Cohomology ring - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain In Mathematics, specifically in Algebraic topology, the cup product is a method of adjoining two Cocycles of degree p and q to form Here 'cohomology' is usually understood as singular cohomology, but the ring structure is also present in other theories such as de Rham cohomology. In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable It is also functorial: for a continuous mapping of spaces one obtains a ring homomorphism on cohomology rings, which is contravariant. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication

Specifically, given a sequence of cohomology groups H^k(X;R) on X with coefficients in a commutative ring R (typically R is Z_n, Z, Q, R, or C) one can define the cup product, which takes the form

$H^k(X;R) \times H^\ell(X;R) \to H^{k+\ell}(X; R)$ . In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Mathematics, specifically in Algebraic topology, the cup product is a method of adjoining two Cocycles of degree p and q to form

The cup product gives a multiplication on the direct sum of the cohomology groups

$H^\bullet(X;R) = \bigoplus_k H^k(X; R)$ . The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction

This multiplication turns H^•(X;R) into a ring. If fact, it is naturally a Z-graded ring with the integer k serving as the degree. In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure The cup product respects this grading.

The cohomology ring is graded-commutative in the sense that the cup product commutes up to a sign determined by the grading. In Mathematics, a supercommutative algebra is a Superalgebra (i Specifically, for pure elements of degree k and ℓ we have

$(\alpha^k \smile \beta^\ell) = (-1)^{k\ell}(\beta^\ell \smile \alpha^k)$ .

A numerical invariant derived from the cohomology ring is the cup-length, which means the maximum number of graded elements of degree ≥ 1, which when multiplied give a non-zero result. For example a complex projective space has cup-length equal to its complex dimension. In Mathematics, complex projective space, P ( C n +1 P n ( C) or CP n In Mathematics, complex dimension usually refers to the dimension of a Complex manifold M, or complex Algebraic variety V

References

S. P. Novikov, Topology I, General Survey, Springer-Verlag, (1996) ISBN 7-03-016673-6

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