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 (m+n)-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 formula_1 and formula_2 we can define the product map formula_3, the only difficulty is showing that this defines a singular (m+n)-simplex in formula_4. To do this one can subdivide formula_5 into (m+n)-simplices. It is then easy to show that this map induces a map on homology of the form
by proving that if formula_7 and formula_8 are cycles then so is formula_9 and if either formula_7 or formula_8 is a boundary then so is the product.
Definition.
Given an H-space formula_12 with multiplication formula_13 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 formula_15 of the H-space followed by application of the homology functor to obtain a homomorphism on the level of homology. Then formula_16.