint of algebraic topology, category theory and their non-Abelian extensionsPlanetmathPlanetmathPlanetmath in higher dimensional algebra and supercategories

  1. 1.

    Quantum operator algebrasPlanetmathPlanetmathPlanetmath (such as: involution, *-algebras, or *-algebras, von Neumann algebrasMathworldPlanetmathPlanetmath, , JB- and JL- algebras, C* - or C*- algebras,

  2. 2.

    Quantum von Neumann algebra and subfactors; Jone’s towers and subfactors

  3. 3.

    Kac-Moody and K-algebras

  4. 4.

    categorical groups

  5. 5.

    Hopf algebrasPlanetmathPlanetmath, quantum GroupsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and quantum group algebras

  6. 6.

    Quantum groupoidsPlanetmathPlanetmath and weak Hopf C*-algebras

  7. 7.

    Groupoid C*-convolution algebras and *-convolution algebroids

  8. 8.

    Quantum spacetimes and quantum fundamental groupoids

  9. 9.

    Quantum double Algebras

  10. 10.

    Quantum gravity, supersymmetries, supergravity, superalgebras and graded ‘Lie’ algebras

  11. 11.

    Quantum categorical algebra and higher–dimensional, Ł-Mn- Toposes

  12. 12.

    Quantum R-categories, R-supercategories and spontaneous symmetry breaking

  13. 13.

    Non-Abelian Quantum algebraic topology (NA-QAT): closely related to NAAT and HDA.

1.16 Quantum Geometry

  1. 1.

    Quantum Geometry overview (http://planetmath.org/QuantumGeometry2)

  2. 2.

    Quantum non-commutative geometry

1.17 Non-Abelian Algebraic Topology (NAAT)

  1. 1.

    Non-Abelian categories

  2. 2.

    Non-commutative groupoids (including non-Abelian groupsMathworldPlanetmath)

  3. 3.

    Generalized van Kampen theorems (http://planetmath.org/GeneralizedVanKampenTheoremsHigherDimensional)

  4. 4.

    Noncommutative Geometry (NCG) (http://planetmath.org/NoncommutativeGeometry)

  5. 5.

    Non-commutative ‘spaces’ of functions

  6. 6.

    http://planetphysics.org/encyclopedia/NonAbelianAlgebraicTopology5.htmlnon-Abelian Algebraic Topology

1.18 12

  1. 1.

    new1

  2. 2.

    new2

  3. 3.

    new3

  4. 4.

    new4

1.19 13

  1. 1.

    new1

  2. 2.

    new2

  3. 3.

    new3

  4. 4.

    new4

1.20 14

1.21 References

http://planetmath.org/?op=getobj&from=objects&id=10746Bibliography on Category theory, AT and QAT

1.21.1 Textbooks and Expositions:

  1. 1.

    A http://planetmath.org/?op=getobj&from=books&id=172Textbook1

  2. 2.

    A http://planetmath.org/?op=getobj&from=books&id=156Textbook2

  3. 3.

    A http://planetmath.org/?op=getobj&from=books&id=159Textbook3

  4. 4.

    A http://planetmath.org/?op=getobj&from=books&id=160Textbook4

  5. 5.

    A http://planetmath.org/?op=getobj&from=books&id=153Textbook5

  6. 6.

    A http://planetmath.org/?op=getobj&from=lec&id=68Textbook6

  7. 7.

    A http://planetmath.org/?op=getobj&from=books&id=158Textbook7

  8. 8.

    A http://planetmath.org/?op=getobj&from=lec&id=75Textbook8

  9. 9.

    A http://planetmath.org/?op=getobj&from=lec&id=73Textbook9

  10. 10.

    A http://planetmath.org/?op=getobj&from=books&id=174Textbook10

  11. 11.

    A http://planetmath.org/?op=getobj&from=books&id=169Textbook11

  12. 12.

    A http://planetmath.org/?op=getobj&from=books&id=178Textbook12

  13. 13.

    A http://www.math.cornell.edu/ hatcher/VBKT/VB.pdfTextbook13

  14. 14.

    new1

Title topics in algebraic topology
Canonical name TopicsInAlgebraicTopology
Date of creation 2013-03-22 18:23:42
Last modified on 2013-03-22 18:23:42
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 82
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
Synonym category theory
Synonym algebraic geometry
Synonym topology and groupoids
Related topic Category
Related topic Functor
Related topic GrothendieckGroup
Related topic NoncommutativeGeometry
Related topic TopologicalGroup2
Related topic Naturaltransformation
Related topic AlgebraicTopology
Related topic AlgebraicGeometry
Related topic CategoricalOntologyABibliographyOfCategoryTheory
Related topic IndexOfCategories
Related topic OverviewOfTheContentOfPlanetMath
Related topic CategoryOf