Cup product
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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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Definition
In cohomology theory, the cup product is a construction giving a product on the graded cohomology ring <math>H^{\bull}(X)<math> of an object X. In the singular theory, this takes the form of a product of cocycle classes: if <math>c^p \, \!<math> and <math>d^q \, \!<math> are cocycle classes in <math>H^p(X) \, \!<math> and <math>H^q(X) \, \!<math>, respectively, then the cup product is defined by
- <math>(c^p \smile d^q)(\sigma) = c^p(\sigma)(d_0, ..., d_p) \circ d^q(\sigma)(d_p, ..., d_{p + q})<math>
where <math>\sigma \, \!<math> is a <math>(p + q) \, \!<math>-simplex and <math>(d_0, ..., d_p) \, \!<math> and <math>(d_p, ..., d_{p + q}) \, \!<math> are the natural injections into <math>\sigma \, \!<math>.
Equations
The cup product satisfies the identity
- <math>\alpha^p \smile \beta^q = (-1)^{pq}(\beta^q \smile \alpha^p)<math>
so that the corresponding multiplication is anticommutative.
The coboundary of the cup product <math>\alpha^p \smile \beta^q<math> of cocycles <math>\alpha^p<math> and <math>\beta^q<math> is given by
- <math>\delta(\alpha^p \smile \beta^q) = \delta{\alpha^p} \smile \beta^q + (-1)^p(\alpha^p \smile \delta{\beta^q})<math>
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).