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 cup product is a binary operation on the cohomology groups of a topological space. It provides a way to combine cohomology classes to produce new cohomology classes, thereby enriching our understanding of the topology of the space.