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

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

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

Definition[]

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

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

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

References[]

This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (December 2020) (Learn how and when to remove this template message)

• Brown, Kenneth S. (1982). Cohomology of groups. Graduate Texts in Mathematics. Vol. 87. Berlin, New York: Springer-Verlag. ISBN 978-0-387-90688-1. MR 0672956.
• Pontryagin, Lev (1939). "Homologies in compact Lie groups". Recueil Mathématique (Matematicheskii Sbornik). New Series. 6 (48): 389–422. MR 0001563.
• Hatcher, Hatcher (2001). Algebraic topology. Cambridge: Cambridge University Press. ISBN 978-0-521-79160-1.

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