55Nxx Homology and cohomology theories
This subtopic introduces the core ideas in homology and cohomology theories, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.
Specific topics
55N07 Steenrod-Sitnikov homologies
Overview
Steenrod-Sitnikov homologies. This topic develops homology and cohomology theories, axioms, computations, and relationships among generalized theories.
Related Wikipedia Page
Wikipedia: Cohomology
Useful Links
Key Ideas
- axiomatic (co)homology frameworks
- derived invariants and functoriality
- generalized and extraordinary theories
Typical Uses
Used to compute and compare deep invariants of spaces, bundles, and maps.
Applications
- Manifold topology
- Homotopy-theoretic computations
- Mathematical physics and gauge theory
References
Recommended Textbooks
55N10 Singular homology and cohomology theory
Overview
Singular homology and cohomology theory. This topic develops homology and cohomology theories, axioms, computations, and relationships among generalized theories.
Related Wikipedia Page
Wikipedia: Cohomology
Useful Links
Key Ideas
- axiomatic (co)homology frameworks
- derived invariants and functoriality
- generalized and extraordinary theories
Typical Uses
Used to compute and compare deep invariants of spaces, bundles, and maps.
Applications
- Manifold topology
- Homotopy-theoretic computations
- Mathematical physics and gauge theory
References
Recommended Textbooks
55N15 Topological $K$-theory
Overview
Topological $K$-theory. This topic develops homology and cohomology theories, axioms, computations, and relationships among generalized theories.
Related Wikipedia Page
Wikipedia: Cohomology
Useful Links
Key Ideas
- axiomatic (co)homology frameworks
- derived invariants and functoriality
- generalized and extraordinary theories
Typical Uses
Used to compute and compare deep invariants of spaces, bundles, and maps.
Applications
- Manifold topology
- Homotopy-theoretic computations
- Mathematical physics and gauge theory
References
Recommended Textbooks
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
Overview
Generalized (extraordinary) homology and cohomology theories in algebraic topology. This topic develops homology and cohomology theories, axioms, computations, and relationships among generalized theories.
Related Wikipedia Page
Wikipedia: Cohomology
Useful Links
Key Ideas
- axiomatic (co)homology frameworks
- derived invariants and functoriality
- generalized and extraordinary theories
Typical Uses
Used to compute and compare deep invariants of spaces, bundles, and maps.
Applications
- Manifold topology
- Homotopy-theoretic computations
- Mathematical physics and gauge theory
References
Recommended Textbooks
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
Overview
Bordism and cobordism theories and formal group laws in algebraic topology. This topic develops homology and cohomology theories, axioms, computations, and relationships among generalized theories.
Related Wikipedia Page
Wikipedia: Cohomology
Useful Links
Key Ideas
- axiomatic (co)homology frameworks
- derived invariants and functoriality
- generalized and extraordinary theories
Typical Uses
Used to compute and compare deep invariants of spaces, bundles, and maps.
Applications
- Manifold topology
- Homotopy-theoretic computations
- Mathematical physics and gauge theory
References
Recommended Textbooks
55N25 Homology with local coefficients, equivariant cohomology
Overview
Homology with local coefficients, equivariant cohomology. This topic develops homology and cohomology theories, axioms, computations, and relationships among generalized theories.
Related Wikipedia Page
Wikipedia: Cohomology
Useful Links
Key Ideas
- axiomatic (co)homology frameworks
- derived invariants and functoriality
- generalized and extraordinary theories
Typical Uses
Used to compute and compare deep invariants of spaces, bundles, and maps.
Applications
- Manifold topology
- Homotopy-theoretic computations
- Mathematical physics and gauge theory
References
Recommended Textbooks
55N30 Sheaf cohomology in algebraic topology
Overview
Sheaf cohomology in algebraic topology. This topic develops homology and cohomology theories, axioms, computations, and relationships among generalized theories.
Related Wikipedia Page
Wikipedia: Cohomology
Useful Links
Key Ideas
- axiomatic (co)homology frameworks
- derived invariants and functoriality
- generalized and extraordinary theories
Typical Uses
Used to compute and compare deep invariants of spaces, bundles, and maps.
