Cohomology - Cohomology theories

In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. That is, cohomology is defined as the abstract study of cochains, cocycles and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign 'qua ...

Including:

• Cohomology - History
• Cohomology - Cohomology theories
• Cohomology - Eilenberg-Steenrod theories
• Cohomology - Extraordinary cohomology theories
• Cohomology - Other cohomology theories

Read more here: » Cohomology: Encyclopedia - Cohomology

Cohomology - Eilenberg-Steenrod theories. A cohomology theory is a family of contravariant functors from the category of pairs of topological spaces and continuous functions (or some subcategory thereof such as the category of CW complexes) to the category of Abelian groups and group homomorphisms that satisfies the Eilenberg-Steenrod axioms. Some cohomology theories in this sense are: simplicial cohomology singular cohomology de Rham cohomology Čech cohomology sheaf cohomology Coho ...

See also:

Cohomology, Cohomology - History, Cohomology - Cohomology theories, Cohomology - Eilenberg-Steenrod theories, Cohomology - Extraordinary cohomology theories, Cohomology - Other cohomology theories

Read more here: » Cohomology: Encyclopedia II - Cohomology - Cohomology theories

Although cohomology is fundamental to modern algebraic topology, its importance was not seen for some 40 years after the development of homology. The concept of dual cell structure, which Henri Poincaré used in his proof of his Poincaré duality theorem, contained the germ of the idea of cohomology, but this was not seen until later. There were various precursors to cohomology. In the mid-1920s, J.W. Alexander and Lefschetz founded the intersection theory of cycles on manifolds. On an n-dimensional manifold M, a ...

See also:

Cohomology, Cohomology - History, Cohomology - Cohomology theories, Cohomology - Eilenberg-Steenrod theories, Cohomology - Extraordinary cohomology theories, Cohomology - Other cohomology theories

Read more here: » Cohomology: Encyclopedia II - Cohomology - History