Cup Product

The cup product is a product on cohomology classes. In the case of de Rham cohomology, a cohomology class can be represented by a closed form. The cup product of [alpha] and [beta] is represented by the closed form [alpha ^ beta], where  ^ is the wedge product of differential forms. It is the dual operation to intersection in homology.

In general, the cup product is a map

  v :H^p×H^q->H^(p+q)

which satisfies a v b=(-1)^(pq)b v a, where H^k is the kth cohomology group.

See also

Cohomology, Cup, de Rham Cohomology, Homology

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Cup Product." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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