Postnikov square

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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].


[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. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article