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:

$$a \frown b$$