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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. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is also present in other theories such as de Rham cohomology. It is also functorial: for a continuous mapping of spaces one obtains a ring homomorphism on cohomology rings, which is contravariant.
Specifically, given a sequence of cohomology groups Hk(X;R) on X with coefficients in a ring R (typcially R is Zn, Z, Q, R, or C) one can define the cup product, which takes the form
- <math>H^k(X;R) \times H^\ell(X;R) \to H^{k+\ell}(X; R)<math>.
The cup product gives a multiplication the direct sum of the cohomology groups
- <math>H^\bullet(X;R) = \bigoplus_k H^k(X; R)<math>.
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. The cup product product respects this grading.
The cohomology ring is a graded-commutative ring in the sense that the cup product commutes up to a sign determined by the grading. That is, for pure elements of degree k and ℓ we have
- <math>(\alpha^k \smile \beta^\ell) = (-1)^{k\ell}(\beta^k \smile \alpha^\ell)<math>.
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