cap product

Let X be a topological spaceMathworldPlanetmath, (C*⁒(X),βˆ‚) the singular chain complexMathworldPlanetmath, and (C*⁒(X;𝕂),Ξ΄) the singular cochain complexMathworldPlanetmathPlanetmath in any coefficient group 𝕂. We can define a bilinear pairing operationMathworldPlanetmath


in the following way: for each cochain b∈Ci⁒(X;𝕂) and each chain ΟƒβˆˆCn⁒(X) we define their cap product bβŒ’Οƒ as the unique (n-i)-singular chain such that


where ⌣:Cj(X;𝕂)Γ—Ch(X;𝕂)β†’Cj+h(X;𝕂) denotes the cup productMathworldPlanetmath. Combining the definition of cap product with the standard properties of cup product we obtain that


thus there is a corresponding operation in cohomology


that we also call cap product.

Title cap product
Canonical name CapProduct
Date of creation 2013-03-22 16:26:10
Last modified on 2013-03-22 16:26:10
Owner Mazzu (14365)
Last modified by Mazzu (14365)
Numerical id 9
Author Mazzu (14365)
Entry type Definition
Classification msc 55N45
Defines cap product