The Iteration of the Steenrod Squares in Algebraic Topology.

PRINCETON UNIVERSITY Communicated by S. Lefschetz, June 23, 1952 In this work it is proved that the Steenrod squares on cohomology classes satisfy certain identities. These identities are applied to solve particular homotopy problems. Full details will appear elsewhere. 1. Let W(K) be the qth cohomology group of a complex K with the integers mod 2 as coefficient group. The Steenrod squares,' denoted by S, (i = 0, 1, ...), are invariant homomorphisms Sqi: H()+(K) defined for any complex K, satisfying, among several others, the following properties: for any map f of one complex into another, f*Sq' = Sq'f*; Sqo = identity; Sq"u = u -u, (mod 2), if q.= dim u. An iterated square is a composition of two or more of the squares, e.g., Sq'SqiSqk The following relations (mod 2) on iterated squares are well known