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