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.

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 Simplicial Cross Product defined on singular chains in the following way, given maps $f:\Delta ^{m}\to X$ and :$g:\Delta ^{n}\to X$ there existst a well-defined product map $f\times g:\Delta ^{m}\times \Delta ^{n}\to X\times Y$, we then subdivide $\Delta ^{m}\times \Delta ^{n}$ into (n+m)-simplices and consider the formal sum of the restrictions of $f\times g$ to these simplices with corresponding signs.

References

• 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
• Pontrjagin, Lev (1939), "Homologies in compact Lie groups", Recueil Mathématique (Matematicheskii Sbornik) N.S., 6 (48): 389–422, MR 0001563