Cap product

From Topospaces
Jump to: navigation, search

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