cap product

Let $X$ 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

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

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 $(n-i)$ -singular chain such that

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

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

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

thus there is a corresponding operation in cohomology

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

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