TOPICS
Search

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

Explore with Wolfram|Alpha

Cite this as:

Rowland, Todd. "Cup Product." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CupProduct.html

Subject classifications