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 f : Δ m → X {\displaystyle f:\Delta ^{m}\to X} and g : Δ n → Y {\displaystyle g:\Delta ^{n}\to Y} we can define the product map f × g : Δ m × Δ n → X × Y {\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 X × Y {\displaystyle X\times Y} . To do this one can subdivide Δ m × Δ n {\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
H m ( X ; R ) ⊗ H n ( Y ; R ) → H m + n ( X × Y ; R ) {\displaystyle H_{m}(X;R)\otimes H_{n}(Y;R)\to H_{m+n}(X\times Y;R)}
by proving that if f {\displaystyle f} and g {\displaystyle g} are cycles then so is f × g {\displaystyle f\times g} and if either f {\displaystyle f} or g {\displaystyle g} is a boundary then so is the product.
Definition Given an H-space X {\displaystyle X} with multiplication μ : X × X → X {\displaystyle \mu :X\times X\to X} , the Pontryagin product on homology is defined by the following composition of maps
H ∗ ( X ; R ) ⊗ H ∗ ( X ; R ) →
× H ∗ ( X × X ; R ) →
μ ∗ H ∗ ( X ; R ) {\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 X × X → X {\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 H ∗ ( X ; R ) = ⨁ n = 0 ∞ H n ( X ; R ) {\displaystyle H_{*}(X;R)=\bigoplus _{n=0}^{\infty }H_{n}(X;R)} .
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. Pontryagin, Lev (1939). "Homologies in compact Lie groups". Recueil Mathématique (Matematicheskii Sbornik). New Series. 6 (48): 389–422. MR 0001563. Hatcher, Allen (2001). Algebraic topology. Cambridge: Cambridge University Press. ISBN 978-0-521-79160-1.