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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. On an oriented closed manifold, the cup product corresponds under Poincaré duality to intersection of homology classes.

For singular cohomology with coefficients in a commutative ring, 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 (Hatcher 2002, p. 210).

See also

Cohomology, Cup, Cup Length, de Rham Cohomology, Homology, Homology Intersection, Poincaré Duality, Singular Cohomology

Portions of this entry contributed by Todd Rowland

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References

Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, Theorem 3.11, 2002.

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Cup Product

Cite this as:

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

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