Pontryagin product


In mathematics, the Pontryagin product, introduced by, is a product on the homology of a topological space induced by a product on the topological space. Special cases include the Pontryagin product on the homology of an abelian group, the Pontryagin product on an H-space, and the Pontryagin product on a loop space.

Cross Product

In order to define the Pontryagin product we first need a map which sends the direct product of the m-th and n-th homology group to the -th homology group of a space. We therefore define the cross product, starting on the level of singular chains. Given two topological spaces X and Y and two singular simplices and we can define the product map, the only difficulty is showing that this defines a singular -simplex in. To do this one can subdivide into -simplices. It is then easy to show that this map induces a map on homology of the form
by proving that if and are cycles then so is and if either or is a boundary then so is the product.

Definition

Given an H-space with multiplication we define the Pontryagin product on Homology by the following composition of maps
where the first map is the cross product defined above and the second map is given by the multiplication of the H-space, and.