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.