Cup product
From Wikipedia, the free encyclopedia
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. In de Rham cohomology, the cup product is also known as the wedge product and in this sense is a special case of Grassmann's exterior product.
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[edit] Definition
In cohomology theory, the cup product is a construction giving a product on the graded cohomology ring H∗(X) of an object X. In the singular theory, this takes the form of a product of cocycle classes: if cp and dq are cocycle classes in Hp(X) and Hq(X), respectively, then the cup product is defined by
where σ is a (p + q) -simplex and (d0, …, dp) and (dp, …, dp + q) are the natural injections into σ, sometimes called the pth front face and the qth back face, respectively.
[edit] Equations
The cup product satisfies the identity
so that the corresponding multiplication is graded-commutative.
The coboundary of the cup product 
of cocycles αp and βq is given by
[edit] Examples
As singular spaces, the 2-sphere S2 with two disjoint 1-dimensional loops attached by their endpoints to the surface and the torus T have identical cohomology groups in all dimensions, but the multiplication of the cup product distinguishes the associated cohomology rings. In the former case the multiplication of the cochains associated to the loops is degenerate, whereas in the latter case multiplication in the first cohomology group can be used to decompose the torus as a 2-cell diagram, thus having product equal to Z (more generally M where this is the base module).
[edit] Cup product and geometric intersections
When two submanifolds of a smooth manifold intersect transversely, their intersection is again a submanifold. By taking the fundamental homology class of these manifolds, this yields a bilinear product on homology. This product is dual to the cup product, i.e. the homology class of the intersection of two submanifolds is the Poincaré dual of the cup product of their Poincaré duals.
