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

edit

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   and   we can define the product map  , the only difficulty is showing that this defines a singular (m+n)-simplex in  . To do this one can subdivide   into (m+n)-simplices. It is then easy to show that this map induces a map on homology of the form

 

by proving that if   and   are cycles then so is   and if either   or   is a boundary then so is the product.

Definition

edit

Given an H-space   with multiplication  , the Pontryagin product on homology is defined by the following composition of maps

 

where the first map is the cross product defined above and the second map is given by the multiplication   of the H-space followed by application of the homology functor to obtain a homomorphism on the level of homology. Then  .

References

edit
  • 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.