# 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 CapProduct 2013-03-22 16:26:10 2013-03-22 16:26:10 Mazzu (14365) Mazzu (14365) 9 Mazzu (14365) Definition msc 55N45 cap product