# Postnikov square

A cohomology operation of type $0 ( 1 , A , 3 , B )$, where $A$ and $B$ are Abelian groups with a fixed heteromorphism $\eta : A \rightarrow B$, i.e. a mapping such that the function

$$h ( g _ {1} , g _ {2} ) = \eta ( g _ {1} + g _ {2} ) - \eta ( g _ {1} ) - \eta ( g _ {2} )$$

is bilinear and $\eta ( - g ) = \eta ( g)$. Let $\xi : F \rightarrow A$ be an epimorphism and let $F = \oplus \mathbf Z$ be a free Abelian group. The Postnikov square for $1$- cocycles is defined by the formula

$$e ^ {1} \rightarrow \widetilde \eta \widetilde \xi ( e _ {0} ^ {1} \cup \delta e _ {0} ^ {1} ) ,$$

where $e _ {0} ^ {1}$ is a cochain with coefficients in $F$ such that $\xi e _ {0} ^ {1} = e ^ {1}$. A suspension of a Postnikov square is a Pontryagin square. For a simply-connected space $X$, the Postnikov square for which $A = \pi _ {2} ( X)$, $B = \pi _ {3} ( X)$ and $\eta$ is defined by composition with the Hopf mapping $S ^ {3} \rightarrow S ^ {2}$ is used to classify the mappings of three-dimensional polyhedra into $X$. 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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Postnikov_square&oldid=48261
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article