topic entry on axioms and foundations of homology and cohomology theories

1 Axioms for Homology and Cohomology theories

1.
Axioms for homology theory and uniqueness theorems
2.
Cech types
3.
$K$ -theory
4.
Generalized cohomology
5.
Generalized (extraordinary) homology and cohomology theories
6.
Galois Cohomology and Categorical Galois theories
7.
(Co)homology of commutative rings and algebras (e.g., Hochschild, André–Quillen, cyclic, dihedral, etc.)
8.
Bordism and cobordism theories, formal group laws
9.
Homology with local coefficients, equivariant cohomology
10.
Sheaf cohomology
11.
Cohomology in Noncommutative algebraic geometry
12.
Classifying spaces for foliations; Gelfand-Fuks cohomology
13.
Intersection homology and cohomology
14.
Elliptic cohomology
15.
Equivariant homology and cohomology
16.
Homology and homotopy of topological groups and related structures
17.
Homotopy Quantum Field Theories and Axiomatic Quantum Field Theories
18.
Non-Abelian Homological Algebra
19.
Grothendieck’s ‘Anabelian Geometry’
20.
Other homology theories–Your new additions

References

1 Hatcher, A. 2001. http://www.math.cornell.edu/ hatcher/AT/AT.pdfAlgebraic Topology (textbook on line)., Cambridge University Press; Cambridge, UK., 405 pages.
2 Ronald Brown: Topology and Groupoids, BookSurge LLC (2006).
3 Ronald Brown R, P.J. Higgins, and R. Sivera.: “Non-Abelian algebraic topology”, (2008).
4 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).
5 R. Brown and J.-L. Loday: Van Kampen Theorems for diagrams of spaces, Topology, 26: 311-337 (1987).
6 R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. Géom. Diff., 17 (1976), 343-362.
7 Allain Connes: Noncommutative Geometry, Academic Press 1994.
8 May, J.P. 1999, A Concise Course in Algebraic Topology., The University of Chicago Press: Chicago

单词	TopicEntryOnAxiomsAndFoundationsOfHomologyAndCohomologyTheories
释义	topic entry on axioms and foundations of homology and cohomology theories 1 Axioms for Homology and Cohomology theories 1. Axioms for homology theory and uniqueness theorems 2. Cech types 3. $K$ -theory 4. Generalized cohomology 5. Generalized (extraordinary) homology and cohomology theories 6. Galois Cohomology and Categorical Galois theories 7. (Co)homology of commutative rings and algebras (e.g., Hochschild, André–Quillen, cyclic, dihedral, etc.) 8. Bordism and cobordism theories, formal group laws 9. Homology with local coefficients, equivariant cohomology 10. Sheaf cohomology 11. Cohomology in Noncommutative algebraic geometry 12. Classifying spaces for foliations; Gelfand-Fuks cohomology 13. Intersection homology and cohomology 14. Elliptic cohomology 15. Equivariant homology and cohomology 16. Homology and homotopy of topological groups and related structures 17. Homotopy Quantum Field Theories and Axiomatic Quantum Field Theories 18. Non-Abelian Homological Algebra 19. Grothendieck’s ‘Anabelian Geometry’ 20. Other homology theories–Your new additions References 1 Hatcher, A. 2001. http://www.math.cornell.edu/ hatcher/AT/AT.pdfAlgebraic Topology (textbook on line)., Cambridge University Press; Cambridge, UK., 405 pages. 2 Ronald Brown: Topology and Groupoids, BookSurge LLC (2006). 3 Ronald Brown R, P.J. Higgins, and R. Sivera.: “Non-Abelian algebraic topology”, (2008). 4 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). 5 R. Brown and J.-L. Loday: Van Kampen Theorems for diagrams of spaces, Topology, 26: 311-337 (1987). 6 R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. Géom. Diff., 17 (1976), 343-362. 7 Allain Connes: Noncommutative Geometry, Academic Press 1994. 8 May, J.P. 1999, A Concise Course in Algebraic Topology., The University of Chicago Press: Chicago
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