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Cup product, the Glossary

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.^[1]

35 relations: Algebraic topology, Cap product, Chain complex, Closed and exact differential forms, Cocycle, Cohomology, Cohomology operation, Cohomology ring, De Rham cohomology, Diagonal functor, Differentiable manifold, Differential form, Eduard Čech, Embedding, Exterior algebra, Functor, Glen Bredon, Graded ring, Hassler Whitney, Homology (mathematics), Homomorphism, James Waddell Alexander II, Künneth theorem, Linking number, Mapping class group, Massey product, Mathematics, Poincaré duality, Ring homomorphism, Samuel Eilenberg, Simplex, Singular homology, Supercommutative algebra, Topological space, Transversality (mathematics).
Binary operations

Algebraic topology

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

See Cup product and Algebraic topology

Cap product

In algebraic topology the cap product is a method of adjoining a chain of degree p with a cochain of degree q, such that q ≤ p, to form a composite chain of degree p − q. It was introduced by Eduard Čech in 1936, and independently by Hassler Whitney in 1938. Cup product and cap product are algebraic topology, Binary operations and homology theory.

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Chain complex

In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next.

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Closed and exact differential forms

In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero, and an exact form is a differential form, α, that is the exterior derivative of another differential form β.

See Cup product and Closed and exact differential forms

Cocycle

In mathematics a cocycle is a closed cochain. Cup product and cocycle are algebraic topology.

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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 Cup product and Cohomology

Cohomology operation

In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from F to itself. Cup product and cohomology operation are algebraic topology.

See Cup product and Cohomology operation

Cohomology ring

In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Cup product and cohomology ring are homology theory.

See Cup product and Cohomology ring

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 Cup product and De Rham cohomology

Diagonal functor

In category theory, a branch of mathematics, the diagonal functor \mathcal \rightarrow \mathcal \times \mathcal is given by \Delta(a).

See Cup product and Diagonal functor

Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus.

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Differential form

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds.

See Cup product and Differential form

Eduard Čech

Eduard Čech (29 June 1893 – 15 March 1960) was a Czech mathematician.

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Embedding

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

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Exterior algebra

In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v.

See Cup product and Exterior algebra

Functor

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

See Cup product and Functor

Glen Bredon

Glen Eugene Bredon (August 24, 1932 in Fresno, California – May 8, 2000, in North Fork, California) was an American mathematician who worked in the area of topology.

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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 Cup product and Graded ring

Hassler Whitney

Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician.

See Cup product and Hassler Whitney

Homology (mathematics)

In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Cup product and homology (mathematics) are homology theory.

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Homomorphism

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).

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James Waddell Alexander II

James Waddell Alexander II (September 19, 1888 September 23, 1971) was a mathematician and topologist of the pre-World War II era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others.

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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 Cup product and Künneth theorem

Linking number

In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space.

See Cup product and Linking number

Mapping class group

In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space.

See Cup product and Mapping class group

Massey product

In algebraic topology, the Massey product is a cohomology operation of higher order introduced in, which generalizes the cup product. Cup product and Massey product are algebraic topology.

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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 Cup product and Mathematics

Poincaré duality

In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. Cup product and Poincaré duality are homology theory.

See Cup product and Poincaré duality

Ring homomorphism

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

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Samuel Eilenberg

Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra.

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Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

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Singular homology

In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n(X). Cup product and singular homology are homology theory.

See Cup product and Singular homology

Supercommutative algebra

In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements x, y we have where |x| denotes the grade of the element and is 0 or 1 (in Z) according to whether the grade is even or odd, respectively.

See Cup product and Supercommutative algebra

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 Cup product and Topological space

Transversality (mathematics)

In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position.

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References

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