Cap product

Definition

Let $X$ be a topological space and $R$ a commutative ring. For $i,j$ integers, the cap product is a bilinear map:

$H^i(X)\times H_j(X)\to H_{j-i}(X)$

Equivalently it is a linear map:

$H^i(X)\otimes H_j(X)\to H_{j-i}(X)$

The cap product turns the direct sum of homology groups into a graded module over the cohomology ring, when viewed as a graded $R$-algebra.