Applications
- Manifold topology
- Homotopy-theoretic computations
- Mathematical physics and gauge theory
References
Recommended Textbooks
55N31 Persistent homology and applications, topological data analysis
Overview
Persistent homology and applications, topological data analysis. This topic develops homology and cohomology theories, axioms, computations, and relationships among generalized theories.
Related Wikipedia Page
Wikipedia: Cohomology
Useful Links
Key Ideas
- axiomatic (co)homology frameworks
- derived invariants and functoriality
- generalized and extraordinary theories
Typical Uses
Used to compute and compare deep invariants of spaces, bundles, and maps.
Applications
- Manifold topology
- Homotopy-theoretic computations
- Mathematical physics and gauge theory
References
Recommended Textbooks
55N32 Intersection homology and cohomology in algebraic topology
Overview
Intersection homology and cohomology in algebraic topology. This topic develops homology and cohomology theories, axioms, computations, and relationships among generalized theories.
Related Wikipedia Page
Wikipedia: Cohomology
Useful Links
Key Ideas
- axiomatic (co)homology frameworks
- derived invariants and functoriality
- generalized and extraordinary theories
Typical Uses
Used to compute and compare deep invariants of spaces, bundles, and maps.
Applications
- Manifold topology
- Homotopy-theoretic computations
- Mathematical physics and gauge theory
References
Recommended Textbooks
55N33 Intersection cohomology and motivic cohomology
Overview
Intersection cohomology and motivic cohomology. This topic develops homology and cohomology theories, axioms, computations, and relationships among generalized theories.
Related Wikipedia Page
Wikipedia: Cohomology
Useful Links
Key Ideas
- axiomatic (co)homology frameworks
- derived invariants and functoriality
- generalized and extraordinary theories
Typical Uses
Used to compute and compare deep invariants of spaces, bundles, and maps.
Applications
- Manifold topology
- Homotopy-theoretic computations
- Mathematical physics and gauge theory
References
Recommended Textbooks
55N34 Elliptic cohomology
Overview
Elliptic cohomology. This topic develops homology and cohomology theories, axioms, computations, and relationships among generalized theories.
Related Wikipedia Page
Wikipedia: Cohomology
Useful Links
Key Ideas
- axiomatic (co)homology frameworks
- derived invariants and functoriality
- generalized and extraordinary theories
Typical Uses
Used to compute and compare deep invariants of spaces, bundles, and maps.
Applications
- Manifold topology
- Homotopy-theoretic computations
- Mathematical physics and gauge theory
References
Recommended Textbooks
55N35 Other homology theories in algebraic topology
Overview
Other homology theories in algebraic topology. This topic develops homology and cohomology theories, axioms, computations, and relationships among generalized theories.
Related Wikipedia Page
Wikipedia: Cohomology
Useful Links
Key Ideas
- axiomatic (co)homology frameworks
- derived invariants and functoriality
- generalized and extraordinary theories
Typical Uses
Used to compute and compare deep invariants of spaces, bundles, and maps.
Applications
- Manifold topology
- Homotopy-theoretic computations
- Mathematical physics and gauge theory
References
Recommended Textbooks
55N40 Axioms for homology theory and uniqueness theorems in algebraic topology
Overview
Axioms for homology theory and uniqueness theorems in algebraic topology. This topic develops homology and cohomology theories, axioms, computations, and relationships among generalized theories.
Related Wikipedia Page
Wikipedia: Cohomology
Useful Links
Key Ideas
- axiomatic (co)homology frameworks
- derived invariants and functoriality
- generalized and extraordinary theories
Typical Uses
Used to compute and compare deep invariants of spaces, bundles, and maps.
Applications
- Manifold topology
- Homotopy-theoretic computations
- Mathematical physics and gauge theory
References
Recommended Textbooks
55N45 Products and intersections in homology and cohomology
Overview
Products and intersections in homology and cohomology. This topic develops homology and cohomology theories, axioms, computations, and relationships among generalized theories.
Related Wikipedia Page
Wikipedia: Cohomology
Useful Links
Key Ideas
- axiomatic (co)homology frameworks
- derived invariants and functoriality
- generalized and extraordinary theories
Typical Uses
Used to compute and compare deep invariants of spaces, bundles, and maps.
Applications
- Manifold topology
- Homotopy-theoretic computations
- Mathematical physics and gauge theory
References
Recommended Textbooks