# Postnikov square

From Encyclopedia of Mathematics

A cohomology operation of type , where and are Abelian groups with a fixed heteromorphism , i.e. a mapping such that the function

is bilinear and . Let be an epimorphism and let be a free Abelian group. The Postnikov square for -cocycles is defined by the formula

where is a cochain with coefficients in such that . A suspension of a Postnikov square is a Pontryagin square. For a simply-connected space , the Postnikov square for which , and is defined by composition with the Hopf mapping is used to classify the mappings of three-dimensional polyhedra into . Postnikov squares were introduced by M.M. Postnikov [1].

#### References

[1] | 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) |

**How to Cite This Entry:**

Postnikov square. A.F. Kharshiladze (originator),

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Postnikov_square&oldid=16646

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098