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.

The cap product of $a$ and $b$ is denoted as:

$a\frown b$