Pontryagin product explained

In mathematics, the Pontryagin product, introduced by, 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

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

by proving that if

f and g are cycles then so is f x g and if either f or g is a boundary then so is the product.

Definition

X with multiplication \mu:X x X\to X , the Pontryagin product on homology is defined by the following composition of mapsH _{*(X;R) ⊗} H _{*(X;R)\xrightarrow[]{ x }} H _{*(X x} 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

X x X\to X of the H-space followed by application of the homology functor to obtain a homomorphism on the level of homology. Then H _{*(X;R)} = o plus ^{inf ty} _{n =0} H _{n(X;R)} .

References

• Book: Kenneth Brown (mathematician)

. Brown . Kenneth S. . Kenneth Brown (mathematician) . Cohomology of groups . . Berlin, New York . . 978-0-387-90688-1 . 672956 . 1982 . 87.

• Pontryagin . Lev . Lev Pontryagin . Homologies in compact Lie groups . 0001563 . 1939 . Recueil Mathématique (Matematicheskii Sbornik) . New Series . 6 . 48 . 389–422.
• Book: Allen Hatcher

. Hatcher . Hatcher . Allen Hatcher . Algebraic topology . . Cambridge . 2001 . 978-0-521-79160-1.