PlanetPhysics/Algebraic Topology

Algebraic topology

Introduction

Algebraic topology (AT) utilizes [[../CoIntersections/|algebraic]] approaches to solve [[../CoIntersections/|topological]] problems, such as the [[../TrivialGroupoid/|classification]] of surfaces, proving [[../TrivialGroupoid/|duality]] [[../Formula/|theorems]] for [[../NoncommutativeGeometry4/|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 [[../ThinEquivalence/|homotopy]], homology and [[../CohomologyTheoryOnCWComplexes/|cohomology groups]]. There are close connections between algebraic topology, Algebraic Geometry (AG), [[../NAQAT2/|non-commutative geometry]] and, of course, its most recent development-- non-Abelian Algebraic Topology (NAAT). On the other hand, there are also close ties between algebraic geometry and number theory.

Outline

1. Homotopy theory and [[../HomotopyCategory/|fundamental groups]] #Topology and [[../QuantumOperatorAlgebra5/|groupoids]]; van Kampen theorem
2. Homology and [[../NoncommutativeGeometry4/|cohomology theories]]
3. Duality
4. [[../TrivialGroupoid/|category theory applications]] in algebraic topology
5. [[../IndexOfCategories/|indexes of category]], [[../TrivialGroupoid/|functors]] and [[../VariableCategory2/|natural transformations]]
6. Grothendieck's Descent theory
7. `[[../IsomorphismClass/|Anabelian Geometry]]' #Categorical Galois theory
8. [[../2Groupoid2/|higher dimensional algebra]] ([[../2Groupoid2/|HDA]])
9. [[../NonAbelianQuantumAlgebraicTopology3/|Non-Abelian Quantum Algebraic Topology]] (NAQAT)
10. Quantum Geometry
11. Non-Abelian algebraic topology (NAAT)

Homotopy theory and fundamental groups

1. Homotopy
2. Fundamental group of a space
3. Fundamental theorems
4. [[../VanKampenTheorems/|Van Kampen theorem]] #Whitehead [[../TrivialGroupoid/|groups]], torsion and towers
5. Postnikov towers

Topology and Groupoids

1. Topology definition, axioms and basic [[../PreciseIdea/|concepts]] #Fundamental groupoid
2. [[../GroupoidHomomorphism2/|topological groupoid]] #van Kampen theorem for groupoids
3. Groupoid [[../Pushout/|pushout]] theorem
4. [[../WeakHomotopy/|double groupoids]] and crossed modules
5. new4

Homology theory

1. [[../ExtendedHurewiczFundamentalTheorem/|homology group]] #Homology sequence
2. Homology complex
3. 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. [[../TrivialGroupoid/|categorical duality]] #[[../DualityAndTriality/|tangled duality]] #DA5
4. DA6
5. DA7

Category theory applications

1. [[../AbelianCategory2/|abelian categories]]
2. Topological [[../Cod/|category]] #Fundamental groupoid functor
3. Categorical Galois theory
4. Non-Abelian algebraic topology
5. Group category
6. [[../GroupoidCategory3/|groupoid category]] #$𝒯op$ category
7. [[../GrothendieckTopos/|topos]] and topoi axioms
8. [[../ManyValuedLogicSubobjectClassifiers/|generalized toposes]] #Categorical logic and algebraic topology
9. [[../MetaTheorems/|meta-theorems]] #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. [[../CategoryOfRiemannianManifolds/|category of Riemannian manifolds]] #Category of CW-complexes
6. Category of Hausdorff spaces
7. [[../CategoryOfBorelSpaces/|category of Borel spaces]] #Category of CR-complexes
8. Category of [[../Cod/|graphs]] #Category of [[../SimplicialCWComplex/|spin networks]] #Category of groups
9. Galois category
10. Category of fundamental groups
11. Category of [[../PolishGroup/|Polish groups]]
12. Groupoid category
13. [[../GroupoidCategory4/|category of groupoids]] (or groupoid category)
14. [[../CategoryOfBorelGroupoids/|category of Borel groupoids]] #Category of fundamental groupoids
15. Category of functors (or [[../TrivialGroupoid/|functor category]])
16. Double groupoid category
17. [[../HorizontalIdentities/|double category]] #[[../CategoryOfHilbertSpaces/|category of Hilbert spaces]] #[[../CategoryOfQuantumAutomata/|category of quantum automata]] #[[../RCategory/|R-category]] #Category of [[../Algebroids/|algebroids]] #Category of [[../GeneralizedSuperalgebras/|double algebroids]]
18. Category of [[../ContinuousGroupoidHomomorphism/|dynamical systems]]

Index of functors

The following is a contributed listing of functors:

1. Covariant functors
2. Contravariant functors
3. [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|adjoint functors]]
4. [[../PreadditiveFunctor/|preadditive functors]]
5. Additive functor
6. [[../CategoryOfLogicAlgebras/|representable functors]]
7. Fundamental groupoid functor
8. Forgetful functors
9. Grothendieck group functor
10. Exact functor
11. Multi-functor
12. [[../RightAdjointFunctor/|section functors]]
13. NT2
14. NT3

Index of natural transformations

The following is a contributed listing of natural transformations:

1. [[../IsomorphismClass/|natural equivalence]] #Natural transformations in a [[../2Category/|2-category]] #NT3
2. NT1
3. NT2
4. NT3

