Pontryagin product

In mathematics, the Pontryagin product, introduced by Lev Pontryagin (1939), 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 $f:\Delta ^{m}\to X$ and $g:\Delta ^{n}\to Y$ we can define the product map $f\times g:\Delta ^{m}\times \Delta ^{n}\to X\times Y$ , the only difficulty is showing that this defines a singular (m+n)-simplex in $X\times Y$ . To do this one can subdivide $\Delta ^{m}\times \Delta ^{n}$ into (m+n)-simplices. It is then easy to show that this map induces a map on homology of the form

H_{m}(X;R)\otimes H_{n}(Y;R)\to H_{m+n}(X\times Y;R)

by proving that if $f$ and $g$ are cycles then so is $f\times g$ and if either $f$ or $g$ is a boundary then so is the product.

Definition

Given an H-space $X$ with multiplication $\mu :X\times X\to X$ we define the Pontryagin product on homology by the following composition of maps

H_{*}(X;R)\otimes H_{*}(X;R){\xrightarrow[{}]{\times }}H_{*}(X\times X;R){\xrightarrow[{}]{\mu _{*}}}H_{*}(X;R)

where the first map is the cross product defined above and the second map is given by the multiplication $X\times X\to X$ of the H-space followed by application of the homology functor to obtain a homomorphism on the level of homology. Then $H_{*}(X;R)=\bigoplus _{n=0}^{\infty }H_{n}(X;R)$ .