PlanetMath: cap product

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cap product

(Definition)

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

$\displaystyle \frown : C^i(X;\mathbb{K})\times C_n(X)\rightarrow C_{n-i}(X),\ \ \ (n\geq i)$

in the following way: for each cochain $b\in C^i(X;\mathbb{K})$ and each chain $\sigma\in C_n(X)$ we define their cap product $b\frown \sigma$ as the unique

-singular chain such that

$\displaystyle a(b\frown \sigma)=(a\smile b)(\sigma),$

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

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

thus there is a corresponding operation in cohomology

$\displaystyle \frown : H^i(X;\mathbb{K})\otimes H_n(X)\rightarrow H_{n-i}(X),\ \ \ (n\geq i)$

that we also call cap product.

"cap product" is owned by Mazzu.

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cap product

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Cross-references: cohomology, properties, cup product, chain, operation, bilinear pairing, group, coefficient, cochain complex, chain complex, singular, topological space
There are 2 references to this entry.

This is version 6 of cap product, born on 2006-11-28, modified 2006-11-28.
Object id is 8589, canonical name is CapProduct.
Accessed 1296 times total.

Classification:

AMS MSC:

55N45 (Algebraic topology :: Homology and cohomology theories :: Products and intersections)

Pending Errata and Addenda

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