Grothendieck proposals

1. Esquisse d'un Programme

\item Pursuing Stacks

1. S2
2. S3
3. 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. [[../SuperCategory6/|super-category]] #weak [[../InfinityGroupoid/|n-categories]] #Bi-dimensional Geometry
9. [[../NoncommutativeGeometry/|Noncommutative geometry]]
10. Higher-Homotopy theories
11. Higher-Homotopy Generalized van Kampen Theorem (HGvKT)
12. H1
13. H2
14. H3
15. 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

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

Non-Abelian Algebraic Topology (NAAT)

1. An overview of Nonabelian Algebraic Topology
2. [[../AbelianCategory3/|non-Abelian categories]]
3. [[../AbelianCategory3/|non-commutative]] groupoids (including non-Abelian groups)
4. Generalized van Kampen theorems
5. [[../NoncommutativeGeometry/|Noncommutative Geometry (NCG)]]
6. Non-commutative `spaces' of [[../Bijective/|functions]] #[[../NonAbelianAlgebraicTopology5/|Non-Abelian Algebraic Topology textbook]]

References for NAAT

1. [1] M. Alp and C. D. Wensley, XMod, Crossed modules and Cat1--groups: a GAP4 package,(2004) (http://www.maths.bangor.ac.uk/chda/)

1. [2] R. Brown, Elements of Modern Topology, McGraw Hill, Maidenhead, 1968. second edition as Topology: a geometric account of general topology, homotopy [[../Bijective/|types]], and the fundamental groupoid, Ellis Horwood, Chichester (1988) 460 pp.

1. [3] R. Brown, `Higher dimensional group theory'

1. [4] R. Brown[[../SingularComplexOfASpace/|.`crossed complexes]] and homotopy groupoids as non commutative tools for higher dimensional local--to--global problems', Proceedings of the [[../CosmologicalConstant/|fields]] Institute Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories, September 23--28, 2002, Contemp. Math. (2004). (to appear), UWB Math Preprint

02.26[[../LebesgueMeasure/|.pdf]] (30 pp.)

1. [5] R. Brown and P. J. Higgins, On the connection between the second relative [[../ExtendedHurewiczFundamentalTheorem/|homotopy groups]] of some related spaces, Proc.London Math. Soc., (3) 36 (1978) 193--212.

1. [6] R. Brown and R. Sivera, `Nonabelian algebraic topology', (in preparation) Part I is downloadable from

(http://www.bangor.ac.uk/~mas010/nonab-a-t.html)

1. [7] R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahiers Top. G'/eom.Diff., 17 (1976) 343--362.

1. [8] R. Brown and C. D.Wensley, `[[../LQG2/|computation]] and homotopical applications of induced crossed modules', J. Symbolic Computation, 35 (2003) 59--72.

1. [9] The GAP Group, 2004, GAP --Groups, [[../RecursiveFunction/|algorithms]], and [[../ComputerProgram/|programming]], version 4.4 , Technical report, (http://www.gap-system.org)

1. [10] A. Grothendieck, `Pursuing [[../GrothendiecksEsquisseDunProgramme/|stacks',]] 600p, 1983, distributed from Bangor. Now being edited by G. Maltsiniotis for the SMF.

1. [11] P. J. Higgins, 1971, Categories and Groupoids,

Van Nostrand, New York. Reprint Series, Theory and Appl. Categories (to appear).

1. [12] V. Sharko, 1993, Functions on manifolds: algebraic and topological aspects, number 131 in Translations of Mathematical Monographs, American Mathematical Society.

1. new1

1. new2
2. new3
3. new4

13

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

References

Bibliography on Category theory, AT and QAT

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

Algebraic Topology and Groupoids

1. Ronald Brown: Topology and Groupoids, BookSurge LLC (2006).
2. Ronald Brown R, P.J. Higgins, and R. Sivera.: "Non-Abelian algebraic topology" .

http://www. bangor.ac.uk/mas010/nonab-a-t.html; http://www.bangor.ac.uk/mas010/nonab-t/partI010604.pdf , Springer: in press (2010).

1. R. Brown and J.-L. Loday: Homotopical excision, and [[../ModuleAlgebraic/|Hurewicz theorems]], for n-cubes of spaces, Proc. London Math. Soc., 54:(3), 176--192, (1987).
2. R. Brown and J.-L. Loday: Van Kampen Theorems for [[../TrivialGroupoid/|diagrams]] of spaces, Topology, 26: 311-337 (1987).
3. R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales-Bangor, Maths Preprint, 1986.
4. R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. G\'eom. Diff. 17 (1976), 343--362.
5. Madalina (Ruxi) Buneci.: [[../GroupoidRepresentations/|groupoid representations]]., Ed. Mirton: Timisoara (2003).
6. Allain Connes: [[../NoncommutativeGeometry4/|noncommutative geometry]], Academic Press 1994.

Non--Abelian Algebraic Topology and Higher Dimensional Algebra

1. Ronald Brown: [[../ModuleAlgebraic/|non--Abelian algebraic topology]], vols. I and II. 2010. (in press: Springer): Nonabelian Algebraic Topology:filtered spaces, crossed complexes, cubical higher homotopy groupoids

1. Higher Dimensional Algebra: An Introduction

1. Higher Dimensional Algebra and Algebraic Topology., 282 pages, Feb. 10, 2010