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

[math]\displaystyle{ H_m(X;R)\otimes H_n(Y;R)\to H_{m+n}(X\times Y;R) }[/math]

by proving that if [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] are cycles then so is [math]\displaystyle{ f\times g }[/math] and if either [math]\displaystyle{ f }[/math] or [math]\displaystyle{ g }[/math] is a boundary then so is the product.

Definition

Given an H-space [math]\displaystyle{ X }[/math] with multiplication [math]\displaystyle{ \mu:X\times X\to X }[/math] we define the Pontryagin product on homology by the following composition of maps

[math]\displaystyle{ H_*(X;R)\otimes H_*(X;R)\xrightarrow[]{\times} H_*(X\times X;R) \xrightarrow[]{\mu_*} H_*(X;R) }[/math]

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

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.