View source for Postnikov square

A [[Cohomology operation|cohomology operation]] of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p0740701.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p0740702.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p0740703.png" /> are Abelian groups with a fixed heteromorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p0740704.png" />, i.e. a mapping such that the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p0740705.png" /></td> </tr></table>

is bilinear and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p0740706.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p0740707.png" /> be an epimorphism and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p0740708.png" /> be a free Abelian group. The Postnikov square for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p0740709.png" />-cocycles is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p07407010.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p07407011.png" /> is a cochain with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p07407012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p07407013.png" />. A [[Suspension|suspension]] of a Postnikov square is a [[Pontryagin square|Pontryagin square]]. For a simply-connected space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p07407014.png" />, the Postnikov square for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p07407015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p07407016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p07407017.png" /> is defined by composition with the Hopf mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p07407018.png" /> is used to classify the mappings of three-dimensional polyhedra into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074070/p07407019.png" />. Postnikov squares were introduced by M.M. Postnikov [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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)</TD></TR></table>