mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Pontryagin product, introduced by , is a product on the

homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...

of a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

induced by a product on the topological space. Special cases include the Pontryagin product on the homology of an

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, the Pontryagin product on an

H-space In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed. Definition An H-space consists of a topological space , together wit ...

, and the Pontryagin product on a

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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:\Delta^m\to X

and

g:\Delta^n\to Y

we can define the product map

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\times Y

. To do this one can subdivide

\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)\otimes H_n(Y;R)\to H_(X\times Y;R)

by proving that if

f

and

g

are cycles then so is

f\times g

and if either

f

g

is a boundary then so is the product.

Definition

Given an

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[] H_*(X\times X;R) \xrightarrow[] H_*(X;R)

where the first map is the cross product defined above and the second map is given by the multiplication

X\times X\to X

of the

followed by application of the homology functor to obtain a homomorphism on the level of homology. Then

H_*(X;R) = \bigoplus_^\infty H_n(X;R)

References

* * * Homology theory Group theory {{topology-stub