cup product

Let X be a topological spaceMathworldPlanetmath and R be a commutative ring. The diagonal map Δ:XX×X induces a chain map between singular cochain complexesMathworldPlanetmathPlanetmath:



Let h:C*(X;R)C*(X;R)C*(X×X;R)

denote the chain homotopy equivalence associated with the Kunneth .

Given αCp(X;R) and βCq(X;R) we define


As Δ* and h are chain maps, induces a well defined productPlanetmathPlanetmathPlanetmathPlanetmath on cohomology groupsPlanetmathPlanetmath, known as the cup productMathworldPlanetmath. Hence the direct sumMathworldPlanetmathPlanetmath of the cohomology groups of X has the structureMathworldPlanetmath of a ring. This is called the cohomology ring of X.

Title cup product
Canonical name CupProduct
Date of creation 2013-03-22 15:37:42
Last modified on 2013-03-22 15:37:42
Owner whm22 (2009)
Last modified by whm22 (2009)
Numerical id 7
Author whm22 (2009)
Entry type Definition
Classification msc 55N45