cap product

Let $X$ be a topological space, $(C_{*}(X),\partial )$ the singular chain complex, and $(C^{*}(X;𝕂),\delta )$ the singular cochain complex in any coefficient group $𝕂$. We can define a bilinear pairing operation

$$⌢:C^i(X;𝕂)\times C_n(X)\to C_{n-i}(X),(n\ge i)$$

in the following way: for each cochain $b\in C^i(X;𝕂)$ and each chain $\sigma \in C_n(X)$ we define their cap product $b⌢\sigma$ as the unique $(n-i)$-singular chain such that

$$a(b⌢\sigma )=(a⌣b)(\sigma ),$$

where $⌣:C^j(X;𝕂)\times C^h(X;𝕂)\to C^{j+h}(X;𝕂)$ denotes the cup product. Combining the definition of cap product with the standard properties of cup product we obtain that

$$\partial (b⌢\xi )=(\partial b)⌢\xi +{(-1)}^{\dim (b)}b⌢\partial (\xi ),$$

thus there is a corresponding operation in cohomology

$$⌢:H^i(X;𝕂)\otimes H_n(X)\to H_{n-i}(X),(n\ge i)$$

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