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Postnikov square

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A cohomology operation of type , 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