The cohomology ring is the graded ring whose multiplication is the cup product. It records intersection information that the graded cohomology groups alone do not detect.
Choose degree-one classes dual to a symplectic basis of one-cycles and let generate . The only nonzero products of positive-degree basis elements are
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In algebraic topology, the cohomology ring is an important algebraic structure associated with a topological space. It is formed from the cohomology groups of the space, which provide algebraic invariants that help in understanding the topological properties of spaces.