Cohomology ring, the Glossary

Table of Contents

1. 18 relations: Algebraic topology, Cohomology, Commutative ring, Complex dimension, Complex projective space, Continuous function, Cup product, De Rham cohomology, Direct sum of modules, Functor, Graded ring, Graded-commutative ring, Künneth theorem, Mathematics, Quantum cohomology, Ring (mathematics), Ring homomorphism, Topological space.

Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.

See Cohomology ring and Algebraic topology

Cohomology

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex.

See Cohomology ring and Cohomology

Commutative ring

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative.

See Cohomology ring and Commutative ring

Complex dimension

In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety.

See Cohomology ring and Complex dimension

Complex projective space

In mathematics, complex projective space is the projective space with respect to the field of complex numbers.

See Cohomology ring and Complex projective space

Continuous function

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function.

See Cohomology ring and Continuous function

Cup product

In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space X into a graded ring, H∗(X), called the cohomology ring. Cohomology ring and cup product are homology theory.

See Cohomology ring and Cup product

De Rham cohomology

In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes.

See Cohomology ring and De Rham cohomology

Direct sum of modules

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module.

See Cohomology ring and Direct sum of modules

Functor

In mathematics, specifically category theory, a functor is a mapping between categories.

See Cohomology ring and Functor

Graded ring

In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that.

See Cohomology ring and Graded ring

Graded-commutative ring

In algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements x, y satisfy where |x | and |y | denote the degrees of x and y. A commutative (non-graded) ring, with trivial grading, is a basic example.

See Cohomology ring and Graded-commutative ring

Künneth theorem

In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem, also called a Künneth formula, is a statement relating the homology of two objects to the homology of their product.

See Cohomology ring and Künneth theorem

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Cohomology ring and Mathematics

Quantum cohomology

In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold.

See Cohomology ring and Quantum cohomology

Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist.

See Cohomology ring and Ring (mathematics)

Ring homomorphism

In mathematics, a ring homomorphism is a structure-preserving function between two rings.

See Cohomology ring and Ring homomorphism

Topological space

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.

See Cohomology ring and Topological space

References

[1] https://en.wikipedia.org/wiki/Cohomology_ring

Also known as Cup length, Cup-length.