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

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

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, and ${\displaystyle H_{*}(X;R)=\bigoplus _{n=0}^{\infty }H_{n}(X;R)}$.

References

• Brown, Kenneth S. (1982), Cohomology of groups, Graduate Texts in Mathematics, 87, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90688-1, MR 0672956
• Pontrjagin, Lev (1939), "Homologies in compact Lie groups", Recueil Mathématique (Matematicheskii Sbornik) N.S., 6 (48): 389–422, MR 0001563
• Allen Hatcher, Algebraic topology. Cambridge University Presses, Cambridge, 2001. xii+544 pp. ISBN 978-0-521-79160-1 and ISBN 0-521-79540-0