Pontryagin product

Short description: Product on the homology of a topological space induced by a product on the topological space

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 $f:{\mathrm{\Delta }}^m\to X$ and $g:{\mathrm{\Delta }}^n\to Y$ we can define the product map $f\times g:{\mathrm{\Delta }}^m\times {\mathrm{\Delta }}^n\to X\times Y$, the only difficulty is showing that this defines a singular (m+n)-simplex in $X\times Y$. To do this one can subdivide ${\mathrm{\Delta }}^m\times {\mathrm{\Delta }}^n$ 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

$H_{*}(X;R)\otimes H_{*}(X;R)\underset{}{\overset{\times }{\to }}{H}_{*}(X\times X;R)\underset{}{\overset{{\mu }_{*}}{\to }}{H}_{*}(X;R)$

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 $H_{*}(X;R)=\underset{n=0}{\overset{\mathrm{\infty }}{⨁}}{H}_n(X;R)$.

References

• Brown, Kenneth S. (1982). Cohomology of groups. Graduate Texts in Mathematics. 87. Berlin, New York: Springer-Verlag. ISBN 978-0-387-90688-1. https://books.google.com/books?id=PMqb2DppvCsC. 
• Pontryagin, Lev (1939). "Homologies in compact Lie groups". Recueil Mathématique (Matematicheskii Sbornik). New Series 6 (48): 389–422. 
• Hatcher, Hatcher (2001). Algebraic topology. Cambridge: Cambridge University Press. ISBN 978-0-521-79160-1. http://www.math.cornell.edu/~hatcher/AT/ATpage.html.