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. The cup product was introduced in work of J. W. Alexander, Eduard Čech and Hassler Whitney from 1935–1938, and, in full generality, by Samuel Eilenberg in 1944.

Contents

• 1 Definition
• 2 Properties
• 3 Interpretation
• 4 Examples
• 5 Other definitions
  • 5.1 Cup product and differential forms
  • 5.2 Cup product and geometric intersections
• 6 Massey products
• 7 See also
• 8 References

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