algebraic topology

1 Algebraic topology

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, 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.

1.2 Outline

1. Homotopy theory and fundamental groups
2. Topology and groupoids; van Kampen theorem
3. Homology and cohomology theories
4. Duality
5. category theory applications in algebraic topology
6. indexes of category, functors and natural transformations
7. Grothendieck’s Descent theory
8. ‘Anabelian geometry’
9. Categorical Galois theory
10. higher dimensional algebra (HDA)
11. non-Abelian Quantum Algebraic Topology (NAQAT)
12. quantum geometry
13. Non-Abelian algebraic topology (NAAT)

1.3 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

1.4 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

1.5 Homology theory

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

1.6 Cohomology theory

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

1.7 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

1.8 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. 𝒯 op category
9. topos and topoi axioms
10. generalized toposes
11. Categorical logic and algebraic topology
12. meta-theorems
13. Duality between spaces and algebras

1.9 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

1.10 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

1.11 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

1.12 Grothendieck proposals

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

1.13 Descent theory

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

1.14 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

1.14.1 Axioms of cohomology theory

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

1.14.2 Axioms of homology theory

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

1.15 Non-Abelian Algebraic Topology (NAAT)

1. An overview of Nonabelian Algebraic Topology
2. non-Abelian categories
3. non-commutative groupoids (including non-Abelian groups)
4. Generalized van Kampen theorems
5. Noncommutative Geometry (NCG)
6. Non-commutative ‘spaces’ of functions
7. Non-Abelian Algebraic Topology textbook

1.15.1 References for NAAT

1. M. Alp and C. D. Wensley, XMod, Crossed modules and Cat1–groups: a GAP4 package,(2004) (http://www.maths.bangor.ac.uk/chda/)
2. R. Brown, Elements of Modern Topology, McGraw Hill, Maidenhead, 1968. second edition as Topology: a geometric account of general topology, homotopy types, and the fundamental groupoid, Ellis Horwood, Chichester (1988) 460 pp.
3. R. Brown.‘crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local–to–global problems’, Proceedings of the 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.pdf (30 pp.)
4. R. Brown and P. J. Higgins, On the connection between the second relative homotopy groups of some related spaces, Proc.London Math. Soc., (3) 36 (1978) 193–212.
5. 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)
6. R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahiers Top. G’/eom.Diff., 17 (1976) 343–362.
7. R. Brown and C. D.Wensley, ‘computation and homotopical applications of induced crossed modules’, J. Symbolic Computation, 35 (2003) 59–72.
8. The GAP Group, 2004, GAP –Groups, algorithms, and programming, version 4.4 , Technical report, (http://www.gap-system.org)
9. A. Grothendieck, ‘Pursuing stacks’, 600p, 1983, distributed from Bangor. Now being edited by G. Maltsiniotis for the SMF.
10. P. J. Higgins, 1971, Categories and Groupoids, Van Nostrand, New York. Reprint Series, Theory and Appl. Categories (to appear).
11. V. Sharko, 1993, Functions on manifolds: algebraic and topological aspects, number 131 in Translations of Mathematical Monographs, American Mathematical Society.

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

1.16 13

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

1.17 14

1.18 References

Bibliography on Category theory, AT and QAT

1.18.1 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

1.19 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).
3. R. Brown and J.-L. Loday: Homotopical excision, and Hurewicz theorems, for n-cubes of spaces, Proc. London Math. Soc., 54:(3), 176–192, (1987).
4. R. Brown and J.-L. Loday: Van Kampen Theorems for diagrams of spaces, Topology, 26: 311-337 (1987).
5. R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales-Bangor, Maths Preprint, 1986.
6. R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. Géom. Diff. 17 (1976), 343–362.
7. Madalina (Ruxi) Buneci.: groupoid representations., Ed. Mirton: Timisoara (2003).
8. Allain Connes: noncommutative geometry, Academic Press 1994.

1.20 Non–Abelian Algebraic Topology and Higher Dimensional Algebra

1. Ronald Brown: non–Abelian algebraic topology, vols. I and II. 2010. (in press: Springer): Nonabelian Algebraic Topology:filtered spaces, crossed complexes, cubical higher homotopy groupoids
2. Higher Dimensional Algebra: An Introduction
3. Higher Dimensional Algebra and Algebraic Topology., 282 pages, Feb. 10, 2010