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)