Pontryagin square

A cohomology operation of type , i.e. a functorial mapping

defined for any pair of topological spaces and such that for any continuous mapping the equality (naturality) holds.

Pontryagin squares have the following properties:

1) , where is the natural imbedding.

2) and , where is the quotient homomorphism modulo .

3) , where is the suspension mapping and is the Postnikov square (in other words, the cohomology suspension of is ). If

and

are the representing mappings, then .

The properties 1), 2) uniquely characterize the Pontryagin square and thus can be taken as an axiomatic definition of it. Constructively the Pontryagin square is defined by the formula

where is a cocycle modulo (for the -products see Steenrod square).

There exists (see [5], [6]) a generalization of the Pontryagin square to the case when is an arbitrary odd prime number. This generalization is a cohomology operation of type and is called the -th Pontryagin power . The operation is uniquely defined by the following formulas:

where is the natural imbedding; and

where is the quotient homomorphism modulo generalizing the corresponding formulas for . The analogue of formula 3) for has the form , which means that the cohomology suspension of for is zero. For the equality holds, the multiplication may be taken both as outer (-multiplication) or inner (-multiplication). For the corresponding equality is valid only up to summands of order 2.

In the most general way the Pontryagin square is defined for cohomology with coefficients in an arbitrary finitely-generated Abelian group (see [2], [3]). In final form this generalization is as follows (see [6]). The Pontryagin square is a ring homomorphism

where is a functor which associates a ring with divided powers to an Abelian group. For , the -th component of this homomorphism coincides with the -th Pontryagin power (for with the Pontryagin square ).

References

[1]	L.S. Pontryagin, "Mappings of a 3-dimensional sphere into an -dimensional complex" Dokl. Akad. Nauk SSSR , 34 (1942) pp. 35–37 (In Russian)
[2]	V.G. Boltyanskii, "The homotopy theory of continuous mapping and vector fields" , Moscow (1955) (In Russian)
[3]	M.M. Postnikov, "The classification of continuous mappings of a three-dimensional polyhedron into a simply connected polyhedron of arbitrary dimension" Dokl. Akad. Nauk SSSR , 64 : 4 (1949) pp. 461–462 (In Russian)
[4]	W. Browder, E. Thomas, "Axioms for the generalized Pontryagin cohomology operations" Quart. J. Math. , 13 (1962) pp. 55–60
[5]	E. Thomas, "A generalization of the Pontrjagin square cohomology operation" Proc. Nat. Acad. Sci. USA , 42 (1956) pp. 266–269
[6]	E. Thomas, "The generalized Pontryagin cohomology operations and rings with divided powers" , Amer. Math. Soc. (1957)

Comments

For a definition of see Ring with divided powers.